diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index eb8d61f..600363f 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -10,11351 +10,12439 @@ initially derived with permission from Nelson Beebe's collection. The second section contains references from Axiom to the literature. The third section sorts papers by topic. \chapter{The Bibliography} -\section{Special Topics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Linear Algebra} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{axiom.bib} -@inproceedings{Bro86, - author = "Bronstein, Manuel", - title = "Gsolve: a faster algorithm for solving systems of algebraic - equations", - booktitle = "Proc of 5th ACM SYMSAC", - year = "1986", - pages = "247-249", - isbn = "0-89791-199-7", - abstract = " - We apply the elimination property of Gr{\"o}bner bases with respect to - pure lexicographic ordering to solve systems of algebraic equations. - We suggest reasons for this approach to be faster than the resultant - technique, and give examples and timings that show that it is indeed - faster and more correct, than MACSYMA's solve." +@Unpublished{Kalt01, + author = "Kaltofen, E.", + title = "Algorithms for sparse and black box matrices + over finite fields (Invited talk)", + year = "2001", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/Ka01_Fq6.pdf", + paper = "Kalt01.pdf" } \end{chunk} -\subsection{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -{Bro99, - author = "Bronstein, Manuel", - title = "Fast Deterministic Computation of Determinants of Dense Matrices", - url = "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html", - paper = "Bro99.pdf", - abstract = " - In this paper we consider deterministic computation of the exact - determinant of a dense matrix $M$ of integers. We present a new - algorithm with worst case complexity - $O(n^4(log n+ log \verb?||M||?)+x^3 log^2 \verb?||M||?)$, - where $n$ is the dimension of the matrix - and \verb?||M||? is a bound on the entries in $M$, but with - average expected complexity - $O(n^4+m^3(log n + log \verb?||M||?)^2)$, - assuming some plausible properties about the distribution of $M$. - We will also describe a practical version of the algorithm and include - timing data to compare this algorithm with existing ones. Our result - does not depend on fast'' integer or matrix techniques." +\begin{chunk}{axiom.bib} +@Article{Chen02, + author = "Chen, L. and Eberly, W. and Kaltofen, E. + and Saunders, B. D. and Turner, W. J. and Villard, G.", + title = "Efficient Matrix Preconditioners for Black Box Linear Algebra", + journal = "Linear Algebra and Applications", + year = "2002", + volume = "343--344", + pages = "119--146", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/CEKSTV02.pdf", + paper = "Chen02.pdf" } \end{chunk} -\begin{chunk}{ignore} -{Kel00, - author = "Kelsey, Tom", - title = "Exact Numerical Computation via Symbolic Computation", - url = "http://tom.host.cs.st-andrews.ac.uk/pub/ccapaper.pdf", - paper = "Kel00.pdf", - abstract = " - We provide a method for converting any symbolic algebraic expression - that can be converted into a floating point number into an exact - numeric representation. We use this method to demonstrate a suite of - procedures for the representation of, and arithmetic over, exact real - numbers in the Maple computer algebra system. Exact reals are - represented by potentially infinite lists of binary digits, and - interpreted as sums of negative powers of the golden ratio." +\begin{chunk}{axiom.bib} +@InCollection{Kalt11d, + author = "Kaltofen, Erich and Storjohann, Arne", + title = "The Complexity of Computational Problems in Exact Linear Algebra", + booktitle = "Encyclopedia of Applied and Computational Mathematics", + crossref = "EACM", + year = "2011", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KS11.pdf", + paper = "Kalt11d.pdf" } \end{chunk} -\begin{chunk}{ignore} -{Yang14, - author ="Yang, Xiang and Mittal, Rajat", - title = "Acceleration of the Jacobi iterative method by factors exceeding 100 - using scheduled relation", - url = -"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf", - paper = "Yang14.pdf" +\begin{chunk}{axiom.bib} +@Article{Come12, + author = "Comer, Matthew T. and Kaltofen, Erich L.", + title = "On the {Berlekamp}/{Massey} Algorithm and Counting Singular {Hankel} + Matrices over a Finite Field", + year = "2012", + month = "April", + journal = "Journal of Symbolic Computation", + volume = "47", + number = "4", + pages = "480--491", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/CoKa10.pdf", + paper = "Come12.pdf" } \end{chunk} -\subsection{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -{Corl0, - author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M. - and Bradford, Russell and Davenport, James H.", - title = "Reasoning about the elementary functions of complex analysis", - url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002-amai-reasoning.pdf", - paper = "Corl05.pdf", - abstract = " - There are many problems with the simplification of elementary - functions, particularly over the complex plane. Systems tend to make - howlers'' or not to simplify enough. In this paper we outline the - unwinding number'' approach to such problems, and show how it can be - used to prevent errors and to systematise such simplification, even - though we have not yet reduced the simplification process to a - complete algorithm. The unsolved problems are probably more amenable - to the techniques of artificial intelligence and theorem proving than - the original problem of complex-variable analysis." +\begin{chunk}{axiom.bib} +@Article{Kalt13a, + author = "Kaltofen, Erich and Yuhasz, George", + title = "A Fraction Free Matrix {Berlekamp}/{Massey} Algorithm", + journal = "Linear Algebra and Applications", + year = "2013", + volume = "439", + number = "9", + month = "November", + pages = "2515--2526", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaYu08.pdf", + paper = "Kalt13a.pdf" } \end{chunk} -\begin{chunk}{ignore} -{Ng68, - author = "Ng, Edward W. and Geller, Murray", - title = "A Table of Integrals of the Error functions", - url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf", - paper = "Ng68.pdf", - abstract = " - This is a compendium of indefinite and definite integrals of products - of the Error functions with elementary and transcendental functions." +\begin{chunk}{axiom.bib} +@Article{Kalt13, + author = "Kaltofen, Erich and Yuhasz, George", + title = "On The Matrix {Berlekamp}-{Massey} Algorithm", + year = "2013", + volume = "9", + number = "4", + month = "September", + journal = "ACM Trans. Algorithms", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaYu06.pdf", + paper = "Kalt13.pdf" } \end{chunk} -\subsection{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -{Gell69, - author = "Geller, Murray and Ng, Edward W.", - title = "A Table of Integrals of the Exponential Integral", - url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf", - paper = "Gell69.pdf", - abstract = " - This is a compendium of indefinite and definite integrals of products - of the Exponential Integral with elementary or transcendental functions." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt02a, + author = "Kaltofen, Erich", + title = "An output-sensitive variant of the baby steps/\allowbreak + giant steps determinant algorithm", + booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC02", + pages = "138--144", + year = "2002", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Ka02.pdf", + paper = "Kalt02a.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@techreport{Segl98, - author = "Segletes, S.B.", - title = "A compact analytical fit to the exponential integral $E_1(x)$", - year = "1998", - institution = "U.S. Army Ballistic Research Laboratory, - Aberdeen Proving Ground, MD", - type = "Technical Report", - number = "ARL-TR-1758", - paper = "Segl98.pdf", - abstract = " - A four-parameter fit is developed for the class of integrals known as - the exponential integral (real branch). Unlike other fits that are - piecewise in nature, the current fit to the exponential integral is - valid over the complete domain of the function (compact) and is - everywhere accurate to within $\pm 0.0052\%$ when evaluating the first - exponential integral, $E_1$. To achieve this result, a methodology - that makes use of analytically known limiting behaviors at either - extreme of the domain is employed. Because the fit accurately captures - limiting behaviors of the $E_1$ function, more accuracy is retained - when the fit is used as part of the scheme to evaluate higher-order - exponential integrals, $E_n$, as compared with the use of brute-force - fits to $E_1$, which fail to accurately model limiting - behaviors. Furthermore, because the fit is compact, no special - accommodations are required (as in the case of spliced piecewise fits) - to smooth the value, slope, and higher derivatives in the transition - region between two piecewise domains. The general methodology employed - to develop this fit is outlined, since it may be used for other - problems as well." +@InProceedings{Kalt01a, + author = "Kaltofen, E. and Villard, G.", + title = "On the complexity of computing determinants", + booktitle = "Proc. Fifth Asian Symposium on Computer Mathematics + (ASCM 2001)", + crossref = "ASCM01", + pages = "13--27", + isbn = "981-02-4763-X", + year = "2001", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/KaVi01.pdf", + paper = "Kalt01a.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@techreport{Se09, - author = "Segletes, S.B.", - title = "Improved fits for $E_1(x)$ {\sl vis-\'a-vis} those presented - in ARL-TR-1758", - type = "Technical Report", - number = "ARL-TR-1758", - institution ="U.S. Army Ballistic Research Laboratory, - Aberdeen Proving Ground, MD", - year = "1998", - month = "September", - paper = "Se09.pdf", - abstract = " - This is a writeup detailing the more accurate fits to $E_1(x)$, - relative to those presented in ARL-TR-1758. My actual fits are to - $F1 =[x\ exp(x) E_1(x)]$ which spans a functional range from 0 to 1. - The best accuracy I have been yet able to achieve, defined by limiting - the value of $[(F1)_{fit} - F1]/F1$ over the domain, is - approximately 3.1E-07 with a 12-parameter fit, which unfortunately - isn't quite to 32-bit floating-point accuracy. Nonetheless, the fit - is not a piecewise fit, but rather a single continuous function over - the domain of nonnegative x, which avoids some of the problems - associated with piecewise domain splicing." +@Article{Kalt04a, + author = "Kaltofen, Erich and Villard, Gilles", + title = "On the Complexity of Computing Determinants", + journal = "Computational Complexity", + volume = "13", + number = "3-4", + year = "2004", + pages = "91--130", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/04/KaVi04_2697263.pdf", + paper = "Kalt04a.pdf" } \end{chunk} -\subsection{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{axiom.bib} +@InProceedings{Kalt97b, + author = "Eberly, W. and Kaltofen, E.", + title = "On Randomized {Lanczos} Algorithms", + booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.", + year = "1997", + crossref = "ISSAC97", + pages = "176--183", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/EbKa97.pdf", + paper = "Kalt97b.pdf" -\begin{chunk}{ignore} -\bibitem[Knuth 71]{ST-PGCD-Knu71} Knuth, Donald -The Art of Computer Programming'' -2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing, -Addison-Wesley 1971, section 4.6 pp399-505 +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Ma 90]{ST-PGCD-Ma90} Ma, Keju; Gathen, Joachim von zur -Analysis of Euclidean Algorithms for Polynomials over Finite Fields'' -J. Symbolic Computation (1990) Vol 9 pp429-455\hfill{} -\verb|www.researchgate.net/publication/220161718_Analysis_of_Euclidean_| -\verb|Algorithms_for_Polynomials_over_Finite_Fields/file/| -\verb|60b7d52b326a1058e4.pdf| -%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ma90.pdf| - abstract = " - This paper analyzes the Euclidean algorithm and some variants of it - for computing the greatest common divisor of two univariate polynomials - over a finite field. The minimum, maximum, and average number of - arithmetic operations both on polynomials and in the ground field - are derived." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt94b, + author = "Kaltofen, E.", + title = "Asymptotically fast solution of {Toeplitz}-like singular + linear systems", + booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.", + pages = "297--304", + crossref = "ISSAC94", + year = "1994", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf", + paper = "Kalt94b.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Naylor 00a]{N00} Naylor, Bill -Polynomial GCD Using Straight Line Program Representation'' -PhD. Thesis, University of Bath, 2000 -\verb|www.sci.csd.uwo.ca/~bill/thesis.ps| -%\verb|axiom-developer.org/axiom-website/papers/N00.pdf| - abstract = " - This thesis is concerned with calculating polynomial greatest common - divisors using straight line program representation. - - In the Introduction chapter, we introduce the problem and describe - some of the traditional representations for polynomials, we then talk - about some of the general subjects central to the thesis, terminating - with a synopsis of the category theory which is central to the Axiom - computer algebra system used during this research. - - The second chapter is devoted to describing category theory. We follow - with a chapter detailing the important sections of computer code - written in order to investigate the straight line program subject. - The following chapter on evalution strategies and algorithms which are - dependant on these follows, the major algorith which is dependant on - evaluation and which is central to our theis being that of equality - checking. This is indeed central to many mathematical problems. - Interpolation, that is the determination of coefficients of a - polynomial is the subject of the next chapter. This is very important - for many straight line program algorithms, as their non-canonical - structure implies that it is relatively difficult to determine - coefficients, these being the basic objects that many algorithms work - on. We talk about three separate interpolation techniques and compare - their advantages and disadvantages. The final two chapters describe - some of the results we have obtained from this research and finally - conclusions we have drawn as to the viability of the straight line - program approach and possible extensions. - - Finally we terminate with a number of appendices discussing side - subjects encountered during the thesis." +\begin{chunk}{axiom.bib} +@Article{Kalt99, + author = "Kaltofen, E. and Lobo, A", + title = "Distributed matrix-free solution of large sparse linear systems over + finite fields", + journal = "Algorithmica", + year = "1999", + pages = "331--348", + month = "July--Aug.", + volume = "24", + number = "3--4", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaLo99.pdf", + paper = "Kalt99.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Shoup 93]{ST-PGCD-Sh93} Shoup, Victor -Factoring Polynomials over Finite Fields: Asymptotic Complexity vs -Reality*'' -Proc. IMACS Symposium, Lille, France, (1993) -\verb|www.shoup.net/papers/lille.pdf| -%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Sh93.pdf| - abstract = " - This paper compares the algorithms by Berlekamp, Cantor and - Zassenhaus, and Gathen and Shoup to conclude that (a) if large - polynomials are factored the FFT should be used for polynomial - multiplication and division, (b) Gathen and Shoup should be used if - the number of irreducible factors of $f$ is small. (c) if nothing is - know about the degrees of the factors then Berlekamp's algorithm - should be used." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt96a, + author = "Kaltofen, E. and Lobo, A.", + title = "Distributed matrix-free solution of large sparse linear systems + over finite fields", + booktitle = "Proc. High Performance Computing '96", + year = "1996", + editor = "A. M. Tentner", + pages = "244--247", + organization = "Society for Computer Simulation", + publisher = "Simulation Councils, Inc.", + address = "San Diego, CA", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf", + paper = "Kalt96a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gathen 01]{ST-PGCD-Ga01} Gathen, Joachim von zur; Panario, Daniel -Factoring Polynomials Over Finite Fields: A Survey'' -J. Symbolic Computation (2001) Vol 31, pp3-17\hfill{} -\verb|people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.pdf| -%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ga01.pdf| - keywords = "survey", - abstract = " - This survey reviews several algorithms for the factorization of - univariate polynomials over finite fields. We emphasize the main ideas - of the methods and provide and up-to-date bibliography of the problem. - This paper gives algorithms for {\sl squarefree factorization}, - {\sl distinct-degree factorization}, and {\sl equal-degree factorization}. - The first and second algorithms are deterministic, the third is - probabilistic." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt94a, + author = "Kaltofen, E. and Lobo, A.", + title = "Factoring high-degree polynomials by the black box + Berlekamp algorithm", + booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC94", + pages = "90--98", + year = "1994", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz", + paper = "Kalt94a.ps" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael -Algorithms for Polynomial GCD Computation over Algebraic Function Fields'' -\verb|www.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Hoeij04.pdf| - abstract = " - Let $L$ be an algebraic function field in $k \ge 0$ parameters - $t_1,\ldots,t)k$. Let $f_1$, $f_2$ be non-zero polynomials in - $L[x]$. We give two algorithms for computing their gcd. The first, a - modular GCD algorithm, is an extension of the modular GCD algorithm - for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf - Q}$(\alpha[x])$ to function fields. The second, a fraction-free - algorithm, is a modification of the Moreno Maza and Rioboo algorithm - for computing gcds over triangular sets. The modification reduces - coefficient grownth in $L$ to be linear. We give an empirical - comparison of the two algorithms using implementations in Maple." +\begin{chunk}{axiom.bib} +@Article{Kalt95, + author = "Kaltofen, E.", + title = "Analysis of {Coppersmith}'s block {Wiedemann} algorithm for the + parallel solution of sparse linear systems", + journal = "Math. Comput.", + year = "1995", + volume = "64", + number = "210", + pages = "777--806", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_mathcomp.pdf", + paper = "Kalt95.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Wang 78]{Wang78} Wang, Paul S. -An Improved Multivariate Polynomial Factoring Algorithm'' -Mathematics of Computation, Vol 32, No 144 Oct 1978, pp1215-1231 -\verb|www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0568284-3/| -\verb|S0025-5718-1978-0568284-3.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Wang78.pdf| - abstract = " - A new algorithm for factoring multivariate polynomials over the - integers based on an algorithm by Wang and Rothschild is described. - The new algorithm has improved strategies for dealing with the known - problems of the original algorithm, namely, the leading coefficient - problem, the bad-zero problem and the occurence of extraneous factors. - It has an algorithm for correctly predetermining leading coefficients - of the factors. A new and efficient p-adic algorith named EEZ is - described. Basically it is a linearly convergent variable-by-variable - parallel construction. The improved algorithm is generally faster and - requires less store than the original algorithm. Machine examples with - comparative timing are included." +\begin{chunk}{axiom.bib} +@Article{Kalt90a, + author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.", + title = "Parallel algorithms for matrix normal forms", + journal = "Linear Algebra and Applications", + year = "1990", + volume = "136", + pages = "189--208", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf", + paper = "Kalt90a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Wiki 4]{Wiki4}. -Polynomial greatest common divisor'' -\verb|en.wikipedia.org/wiki/Polynomial_greatest_common_divisor| +\begin{chunk}{axiom.bib} +@Article{Kalt87, + author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.", + title = "Fast parallel computation of Hermite and Smith forms of + polynomial matrices", + journal = "SIAM J. Alg. Discrete Math.", + year = "1987", + volume = "8", + pages = "683--690", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf", + paper = "Kalt87.pdf" +} \end{chunk} -\subsection{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike -Physics, Topology, Logic and Computation: A Rosetta Stone'' -\verb|arxiv.org/pdf/0903.0340v3.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Baez09.pdf| - abstract = " - In physics, Feynman diagrams are used to reason about quantum - processes. In the 1980s, it became clear that underlying these - diagrams is a powerful analogy between quantum physics and - topology. Namely, a linear operator behaves very much like a - cobordism'': a manifold representing spacetime, going between two - manifolds representing space. But this was just the beginning: simiar - diagrams can be used to reason about logic, where they represent - proofs, and computation, where they represent programs. With the rise - of interest in quantum cryptography and quantum computation, it became - clear that there is an extensive network of analogies between physics, - topology, logic and computation. In this expository paper, we make - some of these analogies precise using the concept of closed - symmetric monodial category''. We assume no prior knowledge of - category theory, proof theory or computer science." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt92, + author = "Kaltofen, E. and Pan, V.", + title = "Processor-efficient parallel solution of linear systems {II}: + the positive characteristic and singular cases", + booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.", + year = "1992", + pages = "714--723", + publisher = "IEEE Computer Society Press", + address = "Los Alamitos, California", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf", + paper = "Kalt92.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross -Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire'' -\verb|eprints.eemcs.utwente.nl/7281/01/db-utwente-40501F46.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Meij91.pdf| - abstract = " - We develop a calculus for lazy functional programming based on - recursion operators associated with data type definitions. For these - operators we derive various algebraic laws that are useful in deriving - and manipulating programs. We shall show that all example functions in - Bird and Wadler's Introduction to Functional Programming'' can be - expressed using these operators." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt91c, + author = "Kaltofen, E. and Pan, V.", + title = "Processor efficient parallel solution of linear systems over + an abstract field", + booktitle = "Proc. SPAA '91 3rd Ann. ACM Symp. Parallel Algor. Architecture", + pages = "180--191", + publisher = "ACM Press", + year = "1991", + address = "New York, N.Y.", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf", + paper = "Kalt91c.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Youssef 04]{You04} Youssef, Saul -Prospects for Category Theory in Aldor'' -October 2004 -%\verb|axiom-developer.org/axiom-website/papers/You04.pdf| - abstract = " - Ways of encorporating category theory constructions and results into - the Aldor language are discussed. The main features of Aldor which - make this possible are identified, examples of categorical - constructions are provided and a suggestion is made for a foundation - for rigorous results." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt91, + author = "Kaltofen, E. and Saunders, B.D.", + editor = "H. F. Mattson and T. Mora and T. R. N. Rao", + title = "On {Wiedemann's} method of solving sparse linear systems", + booktitle = "Proc. AAECC-9", + series = "Lect. Notes Comput. Sci.", + volume = "539", + pages = "29--38", + publisher = "Springer-Verlag", + year = "1991", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf", + paper = "Kalt91.pdf" +} \end{chunk} -\subsection{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Algebraic Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{chunk}{ignore} -\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.; -Martin, U. -Automated theorem proving in support of computer algebra:'' - symbolic definite integration as a case study'' -%\verb|axiom-developer.org/axiom-website/papers/Adam99.pdf| - abstract = " - We assess the current state of research in the application of computer - aided formal reasoning to computer algebra, and argue that embedded - verification support allows users to enjoy its benefits without - wrestling with technicalities. We illustrate this claim by considering - symbolic definite integration, and present a verifiable symbolic - definite integral table look up: a system which matches a query - comprising a definite integral with parameters and side conditions, - against an entry in a verifiable table and uses a call to a library of - lemmas about the reals in the theorem prover PVS to aid in the - transformation of the table entry into an answer. We present the full - model of such a system as well as a description of our prototype - implementation showing the efficacy of such a system: for example, the - prototype is able to obtain correct answers in cases where computer - algebra systems [CAS] do not. We extend upon Fateman's web-based table - by including parametric limits of integration and queries with side - conditions." +\begin{chunk}{axiom.bib} +@InCollection{Diaz97, + author = "Diaz, A. and Kaltofen, E. and Pan, V.", + title = "Algebraic Algorithms", + booktitle = "The Computer Science and Engineering Handbook", + publisher = "CRC Press", + year = "1997", + editor = "A. B. Tucker", + pages = "226--248", + address = "Boca Raton, Florida", + chapter = "10", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/DKP97.ps.gz", + paper = "Diaz97.ps" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne; -Kelsey, Tom; Martin, Ursula; Owre, Sam -Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS'' -\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Adam01.pdf| - abstract = " - We describe an interface between version 6 of the Maple computer - algebra system with the PVS automated theorem prover. The interface is - designed to allow Maple users access to the robust and checkable proof - environment of PVS. We also extend this environment by the provision - of a library of proof strategies for use in real analysis. We - demonstrate examples using the interface and the real analysis - library. These examples provide proofs which are both illustrative and - applicable to genuine symbolic computation problems." +\begin{chunk}{axiom.bib} +@InCollection{Diaz99, + author = "Diaz, A. and Emiris, I. and Kaltofen, E. and Pan, V.", + title = "Algebraic Algorithms", + booktitle = "Algorithms \& Theory of Computation Handbook", + publisher = "CRC Press", + year = "1999", + editor = "M. J. Atallah", + address = "Boca Raton, Florida", + pages = "16.1--16.27", + isbn = "0-8493-2649-4", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/DEKP99.ps.gz", + paper = "Diaz99.ps" +} \end{chunk} \begin{chunk}{axiom.bib} -@article{Mahb06, - author = "Mahboubi, Assia", - title = "Proving Formally the Implementation of an Efficient gcd - Algorithm for Polynomials", - journal = "Lecture Notes in Computer Science", - volume = "4130", - year = "2006", - pages = "438-452", - paper = "Mahb06.pdf", - abstract = " - We describe here a formal proof in the Coq system of the structure - theorem for subresultants which allows to prove formally the - correctness of our implementation of the subresultants algorithm. - Up to our knowledge it is the first mechanized proof of this result." +@InCollection{Kalt87a, + author = "Kaltofen, E.", + editor = "J. F. Traub", + title = "Computer algebra algorithms", + booktitle = "Annual Review in Computer Science", + pages = "91--118", + publisher = "Annual Reviews Inc.", + year = "1987", + volume = "2", + address = "Palo Alto, California", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf", + paper = "Kalt87a.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C. -A Pragmatic Approach to Extending Provers by Computer Algebra -- - with Applications to Coding Theory'' -\verb|www.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Ball99.pdf| - abstract = " - The use of computer algebra is usually considered beneficial for - mechanised reasoning in mathematical domains. We present a case study, - in the application domain of coding theory, that supports this claim: - the mechanised proofs depend on non-trivial algorithms from computer - algebra and increase the reasoning power of the theorem prover. +\section{Sparse Linear Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - The unsoundness of computer algebra systems is a major problem in - interfacing them to theorem provers. Our approach to obtaining a sound - overall system is not blanket distrust but based on the distinction - between algorithms we call sound and {\sl ad hoc} respectively. This - distinction is blurred in most computer algebra systems. Our - experimental interface therefore uses a computer algebra library. It - is based on formal specifications for the algorithms, and links the - computer algebra library Sumit to the prover Isabelle. +\begin{chunk}{axiom.bib} +@InProceedings{Kalt96b, + author = "Kaltofen, E.", + title = "Blocked iterative sparse linear system solvers for finite fields", + booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular + Applic. (Stratagem '96)", + editor = "C. Roucairol", + publisher = "INRIA", + address = "Sophia Antipolis, France", + pages = "91--95", + year = "1996", + keywords = "survey", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz", + paper = "Kalt96b.ps" +} - We give details of the interface, the use of the computer algebra - system on the tactic-level of Isabelle and its integration into proof - procedures." +\end{chunk} + +\section{Matrix Determinants} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@Article{Kalt04, + author = "Kaltofen, E. and Villard, G.", + title = "Computing the sign or the value of the determinant of an integer + matrix, a complexity survey", + journal = "J. Computational Applied Math.", + volume = "162", + number = "1", + month = "January", + pages = "133--146", + year = "2004", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KaVi02.pdf", + paper = "Kalt04.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre -Interactive Theorem Proving and Program Development'' -Springer ISBN 3-540-20854-2 - abstract = " - Coq is an interactive proof assistant for the development of - mathematical theories and formally certified software. It is based on - a theory called the calculus of inductive constructions, a variant of - type theory. - This book provides a pragmatic introduction to the development of - proofs and certified programs using Coq. With its large collection of - examples and exercies it is an invaluable tool for researchers, - students, and engineers interested in formal methods and the - development of zero-fault software." +\section{Open Problems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@Article{Kalt00, + author = "Kaltofen, E.", + title = "Challenges of Symbolic Computation My Favorite Open Problems", + journal = "Journal of Symbolic Computation", + volume = "29", + number = "6", + pages = "891--919", + year = "2000", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/Ka2K.pdf", + paper = "Kalt00.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R. -Polymorphic Data Types, Objects, Modules and Functors,: is it too much?'' -%\verb|axiom-developer.org/axiom-website/papers/BHR00.pdf| - abstract = " - Abstraction is a powerful tool for developers and it is offered by - numerous features such as polymorphism, classes, modules, and - functors, $\ldots$ A working programmer may be confused by this - abundance. We develop a computer algebra library which is being - certificed. Reporting this experience made with a language (Ocaml) - offering all these features, we argue that the are all needed - together. We compare several ways of using classes to represent - algebraic concepts, trying to follow as close as possible mathematical - specification. Thenwe show how to combine classes and modules to - produce code having very strong typing properties. Currently, this - library is made of one hundred units of functional code and behaves - faster than analogous ones such as Axiom." +\section{Parallel Evaluation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@InCollection{Kalt93a, + author = "Kaltofen, E.", + editor = "J. Reif", + title = "Dynamic parallel evaluation of computation {DAG}s", + booktitle = "Synthesis of Parallel Algorithms", + pages = "723--758", + publisher = "Morgan Kaufmann Publ.", + year = "1993", + address = "San Mateo, California", + keywords = "survey", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz", + paper = "Kalt93a.ps" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Boulme 01]{BHHMR01} -Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissier-Morain, V.; Rioboo, R. -On the way to certify Computer Algebra Systems'' -Calculemus-2001 -%\verb|axiom-developer.org/axiom-website/papers/BHHMR01.pdf| - abstract = " - The FOC project aims at supporting, within a coherent software system, - the entire process of mathematical computation, starting with proved - theories, ending with certified implementations of algorithms. In this - paper, we explain our design requirements for the implementation, - using polynomials as a running example. Indeed, proving correctness of - implementations depends heavily on the way this design allows - mathematical properties to be truly handled at the programming level. +\section{Hybrid Symbolic/Numeric} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - The FOC project, started at the fall of 1997, is aimed to build a - programming environment for the development of certified symbolic - computation. The working languages are Coq and Ocaml. In this paper, - we present first the motivations of the project. We then explain why - and how our concern for proving properties of programs has led us to - certain implementation choices in Ocaml. This way, the sources express - exactly the mathematical dependencies between different structures. - This may ease the achievement of proofs." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt06, + author = "Kaltofen, Erich and Zhi, Lihong", + title = "Hybrid Symbolic-Numeric Computation", + year = "2006", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06", + crossref = "ISSAC06", + pages = "7", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaZhi06.pdf", + paper = "Kalt06.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Daly 10]{Daly10} Daly, Timothy -Intel Instruction Semantics Generator'' -\verb|daly.axiom-developer.org/TimothyDaly_files/publications/sei/intel/intel.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Daly10.pdf| - abstract = " - Given an Intel x86 binary, extract the semantics of the instruction - stream as Conditional Concurrent Assignments (CCAs). These CCAs - represent the semantics of each individual instruction. They can be - composed to represent higher level semantics." +\begin{chunk}{axiom.bib} +@InProceedings{Hutt10, + author = "Hutton, Sharon E. and Kaltofen, Erich L. and Zhi, Lihong", + title = "Computing the radius of positive semidefiniteness of a + multivariate real polynomial via a dual of {Seidenberg}'s method", + year = "2010", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'10", + crossref = "ISSAC10", + pages = "227--234", + month = "July", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/HKZ10.pdf", + paper = "Hutt10.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John; -Jansson, Patrik; Gibbons, Jeremy -Fast and Loose Reasoning is Morally Correct'' -ACM POPL'06 January 2005, Charleston, South Carolina, USA -%\verb|axiom-developer.org/axiom-website/papers/Dani06.pdf| - abstract = " - Functional programmers often reason about programs as if they were - written in a total language, expecting the results to carry over to - non-toal (partial) languages. We justify such reasoning. +\begin{chunk}{axiom.bib} +@InProceedings{Kalt09, + author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong", + title = "A Proof of the {Monotone Column Permanent (MCP) Conjecture} for + Dimension 4 via Sums-Of-Squares of Rational Functions", + year = "2009", + booktitle = "Proc. 2009 Internat. Workshop on Symbolic-Numeric Comput.", + crossref = "SNC09", + pages = "65--69", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KYZ09.pdf", + paper = "Kalt09.pdf" +} - Two languages are defined, one total and one partial, with identical - syntax. The semantics of the partial language includes partial and - infinite values, and all types are lifted, including the function - spaces. A partial equivalence relation (PER) is then defined, the - domain of which is the total subset of the partial language. For types - not containing function spaces the PER relates equal values, and - functions are related if they map related values to related values. +\end{chunk} - It is proved that if two closed terms have the same semantics in the - total language, then they have related semantics in the partial - language. It is also shown that the PER gives rise to a bicartesian - closed category which can be used to reason about values in the domain - of the relation." +\begin{chunk}{axiom.bib} +@Article{Kalt12, + author = "Kaltofen, Erich L. and Li, Bin and Yang, Zhengfeng and + Zhi, Lihong", + title = "Exact Certification in Global Polynomial Optimization + Via Sums-Of-Squares of Rational Functions + with Rational Coefficients", + year = "2012", + month = "January", + journal = "Journal of Symbolic Computation", + volume = "47", + number = "1", + pages = "1--15", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KLYZ09.pdf", + paper = "Kalt12.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell; -England, Matthew; Wilson, David -Program Verification in the presence of complex numbers, functions with -branch cuts etc.'' -\verb|arxiv.org/pdf/1212.5417.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Davenp12.pdf| - abstract = " - In considering the reliability of numerical programs, it is normal to - limit our study to the semantics dealing with numerical precision''. - On the other hand, there is a great deal of work on the reliability of - programs that essentially ignores the numerics. The thesis of this - paper is that there is a class of problems that fall between these - two, which could be described as does the low-level arithmetic - implement the high-level mathematics''. Many of these problems arise - because mathematics, particularly the mathematics of the complex - numbers, is more difficult than expected: for example the complex - function log is not continuous, writing down a program to compute an - inverse function is more complicated than just solving an equation, - and many algebraic simplification rules are not universally valid. +\begin{chunk}{axiom.bib} +@InProceedings{Kalt08b, + author = "Kaltofen, Erich and Li, Bin and Yang, Zhengfeng and Zhi, Lihong", + title = "Exact Certification of Global Optimality of Approximate + Factorizations Via Rationalizing Sums-Of-Squares + with Floating Point Scalars", + year = "2008", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08", + crossref = "ISSAC08", + pages = "155--163", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KLYZ08.pdf", + paper = "Kalt08b.pdf" +} - The good news is that these problems are {\sl theoretically} capable - of being solved, and are {\sl practically} close to being solved, but - not yet solved, in several real-world examples. However, there is - still a long way to go before implementations match the theoretical - possibilities." +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt06b, + author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong", + title = "Approximate greatest common divisors of several polynomials + with linearly constrained coefficients and singular polynomials", + year = "2006", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06", + crossref = "ISSAC06", + pages = "169--176", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KYZ06.pdf", + paper = "Kalt06b.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas -Guarded Expressions in Practice'' -\verb|redlog.dolzmann.de/papers/pdf/MIP-9702.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Dolz97.pdf| - abstract = " - Computer algebra systems typically drop some degenerate cases when - evaluating expressions, e.g. $x/x$ becomes 1 dropping the case - $x=0$. We claim that it is feasible in practice to compute also the - degenerate cases yielding {\sl guarded expressions}. We work over real - closed fields but our ideas about handling guarded expressions can be - easily transferred to other situations. Using formulas as guards - provides a powerful tool for heuristically reducing the combinatorial - explosion of cases: equivalent, redundant, tautological, and - contradictive cases can be detected by simplification and quantifier - elimination. Our approach allows to simplify the expressions on the - basis of simplification knowledge on the logical side. The method - described in this paper is implemented in the REDUCE package GUARDIAN, - which is freely available on the WWW." +\begin{chunk}{axiom.bib} +@InCollection{Kalt05, + author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong", + title = "Structured Low Rank Approximation of a {Sylvester} Matrix", + booktitle = "Symbolic-Numeric Computation", + crossref = "SNC06", + pages = "69--83", + year = "2005", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KYZ05.pdf", + paper = "Kalt05.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue -Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants -and Computer Algebra System Framework'' -Calculemus (2011) Springer -\verb|paradise.caltech.edu/~yli/paper/oa-polyml.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DR11.pdf| - abstract = " - This paper presents an ongoing effort to integrate the Axiom family of - computer algebra systems with Poly/ML-based proof assistants in the - same framework. A long term goal is to make a large set of efficient - implementations of algebraic algorithms available to popular proof - assistants, and also to bring the power of mechanized formal - verification to a family of strongly typed computer algebra systems at - a modest cost. Our approach is based on retargeting the code generator - of the OpenAxiom compiler to the Poly/ML abstract machine." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt03a, + author = "Kaltofen, Erich and May, John", + title = "On Approximate Irreducibility of Polynomials in Several Variables", + year = "2003", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'03", + crossref = "ISSAC03", + pages = "161--168", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KM03.pdf", + paper = "Kalt03a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N. -Adding Larch/Aldor Specifications to Aldor'' -%\verb|axiom-developer.org/axiom-website/papers/Dunxx.pdf| - abstract = " - We describe a proposal to add Larch-style annotations to the Aldor - programming language, based on our PhD research. The annotations - are intended to be machine-checkable and may be used for a variety - of purposes ranging from compiler optimizations to verification - condition (VC) generation. In this report we highlight the options - available and describe the changes which would need to be made to - the compiler to make use of this technology." +\begin{chunk}{axiom.bib} +@InProceedings{Gao04a, + author = "Shuhong, Gao and Kaltofen, Erich and May, John P. and + Yang, Zhengfeng and Zhi, Lihong", + title = "Approximate factorization of multivariate polynomials via + differential equations", + year = "2004", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'04", + crossref = "ISSAC04", + pages = "167--174", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/04/GKMYZ04.pdf", + paper = "Gao04a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve; -Martin, Ursula -Lightweight Formal Methods For Computer Algebra Systems'' -\verb|www.cs.st-andrews.ac.uk/~tom/pub/issac98.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Dun98.pdf| - abstract = " - Demonstrates the use of formal methods tools to provide a semantics - for the type hierarchy of the Axiom computer algebra system, and a - methodology for Aldor program analysis and verification. There are - examples of abstract specifications of Axiom primitives." +\begin{chunk}{axiom.bib} +@Article{Kalt08, + author = "Kaltofen, Erich and May, John and Yang, Zhengfeng and Zhi, Lihong", + title = "Approximate Factorization of Multivariate Polynomials Using + Singular Value Decomposition", + year = "2008", + journal = "Journal of Symbolic Computation", + volume = "43", + number = "5", + pages = "359--376", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KMYZ07.pdf", + paper = "Kalt08.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN -Larch/Aldor - A Larch BISL for AXIOM and Aldor'' -PhD Thesis, 1999 -\verb|www.cs.st-andrews.uk/files/publications/Dun99.php| -%\verb|axiom-developer.org/axiom-website/papers/Dun99a.pdf| - abstract = " - In this thesis we investigate the use of lightweight formal methods - and verification conditions (VCs) to help improve the reliability of - components constructed within a computer algebra system. We follow the - Larch approach to formal methods and have designed a new behavioural - interface specification language (BISL) for use with Aldor: the - compiled extension language of Axiom and a fully-featured programming - language in its own right. We describe our idea of lightweight formal - methods, present a design for a lightweight verification condition - generator and review our implementation of a prototype verification - condition generator for Larch/Aldor." +\begin{chunk}{axiom.bib} +@InProceedings{Hitz99, + author = "Hitz, M.A. and Kaltofen, E. and Lakshman, Y.N.", + title = "Efficient Algorithms for Computing the Nearest Polynomial + With A Real Root and Related Problems", + booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC99", + pages = "205--212", + year = "1999", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/HKL99.pdf", + paper = "Hitz99.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula; -Linton, Steve -Formal Methods for Extensions to CAS'' -FME'99, Toulouse, France, Sept 20-24, 1999, pp 1758-1777 -\verb|tom.host.cs.st-andrews.ac.uk/pub/fm99.ps| -%\verb|axiom-developer.org/axiom-website/papers/Dun00.pdf| - abstract = " - We demonstrate the use of formal methods tools to provide a semantics - for the type hierarchy of the AXIOM computer algebra system, and a - methodology for Aldor program analysis and verification. We give a - case study of abstract specifications of AXIOM primitives, and provide - an interface between these abstractions and Aldor code." +\begin{chunk}{axiom.bib} +@InProceedings{Hitz98, + author = "Hitz, M. A. and Kaltofen, E.", + title = "Efficient Algorithms for Computing the Nearest Polynomial + with Constrained Roots", + booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC98", + year = "1998", + pages = "236--243", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/HiKa98.pdf", + paper = "Hitz98.pdf" +} \end{chunk} +\section{Software Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@misc{Hard13, - author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.", - title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator", - url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf", - paper = "Hard13.pdf", - abstract = " - This paper describes an effort to create a library of formally - verified software component models from code that have been compiled - using the Low-Level Virtual Machine (LLVM) intermediate form. The idea - is to build a translator from LLVM to the applicative subset of Common - Lisp accepted by the ACL2 theorem prover. They perform verification of - the component model using ACL2's automated reasoning capabilities." +@InProceedings{Diaz91, + author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.", + title = "DSC A System for Distributed Symbolic Computation", + booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC91", + pages = "323--332", + year = "1991", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf", + paper = "Diaz91.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Hard14, - author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and - McClurg, Jedidiah R.", - title = "Development of a Translator from LLVM to ACL2", - url = "http://arxiv.org/pdf/1406.1566", - paper = "Hard14.pdf", - abstract = " - In our current work a library of formally verified software components - is to be created, and assembled, using the Low-Level Virtual Machine - (LLVM) intermediate form, into subsystems whose top-level assurance - relies on the assurance of the individual components. We have thus - undertaken a project to build a translator from LLVM to the - applicative subset of Common Lisp accepted by the ACL2 theorem - prover. Our translator produces executable ACL2 formal models, - allowing us to both prove theorems about the translated models as well - as validate those models by testing. The resulting models can be - translated and certified without user intervention, even for code with - loops, thanks to the use of the def::ung macro which allows us to - defer the question of termination. Initial measurements of concrete - execution for translated LLVM functions indicate that performance is - nearly 2.4 million LLVM instructions per second on a typical laptop - computer. In this paper we overview the translation process and - illustrate the translator's capabilities by way of a concrete example, - including both a functional correctness theorem as well as a - validation test for that example." +@InProceedings{Chan94, + author = "Chan, K.C. and Diaz, A. and Kaltofen, E.", + editor = "R. J. Lopez", + title = "A distributed approach to problem solving in Maple", + booktitle = "Maple V: Mathematics and its Application", + pages = "13--21", + publisher = {Birkh\"auser}, + year = "1994", + series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)", + address = "Boston", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz", + paper = "Chan94.ps" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Lamport 02]{Lamp02} Lamport, Leslie -Specifying Systems'' -\verb|research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf| -Addison-Wesley ISBN 0-321-14306-X -%\verb|axiom-developer.org/axiom-website/papers/Lamp02.pdf| +\begin{chunk}{axiom.bib} +@InProceedings{Duma02, + author = "Dumas, J.-G. and Gautier, T. and Giesbrecht, M. and Giorgi, P. + and Hovinen, B. and Kaltofen, E. and Saunders, B.D. and + Turner, W.J. and Villard, G.", + title = "{LinBox}: A Generic Library for Exact Linear Algebra", + booktitle = "Proc. First Internat. Congress Math. Software ICMS 2002, + Beijing, China", + crossref = "ICMS02", + pages = "40--50", + year = "2002", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Detal02.pdf", + paper = "Duma02.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D. -Investigating some Embedded Verification Techniques for Computer - Algebra Systems'' -\verb|www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz| -%\verb|axiom-developer.org/axiom-website/papers/Mart97.ps| - abstract = " - This paper reports some preliminary ideas on a collaborative project - between St. Andrews University in the UK and NAG Ltd. The project aims - to use embedded verification techniques to improve the reliability and - mathematical soundness of computer algebra systems. We give some - history of attempts to integrate computer algebra systems and - automated theorem provers and discuss possible advantages and - disadvantages of these approaches. We also discuss some possible case - studies." +\begin{chunk}{axiom.bib} +@InProceedings{Kalt05a, + author = "Kaltofen, Erich and Morozov, Dmitriy and Yuhasz, George", + title = "Generic Matrix Multiplication and Memory Management in {LinBox}", + year = "2005", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05", + crossref = "ISSAC05", + pages = "216--223", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KMY05.pdf", + paper = "Kalt05a.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@book{Maso86, - author = "Mason, Ian A.", - title = "The Semantics of Destructive Lisp", - publisher = "Center for the Study of Language and Information", - year = "1986", - isbn = "0-937073-06-7", - abstract = " - Our basic premise is that the ability to construct and modify programs - will not improve without a new and comprehensive look at the entire - programming process. Past theoretical research, say, in the logic of - programs, has tended to focus on methods for reasoning about - individual programs; little has been done, it seems to us, to develop - a sound understanding of the process of programming -- the process by - which programs evolve in concept and in practice. At present, we lack - the means to describe the techniques of program construction and - improvement in ways that properly link verification, documentation and - adaptability." +@InProceedings{Diaz98, + author = "Diaz, A. and Kaltofen, E.", + title = "{FoxBox}, a System for Manipulating Symbolic Objects in Black Box + Representation", + booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC98", + year = "1998", + pages = "30--37", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/DiKa98.pdf", + paper = "Diaz98.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan; -Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael -Use of Formal Methods at Amazon Web Services'' -\verb|research.microsoft.com/en-us/um/people/lamport/tla/| -\verb|formal-methods-amazon.pdf| - abstract = " - In order to find subtle bugs in a system design, it is necessary to - have a precise description of that design. There are at least two - major benefits to writing a precise design; the author is forced to - think more clearly, which helps eliminate plausible hand-waving'', - and tools can be applied to check for errors in the design, even while - it is being written. In contrast, conventional design documents - consist of prose, static diagrams, and perhaps pseudo-code in an ad - hoc untestable language. Such descriptions are far from precise; they - are often ambiguous, or omit critical aspects such as partial failure - or the granularity of concurrency (i.e. which constructs are assumed - to be atomic). At the other end of the spectrum, the final executable - code is unambiguous, but contains an overwhelming amount of detail. We - needed to be able to capture the essence of a design in a few hundred - lines of precise description. As our designs are unavoidably complex, - we need a highly-expressive language, far above the level of code, but - with precise semantics. That expressivity must cover real-world - concurrency and fault-tolerance. And, as we wish to build services - quickly, we wanted a language that is simple to learn and apply, - avoiding esoteric concepts. We also very much wanted an existing - ecosystem of tools. We found what we were looking for in TLA+, a - formal specification language." +\begin{chunk}{axiom.bib} +@InProceedings{Diaz93, + author = "Diaz, A. and Kaltofen, E. and Lobo, A. and Valente, T.", + editor = "A. Miola", + title = "Process scheduling in {DSC} and the large sparse linear + systems challenge", + booktitle = "Proc. DISCO '93", + series = "Lect. Notes Comput. Sci.", + pages = "66--80", + year = "1993", + volume = "722", + publisher = "Springer-Verlag", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf", + paper = "Diaz93.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Poll 99a]{P99a} Poll, Erik -The Type System of Axiom'' -\verb|www.cs.ru.nl/E.Poll/talks/axiom.pdf| -%\verb|axiom-developer.org/axiom-website/papers/P99a.pdf| - abstract = " - This is a slide deck from a talk on the correspondence between - Axiom/Aldor types and Logic." +\begin{chunk}{axiom.bib} +@Article{Diaz95a, + author = "Diaz, A. and Hitz, M. and Kaltofen, E. and Lobo, A. and + Valtente, T.", + title = "Process scheduling in {DSC} and the large sparse linear + systems challenge", + journal = "Journal of Symbolic Computing", + year = "1995", + volume = "19", + number = "1--3", + pages = "269--282", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DHKLV95.pdf", + paper = "Diaz95a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon -The Type System of Aldor'' -\verb|www.cs.kent.ac.uk/pubs/1999/874/content.ps| -%\verb|axiom-developer.org/axiom-website/papers/PT99.pdf| - abstract = " - This paper gives a formal description of -- at least a part of -- - the type system of Aldor, the extension language of the Axiom. - In the process of doing this a critique of the design of the system - emerges." +\begin{chunk}{axiom.bib} +@Article{Free88, + author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and + Yagati, Lakshman", + title = "DAGWOOD: A system for manipulating polynomials given by + straight-line programs", + journal = "ACM Trans. Math. Software", + year = "1988", + volume = "14", + number = "3", + pages = "218--240", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf", + paper = "Free88.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon -Adding the axioms to Axiom. Toward a system of automated reasoning in -Aldor'' -\verb|citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps| -%\verb|axiom-developer.org/axiom-website/papers/PTxx.pdf| - abstract = " - This paper examines the proposal of using the type system of Axiom to - represent a logic, and thus to use the constructions of Axiom to - handle the logic and represent proofs and propositions, in the same - way as is done in theorem provers based on type theory such as Nuprl - or Coq. - - The paper shows an interesting way to decorate Axiom with pre- and - post-conditions. +\section{The Seven Dwarfs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - The Curry-Howard correspondence used is - \begin{verbatim} - PROGRAMMING LOGIC - Type Formula - Program Proof - Product/record type (...,...) Conjunction - Sum/union type \/ Disjunction - Function type -> Implication - Dependent function type (x:A) -> B(x) Universal quantifier - Dependent product type (x:A,B(x)) Existential quantifier - Empty type Exit Contradictory proposition - One element type Triv True proposition - \end{verbatim}" +\begin{chunk}{axiom.bib} +@InCollection{Kalt10a, + author = "Kaltofen, Erich L.", + title = "The {Seven} {Dwarfs}'' of Symbolic Computation", + booktitle = "Numeric and Symbolic Scientific Computing + Progress and Prospects", + crossref = "LaPau12", + pages = "95--104", + year = "2010", + keywords = "survey", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_7dwarfs.pdf", + paper = "Kalt10a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon -Integrating Computer Algebra and Reasoning through the Type System -of Aldor'' -%\verb|axiom-developer.org/axiom-website/papers/PT00.pdf| +\section{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@inproceedings{Bro86, + author = "Bronstein, Manuel", + title = "Gsolve: a faster algorithm for solving systems of algebraic + equations", + booktitle = "Proc of 5th ACM SYMSAC", + year = "1986", + pages = "247-249", + isbn = "0-89791-199-7", abstract = " - A number of combinations of reasoning and computer algebra systems - have been proposed; in this paper we describe another, namely a way to - incorporate a logic in the computer algebra system Axiom. We examine - the type system of Aldor -- the Axiom Library Compiler -- and show - that with some modifications we can use the dependent types of the - system to model a logic, under the Curry-Howeard isomorphism. We give - a number of example applications of the logi we construct and explain - a prototype implementation of a modified type-checking system written - in Haskell." + We apply the elimination property of Gr{\"o}bner bases with respect to + pure lexicographic ordering to solve systems of algebraic equations. + We suggest reasons for this approach to be faster than the resultant + technique, and give examples and timings that show that it is indeed + faster and more correct, than MACSYMA's solve." +} \end{chunk} -\subsection{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Boehm 86]{Boe86} Boehm, Hans-J.; Cartwright, Robert; Riggle, Mark; -O'Donnell, Michael J. -Exact Real Arithmetic: A Case Study in Higher Order Programming'' -\verb|dev.acm.org/pubs/citations/proceedings/lfp/319838/p162-boehm| -%\verb|axiom-developer.org/axiom-website/papers/Boe86.pdf| - -\end{chunk} +\section{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Briggs 04]{Bri04} Briggs, Keith -Exact real arithmetic'' -\verb|keithbriggs.info/documents/xr-kent-talk-pp.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bri04.pdf| +{Bro99, + author = "Bronstein, Manuel", + title = "Fast Deterministic Computation of Determinants of Dense Matrices", + url = "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html", + paper = "Bro99.pdf", + abstract = " + In this paper we consider deterministic computation of the exact + determinant of a dense matrix $M$ of integers. We present a new + algorithm with worst case complexity + $O(n^4(log n+ log \verb?||M||?)+x^3 log^2 \verb?||M||?)$, + where $n$ is the dimension of the matrix + and \verb?||M||? is a bound on the entries in $M$, but with + average expected complexity + $O(n^4+m^3(log n + log \verb?||M||?)^2)$, + assuming some plausible properties about the distribution of $M$. + We will also describe a practical version of the algorithm and include + timing data to compare this algorithm with existing ones. Our result + does not depend on fast'' integer or matrix techniques." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W. -Computation with the Extended Rational Numbers and an Application to -Interval Arithmetic'' -\verb|www.cs.berkeley.edu/~fateman/papers/extrat.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Fat94.pdf| +{Kel00, + author = "Kelsey, Tom", + title = "Exact Numerical Computation via Symbolic Computation", + url = "http://tom.host.cs.st-andrews.ac.uk/pub/ccapaper.pdf", + paper = "Kel00.pdf", abstract = " - Programming languages such as Common Lisp, and virtually every - computer algebra system (CAS), support exact arbitrary-precision - integer arithmetic as well as exect rational number computation. - Several CAS include interval arithmetic directly, but not in the - extended form indicated here. We explain why changes to the usual - rational number system to include infinity and not-a-number'' may be - useful, especially to support robust interval computation. We describe - techniques for implementing these changes." + We provide a method for converting any symbolic algebraic expression + that can be converted into a floating point number into an exact + numeric representation. We use this method to demonstrate a suite of + procedures for the representation of, and arithmetic over, exact real + numbers in the Maple computer algebra system. Exact reals are + represented by potentially infinite lists of binary digits, and + interpreted as sums of negative powers of the golden ratio." +} \end{chunk} -\begin{chunk}{axiom.bib} -@incollection{Lamb06, - author = "Lambov, Branimir", - title = "Interval Arithmetic Using SSE-2", - booktitle = "Lecture Notes in Computer Science", - publisher = "Springer-Verlag", - year = "2006", - isbn = "978-3-540-85520-0", - pages = "102-113" +\begin{chunk}{ignore} +{Yang14, + author ="Yang, Xiang and Mittal, Rajat", + title = "Acceleration of the Jacobi iterative method by factors exceeding 100 + using scheduled relation", + url = +"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf", + paper = "Yang14.pdf" } \end{chunk} -\subsection{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David -Numerical Solution of Ordinary Differential Equations'' -\verb|homepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Atk09.pdf| +{Corl0, + author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M. + and Bradford, Russell and Davenport, James H.", + title = "Reasoning about the elementary functions of complex analysis", + url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002-amai-reasoning.pdf", + paper = "Corl05.pdf", abstract = " - This book is an expanded version of supplementary notes that we used - for a course on ordinary differential equations for upper-division - undergraduate students and beginning graduate students in mathematics, - engineering, and sciences. The book introduces the numerical analysis - of differential equations, describing the mathematical background for - understanding numerical methods and giving information on what to - expect when using them. As a reason for studying numerical methods as - a part of a more general course on differential equations, many of the - basic ideas of the numerical analysis of differential equations are - tied closely to theoretical behavior associated with the problem being - solved. For example, the criteria for the stability of a numerical - method is closely connected to the stability of the differential - equation problem being solved." - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P. -A practical method for numerical evaluations of solutions of partial - differential equations of heat-conduction type'' -Advances in Computational Mathematics Vol 6 pp207-226 (1996) -\verb|www.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/| -\verb|CNMethod-original.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Cran96.pdf| + There are many problems with the simplification of elementary + functions, particularly over the complex plane. Systems tend to make + howlers'' or not to simplify enough. In this paper we outline the + unwinding number'' approach to such problems, and show how it can be + used to prevent errors and to systematise such simplification, even + though we have not yet reduced the simplification process to a + complete algorithm. The unsolved problems are probably more amenable + to the techniques of artificial intelligence and theorem proving than + the original problem of complex-variable analysis." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien; -Zimmermann, Paul -Worst Cases for the Exponential Function -in the IEEE-754r decimal64 Format'' -in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0 -(2006) pp114-125 +{Ng68, + author = "Ng, Edward W. and Geller, Murray", + title = "A Table of Integrals of the Error functions", + url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf", + paper = "Ng68.pdf", abstract = " - We searched for the worst cases for correct rounding of the - exponential function in the IEEE 754r decimal64 format, and computed - all the bad cases whose distance from a breakpoint (for all rounding - modes) is less than $10^{-15}$ ulp, and we give the worst ones. In - particular, the worst case for - $\vert{}x\vert{} \ge 3 x 10^{-11}$ is - $- exp(9.407822313572878x10^{-2} = - 1.09864568206633850000000000000000278\ldots -$ - This work can be extended to other elementary functions in the decimal64 - format and allows the design of reasonably fast routines that will - evaluate these functions with correct rounding, at least in some - situations." - -\end{chunk} - -\begin{chunk}{axiom.bib} -@book{Hamm62, - author = "Hamming R W.", - title = "Numerical Methods for Scientists and Engineers", - publisher = "Dover", - year = "1973", - isbn = "0-486-65241-6" + This is a compendium of indefinite and definite integrals of products + of the Error functions with elementary and transcendental functions." } \end{chunk} -\subsection{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem [Bostock 14]{Bos14} Bostock, Mike -Visualizing Algorithms'' -\verb|bost.ocks.org/mike/algorithms| +{Gell69, + author = "Geller, Murray and Ng, Edward W.", + title = "A Table of Integrals of the Exponential Integral", + url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf", + paper = "Gell69.pdf", abstract = " - This website hosts various ways of visualizing algorithms. The hope is - that these kind of techniques can be applied to Axiom." + This is a compendium of indefinite and definite integrals of products + of the Exponential Integral with elementary or transcendental functions." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A. -Representation of mathematical object in interactive books'' -%\verb|axiom-developer.org/axiom-website/papers/Leexx.pdf| - abstract = " - We present a model for the representation of mathematical objects in - structured electronic documents, in a way that allows for interaction - with applications such as computer algebra systems and proof checkers. - Using a representation that reflects only the intrinsic information of - an object, and storing application-dependent information in so-called - {\sl application descriptions}, it is shown how the translation from - the internal to an external representation and {\sl vice versa} can be - achieved. Hereby a formalisation of the concept of {\sl context} is - introduced. The proposed scheme allows for a high degree of - application integration, e.g., parallel evaluation of subexpressions - (by different computer algebra systems), or a proof checker using a - computer algebra system to verify an equation involving a symbolic - computation." - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell -The Design of a User Interface for Computer Algebra Systems'' -\verb|www.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD-91-626.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Soif91.pdf| +\begin{chunk}{axiom.bib} +@techreport{Segl98, + author = "Segletes, S.B.", + title = "A compact analytical fit to the exponential integral $E_1(x)$", + year = "1998", + institution = "U.S. Army Ballistic Research Laboratory, + Aberdeen Proving Ground, MD", + type = "Technical Report", + number = "ARL-TR-1758", + paper = "Segl98.pdf", abstract = " - This thesis discusses the design and implementation of natural user - interfaces for Computer Algebra Systems. Such an interface must not - only display expressions generated by the Computer Algebra System in - standard mathematical notation, but must also allow easy manipulation - and entry of expressions in that notation. The user interface should - also assist in understanding of large expressions that are generated - by Computer Algebra Systems and should be able to accommodate new - notational forms." + A four-parameter fit is developed for the class of integrals known as + the exponential integral (real branch). Unlike other fits that are + piecewise in nature, the current fit to the exponential integral is + valid over the complete domain of the function (compact) and is + everywhere accurate to within $\pm 0.0052\%$ when evaluating the first + exponential integral, $E_1$. To achieve this result, a methodology + that makes use of analytically known limiting behaviors at either + extreme of the domain is employed. Because the fit accurately captures + limiting behaviors of the $E_1$ function, more accuracy is retained + when the fit is used as part of the scheme to evaluate higher-order + exponential integrals, $E_n$, as compared with the use of brute-force + fits to $E_1$, which fail to accurately model limiting + behaviors. Furthermore, because the fit is compact, no special + accommodations are required (as in the case of spliced piecewise fits) + to smooth the value, slope, and higher derivatives in the transition + region between two piecewise domains. The general methodology employed + to develop this fit is outlined, since it may be used for other + problems as well." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Victor 11]{Vict11} Victor, Bret -Up and Down the Ladder of Abstraction'' -\verb|worrydream.com/LadderOfAbstraction| +\begin{chunk}{axiom.bib} +@techreport{Se09, + author = "Segletes, S.B.", + title = "Improved fits for $E_1(x)$ {\sl vis-\'a-vis} those presented + in ARL-TR-1758", + type = "Technical Report", + number = "ARL-TR-1758", + institution ="U.S. Army Ballistic Research Laboratory, + Aberdeen Proving Ground, MD", + year = "1998", + month = "September", + paper = "Se09.pdf", abstract = " - This interactive essay presents the ladder of abstraction, a technique for - thinking explicitly about these levels, so a designer can move among - them consciously and confidently. " + This is a writeup detailing the more accurate fits to $E_1(x)$, + relative to those presented in ARL-TR-1758. My actual fits are to + $F1 =[x\ exp(x) E_1(x)]$ which spans a functional range from 0 to 1. + The best accuracy I have been yet able to achieve, defined by limiting + the value of $[(F1)_{fit} - F1]/F1$ over the domain, is + approximately 3.1E-07 with a 12-parameter fit, which unfortunately + isn't quite to 32-bit floating-point accuracy. Nonetheless, the fit + is not a piecewise fit, but rather a single continuous function over + the domain of nonnegative x, which avoids some of the problems + associated with piecewise domain splicing." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Victor 12]{Vict12} Victor, Bret -Inventing on Principle'' -\verb|www.youtube.com/watch?v=PUv66718DII| - abstract = " - This video raises the level of discussion about human-computer - interaction from a technical question to a question of effectively - capturing ideas. In particular, this applies well to Axiom's focus on - literate programming." +\section{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\end{chunk} +\begin{chunk}{axiom.bib} +@InProceedings{Kalt99a, + author = "Kaltofen, E. and Monagan, M.", + title = "On the Genericity of the Modular Polynomial {GCD} Algorithm", + booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC99", + year = "1999", + pages = "59--66", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaMo99.pdf", + paper = "Kalt99a.pdf" +} -\subsection{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\end{chunk} \begin{chunk}{ignore} -\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel; -Petkovsek, Marko -On Polynomial Solutions of Linear Operator Equations'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Abra95.pdf| +\bibitem[Knuth 71]{ST-PGCD-Knu71} Knuth, Donald +The Art of Computer Programming'' +2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing, +Addison-Wesley 1971, section 4.6 pp399-505 \end{chunk} \begin{chunk}{ignore} -\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel -On Solutions of Linear Functional Systems'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Abra01.pdf| +\bibitem[Ma 90]{ST-PGCD-Ma90} Ma, Keju; Gathen, Joachim von zur +Analysis of Euclidean Algorithms for Polynomials over Finite Fields'' +J. Symbolic Computation (1990) Vol 9 pp429-455\hfill{} +\verb|www.researchgate.net/publication/220161718_Analysis_of_Euclidean_| +\verb|Algorithms_for_Polynomials_over_Finite_Fields/file/| +\verb|60b7d52b326a1058e4.pdf| +%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ma90.pdf| abstract = " - We describe a new direct algorithm for transforming a linear system of - recurrences into an equivalent one with nonsingular leading or - trailing matrix. Our algorithm, which is an improvement to the EG - elimination method, uses only elementary linear algebra operations - (ranks, kernels, and determinants) to produce an equation satisfied by - the degress of the solutions with finite support. As a consequence, we - can boudn and compute the polynomial and rational solutions of very - general linear functional systems such as systems of differential or - ($q$)-difference equations." + This paper analyzes the Euclidean algorithm and some variants of it + for computing the greatest common divisor of two univariate polynomials + over a finite field. The minimum, maximum, and average number of + arithmetic operations both on polynomials and in the ground field + are derived." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel -On the Factorization of Linear Ordinary Differential Operators'' -Mathematics and Computers in Simulation 42 pp 387-389 (1996) -%\verb|axiom-developer.org/axiom-website/papers/Bro96b.pdf| +\bibitem[Naylor 00a]{N00} Naylor, Bill +Polynomial GCD Using Straight Line Program Representation'' +PhD. Thesis, University of Bath, 2000 +\verb|www.sci.csd.uwo.ca/~bill/thesis.ps| +%\verb|axiom-developer.org/axiom-website/papers/N00.pdf| abstract = " - After reviewing the arithmetic of linear ordinary differential - operators, we describe the current status of the factorisation - algorithm, specially with respect to factoring over non-algebraically - closed constant fields. We also describe recent results from Singer - and Ulmer that reduce determining the differential Galois group of an - operator to factoring." + This thesis is concerned with calculating polynomial greatest common + divisors using straight line program representation. + + In the Introduction chapter, we introduce the problem and describe + some of the traditional representations for polynomials, we then talk + about some of the general subjects central to the thesis, terminating + with a synopsis of the category theory which is central to the Axiom + computer algebra system used during this research. + + The second chapter is devoted to describing category theory. We follow + with a chapter detailing the important sections of computer code + written in order to investigate the straight line program subject. + The following chapter on evalution strategies and algorithms which are + dependant on these follows, the major algorith which is dependant on + evaluation and which is central to our theis being that of equality + checking. This is indeed central to many mathematical problems. + Interpolation, that is the determination of coefficients of a + polynomial is the subject of the next chapter. This is very important + for many straight line program algorithms, as their non-canonical + structure implies that it is relatively difficult to determine + coefficients, these being the basic objects that many algorithms work + on. We talk about three separate interpolation techniques and compare + their advantages and disadvantages. The final two chapters describe + some of the results we have obtained from this research and finally + conclusions we have drawn as to the viability of the straight line + program approach and possible extensions. + + Finally we terminate with a number of appendices discussing side + subjects encountered during the thesis." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko -An introduction to pseudo-linear algebra'' -Theoretical Computer Science V157 pp3-33 (1966) -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro96a.pdf| +\bibitem[Shoup 93]{ST-PGCD-Sh93} Shoup, Victor +Factoring Polynomials over Finite Fields: Asymptotic Complexity vs +Reality*'' +Proc. IMACS Symposium, Lille, France, (1993) +\verb|www.shoup.net/papers/lille.pdf| +%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Sh93.pdf| abstract = " - Pseudo-linear algebra is the study of common properties of linear - differential and difference operators. We introduce in this paper its - basic objects (pseudo-derivations, skew polynomials, and pseudo-linear - operators) and describe several recent algorithms on them, which, when - applied in the differential and difference cases, yield algorithms for - uncoupling and solving systems of linear differential and difference - equations in closed form." + This paper compares the algorithms by Berlekamp, Cantor and + Zassenhaus, and Gathen and Shoup to conclude that (a) if large + polynomials are factored the FFT should be used for polynomial + multiplication and division, (b) Gathen and Shoup should be used if + the number of irreducible factors of $f$ is small. (c) if nothing is + know about the degrees of the factors then Berlekamp's algorithm + should be used." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel -Computer Algebra Algorithms for Linear Ordinary Differential and -Difference equations'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Broxb.pdf| +\bibitem[Gathen 01]{ST-PGCD-Ga01} Gathen, Joachim von zur; Panario, Daniel +Factoring Polynomials Over Finite Fields: A Survey'' +J. Symbolic Computation (2001) Vol 31, pp3-17\hfill{} +\verb|people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.pdf| +%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ga01.pdf| + keywords = "survey", abstract = " - Galois theory has now produced algorithms for solving linear ordinary - differential and difference equations in closed form. In addition, - recent algorithmic advances have made those algorithms effective and - implementable in computer algebra systems. After introducing the - relevant parts of the theory, we describe the latest algorithms for - solving such equations." + This survey reviews several algorithms for the factorization of + univariate polynomials over finite fields. We emphasize the main ideas + of the methods and provide and up-to-date bibliography of the problem. + This paper gives algorithms for {\sl squarefree factorization}, + {\sl distinct-degree factorization}, and {\sl equal-degree factorization}. + The first and second algorithms are deterministic, the third is + probabilistic." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel -An improved algorithm for factoring linear ordinary differential -operators'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael +Algorithms for Polynomial GCD Computation over Algebraic Function Fields'' +\verb|www.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Hoeij04.pdf| abstract = " - We describe an efficient algorithm for computing the associated - equations appearing in the Beke-Schlesinger factorisation method for - linear ordinary differential operators. This algorithm, which is based - on elementary operations with sets of integers, can be easily - implemented for operators of any order, produces several possible - associated equations, of which only the simplest can be selected for - solving, and often avoids the degenerate case, where the order of the - associated equation is less than in the generic case. We conclude with - some fast heuristics that can produce some factorizations while using - only linear computations." + Let $L$ be an algebraic function field in $k \ge 0$ parameters + $t_1,\ldots,t)k$. Let $f_1$, $f_2$ be non-zero polynomials in + $L[x]$. We give two algorithms for computing their gcd. The first, a + modular GCD algorithm, is an extension of the modular GCD algorithm + for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf + Q}$(\alpha[x])$ to function fields. The second, a fraction-free + algorithm, is a modification of the Moreno Maza and Rioboo algorithm + for computing gcds over triangular sets. The modification reduces + coefficient grownth in $L$ to be linear. We give an empirical + comparison of the two algorithms using implementations in Maple." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel -On Solutions of Linear Ordinary Differential Equations in their -Coefficient Field'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro90.pdf| +\bibitem[Wang 78]{Wang78} Wang, Paul S. +An Improved Multivariate Polynomial Factoring Algorithm'' +Mathematics of Computation, Vol 32, No 144 Oct 1978, pp1215-1231 +\verb|www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0568284-3/| +\verb|S0025-5718-1978-0568284-3.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Wang78.pdf| abstract = " - We describe a rational algorithm for finding the denominator of any - solution of a linear ordinary differential equation in its coefficient - field. As a consequence, there is now a rational algorithm for finding - all such solutions when the coefficients can be built up from the - rational functions by finitely many algebraic and primitive - adjunctions. This also eliminates one of the computational bottlenecks - in algorithms that either factor or search for Liouvillian solutions - of such equations with Liouvillian coefficients." + A new algorithm for factoring multivariate polynomials over the + integers based on an algorithm by Wang and Rothschild is described. + The new algorithm has improved strategies for dealing with the known + problems of the original algorithm, namely, the leading coefficient + problem, the bad-zero problem and the occurence of extraneous factors. + It has an algorithm for correctly predetermining leading coefficients + of the factors. A new and efficient p-adic algorith named EEZ is + described. Basically it is a linearly convergent variable-by-variable + parallel construction. The improved algorithm is generally faster and + requires less store than the original algorithm. Machine examples with + comparative timing are included." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel -$\sum^{IT}$ -- A strongly-typed embeddable computer algebra library'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro96.pdf| - abstract = " - We describe the new computer algebra library $\sum^{IT}$ and its - underlying design. The development of $\sum^{IT}$ is motivated by the - need to provide highly efficient implementations of key algorithms for - linear ordinary differential and ($q$)-difference equations to - scientific programmers and to computer algebra users, regardless of - the programming language or interactive system they use. As such, - $\sum^{IT}$ is not a computer algebra system per se, but a library (or - substrate) which is designed to be plugged'' with minimal efforts - into different types of client applications." +\bibitem[Wiki 4]{Wiki4}. +Polynomial greatest common divisor'' +\verb|en.wikipedia.org/wiki/Polynomial_greatest_common_divisor| \end{chunk} +\section{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel -Solving linear ordinary differential equations over -$C(x,e^{\int{f(x)dx}})$ -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro99a.pdf| +\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike +Physics, Topology, Logic and Computation: A Rosetta Stone'' +\verb|arxiv.org/pdf/0903.0340v3.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Baez09.pdf| abstract = " - We describe a new algorithm for computing the solutions in - $F=C(x,e^{\int{f(x)dx}})$ of linear ordinary differential equations - with coefficients in $F$. Compared to the general algorithm, our - algorithm avoids the computation of exponential solutions of equations - with coefficients in $C(x)$, as well as the solving of linear - differential systems over $C(x)$. Our method is effective and has been - implemented." + In physics, Feynman diagrams are used to reason about quantum + processes. In the 1980s, it became clear that underlying these + diagrams is a powerful analogy between quantum physics and + topology. Namely, a linear operator behaves very much like a + cobordism'': a manifold representing spacetime, going between two + manifolds representing space. But this was just the beginning: simiar + diagrams can be used to reason about logic, where they represent + proofs, and computation, where they represent programs. With the rise + of interest in quantum cryptography and quantum computation, it became + clear that there is an extensive network of analogies between physics, + topology, logic and computation. In this expository paper, we make + some of these analogies precise using the concept of closed + symmetric monodial category''. We assume no prior knowledge of + category theory, proof theory or computer science." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel -On Solutions of Linear Ordinary Differential Equations in their - Coefficient Field'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro00.pdf| +\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross +Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire'' +\verb|eprints.eemcs.utwente.nl/7281/01/db-utwente-40501F46.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Meij91.pdf| abstract = " - We extend the notion of monomial extensions of differential fields, - i.e. simple transcendental extensions in which the polynomials are - closed under differentiation, to difference fields. The structure of - such extensions provides an algebraic framework for solving - generalized linear difference equations with coefficients in such - fields. We then describe algorithms for finding the denominator of any - solution of those equations in an important subclass of monomial - extensions that includes transcendental indefinite sums and - products. This reduces the general problem of finding the solutions of - such equations in their coefficient fields to bounding their - degrees. In the base case, this yields in particular a new algorithm - for computing the rational solutions of $q$-difference equations with - polynomial coefficients." + We develop a calculus for lazy functional programming based on + recursion operators associated with data type definitions. For these + operators we derive various algebraic laws that are useful in deriving + and manipulating programs. We shall show that all example functions in + Bird and Wadler's Introduction to Functional Programming'' can be + expressed using these operators." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien -Solutions of linear ordinary differential equations in terms of -special functions'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bro02.pdf| +\bibitem[Youssef 04]{You04} Youssef, Saul +Prospects for Category Theory in Aldor'' +October 2004 +%\verb|axiom-developer.org/axiom-website/papers/You04.pdf| abstract = " - We describe a new algorithm for computing special function solutions - of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary - differential equations, where $m(x)$ is an arbitrary Liouvillian - function, $\eta(x)$ is an arbitrary rational function, and $F$ - satisfies a given second order linear ordinary differential - equations. Our algorithm, which is base on finding an appropriate - point transformation between the equation defining $F$ and the one to - solve, is able to find all rational transformations for a large class - of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$ - special functions of mathematical physics, such as Airy, Bessel, - Kummer and Whittaker functions. It is also able to identify the values - of the parameters entering those special functions, and can be - generalized to equations of higher order." + Ways of encorporating category theory constructions and results into + the Aldor language are discussed. The main features of Aldor which + make this possible are identified, examples of categorical + constructions are provided and a suggestion is made for a foundation + for rigorous results." \end{chunk} +\section{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M. -A Reduction for Regular Differential Systems'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bro03.pdf| +\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.; +Martin, U. +Automated theorem proving in support of computer algebra:'' + symbolic definite integration as a case study'' +%\verb|axiom-developer.org/axiom-website/papers/Adam99.pdf| abstract = " - We propose a definition of regularity of a linear differential system - with coefficients in a monomial extension of a differential field, as - well as a global and truly rational (i.e. factorisation-free) - iteration that transforms a system with regular finite singularites - into an equivalent one with simple finite poles. We then apply our - iteration to systems satisfied by bases of algebraic function fields, - obtaining algorithms for computing the number of irreducible - components and the genus of algebraic curves." + We assess the current state of research in the application of computer + aided formal reasoning to computer algebra, and argue that embedded + verification support allows users to enjoy its benefits without + wrestling with technicalities. We illustrate this claim by considering + symbolic definite integration, and present a verifiable symbolic + definite integral table look up: a system which matches a query + comprising a definite integral with parameters and side conditions, + against an entry in a verifiable table and uses a call to a library of + lemmas about the reals in the theorem prover PVS to aid in the + transformation of the table entry into an answer. We present the full + model of such a system as well as a description of our prototype + implementation showing the efficacy of such a system: for example, the + prototype is able to obtain correct answers in cases where computer + algebra systems [CAS] do not. We extend upon Fateman's web-based table + by including parametric limits of integration and queries with side + conditions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick -Linear recurrences with polynomial coefficients'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| -%\verb|axiom-developer.org/axiom-website/papers/Bro03a.pdf| +\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne; +Kelsey, Tom; Martin, Ursula; Owre, Sam +Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS'' +\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Adam01.pdf| abstract = " - We relate sequences generated by recurrences with polynomial - coefficients to interleaving and multiplexing of sequences generated - by recurrences with constant coefficients. In the special case of - finite fields, we show that such sequences are periodic and provide - linear complexity estimates for all three constructions." + We describe an interface between version 6 of the Maple computer + algebra system with the PVS automated theorem prover. The interface is + designed to allow Maple users access to the robust and checkable proof + environment of PVS. We also extend this environment by the provision + of a library of proof strategies for use in real analysis. We + demonstrate examples using the interface and the real analysis + library. These examples provide proofs which are both illustrative and + applicable to genuine symbolic computation problems." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min -Picard-Vessiot Extensions for Linear Functional Systems'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bro05.pdf| +\begin{chunk}{axiom.bib} +@article{Mahb06, + author = "Mahboubi, Assia", + title = "Proving Formally the Implementation of an Efficient gcd + Algorithm for Polynomials", + journal = "Lecture Notes in Computer Science", + volume = "4130", + year = "2006", + pages = "438-452", + paper = "Mahb06.pdf", abstract = " - Picard-Vessiot extensions for ordinary differential and difference - equations are well known and are at the core of the associated Galois - theories. In this paper, we construct fundamental matrices and - Picard-Vessiot extensions for systems of linear partial functional - equations having finite linear dimension. We then use those extensions - to show that all the solutions of a factor of such a system can be - completed to solutions of the original system." + We describe here a formal proof in the Coq system of the structure + theorem for subresultants which allows to prove formally the + correctness of our implementation of the subresultants algorithm. + Up to our knowledge it is the first mechanized proof of this result." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport 86]{Dav86} Davenport, J.H. -The Risch Differential Equation Problem'' -SIAM J. COMPUT. Vol 15, No. 4 1986 -%\verb|axiom-developer.org/axiom-website/papers/Dav86.pdf| +\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C. +A Pragmatic Approach to Extending Provers by Computer Algebra -- + with Applications to Coding Theory'' +\verb|www.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Ball99.pdf| abstract = " - We propose a new algorithm, similar to Hermite's method for the - integration of rational functions, for the resolution of Risch - differential equations in closed form, or proving that they have no - resolution. By requiring more of the presentation of our differential - fields (in particular that the exponentials be weakly normalized), we - can avoid the introduction of arbitrary constants which have to be - solved for later. + The use of computer algebra is usually considered beneficial for + mechanised reasoning in mathematical domains. We present a case study, + in the application domain of coding theory, that supports this claim: + the mechanised proofs depend on non-trivial algorithms from computer + algebra and increase the reasoning power of the theorem prover. - We also define a class of fields known as exponentially reduced, and - show that solutions of Risch differential equations which arise from - integrating in these fields satisfy the natural'' degree constraints - in their main variables, and we conjecture (after Risch and Norman) - that this is true in all variables." + The unsoundness of computer algebra systems is a major problem in + interfacing them to theorem provers. Our approach to obtaining a sound + overall system is not blanket distrust but based on the distinction + between algorithms we call sound and {\sl ad hoc} respectively. This + distinction is blurred in most computer algebra systems. Our + experimental interface therefore uses a computer algebra library. It + is based on formal specifications for the algorithms, and links the + computer algebra library Sumit to the prover Isabelle. + + We give details of the interface, the use of the computer algebra + system on the tactic-level of Isabelle and its integration into proof + procedures." \end{chunk} \begin{chunk}{ignore} -\bibitem[Singer 9]{Sing91.pdf} singer, Michael F. -Liouvillian Solutions of Linear Differential Equations with Liouvillian - Coefficients'' -J. Symbolic Computation V11 No 3 pp251-273 (1991) -\verb|www.sciencedirect.com/science/article/pii/S074771710880048X| -%\verb|axiom-developer.org/axiom-website/papers/Sing91.pdf| +\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre +Interactive Theorem Proving and Program Development'' +Springer ISBN 3-540-20854-2 abstract = " - Let $L(y)=b$ be a linear differential equation with coefficients in a - differential field $K$. We discuss the problem of deciding if such an - equation has a non-zero solution in $K$ and give a decision procedure - in case $K$ is an elementary extension of the field of rational - functions or is an algebraic extension of a transcendental liouvillian - extension of the field of rational functions We show how one can use - this result to give a procedure to find a basis for the space of - solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field - and $L(y)$ has coefficients in $K$." - -\end{chunk} + Coq is an interactive proof assistant for the development of + mathematical theories and formally certified software. It is based on + a theory called the calculus of inductive constructions, a variant of + type theory. -\begin{chunk}{ignore} -\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin -Symbolic Solutions of Discontinuous Differential Equations'' -\verb|e-collection.library.ethz.ch/eserv/eth:39463/eth-39463-01.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mohr94.pdf| + This book provides a pragmatic introduction to the development of + proofs and certified programs using Coq. With its large collection of + examples and exercies it is an invaluable tool for researchers, + students, and engineers interested in formal methods and the + development of zero-fault software." \end{chunk} \begin{chunk}{ignore} -\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin -A Normal Form for Function Rings of Piecewise Functions'' -J. Symbolic Computation (1998) Vol 26 pp607-619 -\verb|www.cas.mcmaster.ca/~mohrens/JSC.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mohr98.pdf| +\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R. +Polymorphic Data Types, Objects, Modules and Functors,: is it too much?'' +%\verb|axiom-developer.org/axiom-website/papers/BHR00.pdf| abstract = " - Computer algebra systems often have to deal with piecewise continuous - functions. These are, for example, the absolute value function, - signum, piecewise defined functions but also functions that are the - supremum or infimum of two functions. We present a new algebraic - approach to these types of problems. This paper presents a normal form - for a function ring containing piecewise polynomial functions of an - expression. The main result is that this normal form can be used to - decide extensional equality of two piecewise functions. Also we define - supremum and infimum for piecewise functions; in fact, we show that - the function ring forms a lattice. Additionally, a method to solve - equalities and inequalities in this function ring is - presented. Finally, we give a user interface'' to the algebraic - representation of the piecewise functions." + Abstraction is a powerful tool for developers and it is offered by + numerous features such as polymorphism, classes, modules, and + functors, $\ldots$ A working programmer may be confused by this + abundance. We develop a computer algebra library which is being + certificed. Reporting this experience made with a language (Ocaml) + offering all these features, we argue that the are all needed + together. We compare several ways of using classes to represent + algebraic concepts, trying to follow as close as possible mathematical + specification. Thenwe show how to combine classes and modules to + produce code having very strong typing properties. Currently, this + library is made of one hundred units of functional code and behaves + faster than analogous ones such as Axiom." \end{chunk} \begin{chunk}{ignore} -\bibitem[Weber 06]{Webe06} Weber, Andreas -Quantifier Elimination on Real Closed Fields and Differential Equations'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber2006a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe06.pdf| - keywords = "survey", +\bibitem[Boulme 01]{BHHMR01} +Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissier-Morain, V.; Rioboo, R. +On the way to certify Computer Algebra Systems'' +Calculemus-2001 +%\verb|axiom-developer.org/axiom-website/papers/BHHMR01.pdf| abstract = " - This paper surveys some recent applications of quantifier elimination - on real closed fields in the context of differential - equations. Although polynomial vector fields give rise to solutions - involving the exponential and other transcendental functions in - general, many questions can be settled within the real closed field - without referring to the real exponential field. The technique of - quantifier elimination on real closed fields is not only of - theoretical interest, but due to recent advances on the algorithmic - side including algorithms for the simplification of quantifier-free - formulae the method has gained practical applications, e.g. in the - context of computing threshold conditions in epidemic modeling." + The FOC project aims at supporting, within a coherent software system, + the entire process of mathematical computation, starting with proved + theories, ending with certified implementations of algorithms. In this + paper, we explain our design requirements for the implementation, + using polynomials as a running example. Indeed, proving correctness of + implementations depends heavily on the way this design allows + mathematical properties to be truly handled at the programming level. + + The FOC project, started at the fall of 1997, is aimed to build a + programming environment for the development of certified symbolic + computation. The working languages are Coq and Ocaml. In this paper, + we present first the motivations of the project. We then explain why + and how our concern for proving properties of programs has led us to + certain implementation choices in Ocaml. This way, the sources express + exactly the mathematical dependencies between different structures. + This may ease the achievement of proofs." \end{chunk} \begin{chunk}{ignore} -\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix -Liouvillian solutions of third order differential equations'' -J. Symbolic COmputations 36 pp 855-889 (2003) -\verb|www.sciencedirect.com/science/article/pii/S0747717103000658| -%\verb|axiom-developer.org/axiom-website/papers/Ulm03.pdf| +\bibitem[Daly 10]{Daly10} Daly, Timothy +Intel Instruction Semantics Generator'' +\verb|daly.axiom-developer.org/TimothyDaly_files/publications/sei/intel/intel.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Daly10.pdf| abstract = " - The Kovacic algorithm and its improvements give explicit formulae for - the Liouvillian solutions of second order linear differential - equations. Algorithms for third order differential equations also - exist, but the tools they use are more sophisticated and the - computations more involved. In this paper we refine parts of the - algorithm to find Liouvillian solutions of third order equations. We - show that,except for four finite groups and a reduction to the second - order case, it is possible to give a formula in the imprimitve - case. We also give necessary conditions and several simplifications - for the computation of the minimal polynomial for the remaining finite - set of finite groups (or any known finite group) by extracting - ramification information from the character table. Several examples - have been constructed, illustrating the possibilities and limitations." + Given an Intel x86 binary, extract the semantics of the instruction + stream as Conditional Concurrent Assignments (CCAs). These CCAs + represent the semantics of each individual instruction. They can be + composed to represent higher level semantics." \end{chunk} -\subsection{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Carette 04]{Car04} Carette, Jacques -Understanding Expression Simplification'' -\verb|www.cas.mcmaster.ca/~carette/publications/simplification.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Car04.pdf| +\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John; +Jansson, Patrik; Gibbons, Jeremy +Fast and Loose Reasoning is Morally Correct'' +ACM POPL'06 January 2005, Charleston, South Carolina, USA +%\verb|axiom-developer.org/axiom-website/papers/Dani06.pdf| abstract = " - We give the first formal definition of the concept of {\sl - simplification} for general expressions in the context of Computer - Algebra Systems. The main mathematical tool is an adaptation of the - theory of Minimum Description Length, which is closely related to - various theories of complexity, such as Kolmogorov Complexity and - Algorithmic Information Theory. In particular, we show how this theory - can justify the use of various magic constants'' for deciding - between some equivalent representations of an expression, as found in - implementations of simplification routines." + Functional programmers often reason about programs as if they were + written in a total language, expecting the results to carry over to + non-toal (partial) languages. We justify such reasoning. -\end{chunk} + Two languages are defined, one total and one partial, with identical + syntax. The semantics of the partial language includes partial and + infinite values, and all types are lifted, including the function + spaces. A partial equivalence relation (PER) is then defined, the + domain of which is the total subset of the partial language. For types + not containing function spaces the PER relates equal values, and + functions are related if they map related values to related values. + + It is proved that if two closed terms have the same semantics in the + total language, then they have related semantics in the partial + language. It is also shown that the PER gives rise to a bicartesian + closed category which can be used to reason about values in the domain + of the relation." -\subsection{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\end{chunk} \begin{chunk}{ignore} -\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor -Definite Integration'' -\verb|www.cs.cmu.edu/~adamchik/articles/integr/mj.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Adamxx.pdf| +\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell; +England, Matthew; Wilson, David +Program Verification in the presence of complex numbers, functions with +branch cuts etc.'' +\verb|arxiv.org/pdf/1212.5417.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Davenp12.pdf| + abstract = " + In considering the reliability of numerical programs, it is normal to + limit our study to the semantics dealing with numerical precision''. + On the other hand, there is a great deal of work on the reliability of + programs that essentially ignores the numerics. The thesis of this + paper is that there is a class of problems that fall between these + two, which could be described as does the low-level arithmetic + implement the high-level mathematics''. Many of these problems arise + because mathematics, particularly the mathematics of the complex + numbers, is more difficult than expected: for example the complex + function log is not continuous, writing down a program to compute an + inverse function is more complicated than just solving an equation, + and many algebraic simplification rules are not universally valid. + + The good news is that these problems are {\sl theoretically} capable + of being solved, and are {\sl practically} close to being solved, but + not yet solved, in several real-world examples. However, there is + still a long way to go before implementations match the theoretical + possibilities." \end{chunk} \begin{chunk}{ignore} -\bibitem[Adamchik 97]{Adam97} Adamchik, Victor -A Class of Logarithmic Integrals'' -\verb|www.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Adam97.pdf| +\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas +Guarded Expressions in Practice'' +\verb|redlog.dolzmann.de/papers/pdf/MIP-9702.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Dolz97.pdf| abstract = " - A class of definite integrals involving cyclotomic polynomials and - nested logarithms is considered. The results are given in terms of - derivatives of the Hurwitz Zeta function. Some special cases for which - such derivatives can be expressed in closed form are also considered." + Computer algebra systems typically drop some degenerate cases when + evaluating expressions, e.g. $x/x$ becomes 1 dropping the case + $x=0$. We claim that it is feasible in practice to compute also the + degenerate cases yielding {\sl guarded expressions}. We work over real + closed fields but our ideas about handling guarded expressions can be + easily transferred to other situations. Using formulas as guards + provides a powerful tool for heuristically reducing the combinatorial + explosion of cases: equivalent, redundant, tautological, and + contradictive cases can be detected by simplification and quantifier + elimination. Our approach allows to simplify the expressions on the + basis of simplification knowledge on the logical side. The method + described in this paper is implemented in the REDUCE package GUARDIAN, + which is freely available on the WWW." \end{chunk} \begin{chunk}{ignore} -\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios -Definite Integration using the Generalized Hypergeometric Functions'' -\verb|dspace.mit.edu/handle/1721.1/16269| -%\verb|axiom-developer.org/axiom-websitep/papers/Avgo77.pdf| +\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue +Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants +and Computer Algebra System Framework'' +Calculemus (2011) Springer +\verb|paradise.caltech.edu/~yli/paper/oa-polyml.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DR11.pdf| abstract = " - A design for the definite integration of approximately fifty Special - Functions is described. The Generalized Hypergeometric Functions are - utilized as a basis for the representation of the members of the above - set of Special Functions. Only a relatively small number of formulas - that generally involve Generalized Hypergeometric Functions are - utilized for the integration stage. A last and crucial stage is - required for the integration process: the reduction of the Generalized - Hypergeometric Function to Elementary and/or Special Functions. - - The result of an early implementation which involves Laplace - transforms are given and some actual examples with their corresponding - timing are provided." + This paper presents an ongoing effort to integrate the Axiom family of + computer algebra systems with Poly/ML-based proof assistants in the + same framework. A long term goal is to make a large set of efficient + implementations of algebraic algorithms available to popular proof + assistants, and also to bring the power of mechanized formal + verification to a family of strongly typed computer algebra systems at + a modest cost. Our approach is based on retargeting the code generator + of the OpenAxiom compiler to the Poly/ML abstract machine." \end{chunk} \begin{chunk}{ignore} -\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil -A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite - Terms'' -\verb|www.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bad89.pdf| - abstract = " - The result obtained generalizes Liouville's Theorem by allowing, in - addition to the elementary functions, dilogarithms to appear in the - integral of an elementary function. The basic conclusion is that an - associated function to the dilogarihm, if dilogarithms appear in the - integral, appears linearly, with logarithms appearing in a non-linear - way." - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil -Integration in Finite Terms with Elementary Functions and Dilogarithms'' -\verb|dspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Bad94.pdf| +\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N. +Adding Larch/Aldor Specifications to Aldor'' +%\verb|axiom-developer.org/axiom-website/papers/Dunxx.pdf| abstract = " - In this thesis, we report on a new theorem that generalizes - Liouville's theorem on integration in finite terms. The new theorem - allows dilogarithms to occur in the integral in addition to elementary - functions. The proof is base on two identities for the dilogarithm, - that characterize all the possible algebraic relations among - dilogarithms of functions that are built up from the rational - functions by taking transcendental exponentials, dilogarithms, and - logarithms." + We describe a proposal to add Larch-style annotations to the Aldor + programming language, based on our PhD research. The annotations + are intended to be machine-checkable and may be used for a variety + of purposes ranging from compiler optimizations to verification + condition (VC) generation. In this report we highlight the options + available and describe the changes which would need to be made to + the compiler to make use of this technology." \end{chunk} \begin{chunk}{ignore} -\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil -A Note on Symbolic Integration with Polylogarithms'' -J. Math Vol 8 pp229-241 (2011) -%\verb|axiom-developer.org/axiom-website/papers/Bad10.pdf| +\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve; +Martin, Ursula +Lightweight Formal Methods For Computer Algebra Systems'' +\verb|www.cs.st-andrews.ac.uk/~tom/pub/issac98.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Dun98.pdf| abstract = " - We generalize partially Liouville's theorem on integration in finite - terms to allow polylogarithms of any order to occur in the integral in - addition to elementary functions. The result is a partial - generalization of a theorem proved by the author for the - dilogarithm. It is also a partial proof of a conjecture postulated by - the author in 1994. The basic conclusion is that an associated - function to the nth polylogarithm appears linearly with logarithms - appearing possibly in a polynomial way with non-constant coefficients." + Demonstrates the use of formal methods tools to provide a semantics + for the type hierarchy of the Axiom computer algebra system, and a + methodology for Aldor program analysis and verification. There are + examples of abstract specifications of Axiom primitives." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D. -A contour integral involving legendre polynomial and Meijer's G-function'' -\verb|link.springer.com/article/10.1007/BF03049565| -%\verb|axiom-developer.org/axiom-website/papers/Bajp70.pdf| +\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN +Larch/Aldor - A Larch BISL for AXIOM and Aldor'' +PhD Thesis, 1999 +\verb|www.cs.st-andrews.uk/files/publications/Dun99.php| +%\verb|axiom-developer.org/axiom-website/papers/Dun99a.pdf| abstract = " - In this paper a countour integral involving Legendre polynomial and - Meijer's G-function is evaluated. the integral is of general character - and it is a generalization of results recently given by Meijer, - MacRobert and others. An integral involving regular radial Coulomb - wave function is also obtained as a particular case." + In this thesis we investigate the use of lightweight formal methods + and verification conditions (VCs) to help improve the reliability of + components constructed within a computer algebra system. We follow the + Larch approach to formal methods and have designed a new behavioural + interface specification language (BISL) for use with Aldor: the + compiled extension language of Axiom and a fully-featured programming + language in its own right. We describe our idea of lightweight formal + methods, present a design for a lightweight verification condition + generator and review our implementation of a prototype verification + condition generator for Larch/Aldor." \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 89]{Bro89a} Bronstein, M. -An Algorithm for the Integration of Elementary Functions'' -Lecture Notes in Computer Science Vol 378 pp491-497 (1989) -%\verb|axiom-developer.org/axiom-website/papers/Bro89a.pdf| +\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula; +Linton, Steve +Formal Methods for Extensions to CAS'' +FME'99, Toulouse, France, Sept 20-24, 1999, pp 1758-1777 +\verb|tom.host.cs.st-andrews.ac.uk/pub/fm99.ps| +%\verb|axiom-developer.org/axiom-website/papers/Dun00.pdf| abstract = " - Trager (1984) recently gave a new algorithm for the indefinite - integration of algebraic functions. His approach was rational'' in - the sense that the only algebraic extension computed in the smallest - one necessary to express the answer. We outline a generalization of - this approach that allows us to integrate mixed elementary - functions. Using only rational techniques, we are able to normalize - the integrand, and to check a necessary condition for elementary - integrability." + We demonstrate the use of formal methods tools to provide a semantics + for the type hierarchy of the AXIOM computer algebra system, and a + methodology for Aldor program analysis and verification. We give a + case study of abstract specifications of AXIOM primitives, and provide + an interface between these abstractions and Aldor code." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel -Integration of Elementary Functions'' -J. Symbolic Computation (1990) 9, pp117-173 September 1988 -%\verb|axiom-developer.org/axiom-website/papers/Bro90a.pdf| +\begin{chunk}{axiom.bib} +@misc{Hard13, + author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.", + title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator", + url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf", + paper = "Hard13.pdf", abstract = " - We extend a recent algorithm of Trager to a decision procedure for the - indefinite integration of elementary functions. We can express the - integral as an elementary function or prove that it is not - elementary. We show that if the problem of integration in finite terms - is solvable on a given elementary function field $k$, then it is - solvable in any algebraic extension of $k(\theta)$, where $\theta$ is - a logarithm or exponential of an element of $k$. Our proof considers - an element of such an extension field to be an algebraic function of - one variable over $k$. - - In his algorithm for the integration of algebraic functions, Trager - describes a Hermite-type reduction to reduce the problem to an - integrand with only simple finite poles on the associated Riemann - surface. We generalize that technique to curves over liouvillian - ground fields, and use it to simplify our integrands. Once the - multipe finite poles have been removed, we use the Puiseux expansions - of the integrand at infinity and a generalization of the residues to - compute the integral. We also generalize a result of Rothstein that - gives us a necessary condition for elementary integrability, and - provide examples of its use." + This paper describes an effort to create a library of formally + verified software component models from code that have been compiled + using the Low-Level Virtual Machine (LLVM) intermediate form. The idea + is to build a translator from LLVM to the applicative subset of Common + Lisp accepted by the ACL2 theorem prover. They perform verification of + the component model using ACL2's automated reasoning capabilities." +} \end{chunk} \begin{chunk}{axiom.bib} -@article{Bron90c, - author = "Bronstein, Manuel", - title = "On the integration of elementary functions", - journal = "Journal of Symbolic Computation", - volume = "9", - number = "2", - pages = "117-173", - year = "1990", - month = "February" +@misc{Hard14, + author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and + McClurg, Jedidiah R.", + title = "Development of a Translator from LLVM to ACL2", + url = "http://arxiv.org/pdf/1406.1566", + paper = "Hard14.pdf", + abstract = " + In our current work a library of formally verified software components + is to be created, and assembled, using the Low-Level Virtual Machine + (LLVM) intermediate form, into subsystems whose top-level assurance + relies on the assurance of the individual components. We have thus + undertaken a project to build a translator from LLVM to the + applicative subset of Common Lisp accepted by the ACL2 theorem + prover. Our translator produces executable ACL2 formal models, + allowing us to both prove theorems about the translated models as well + as validate those models by testing. The resulting models can be + translated and certified without user intervention, even for code with + loops, thanks to the use of the def::ung macro which allows us to + defer the question of termination. Initial measurements of concrete + execution for translated LLVM functions indicate that performance is + nearly 2.4 million LLVM instructions per second on a typical laptop + computer. In this paper we overview the translation process and + illustrate the translator's capabilities by way of a concrete example, + including both a functional correctness theorem as well as a + validation test for that example." } \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 93]{REF-BS93} Bronstein, Manuel; Salvy, Bruno -Full partial fraction decomposition of rational functions'' -In Bronstein [Bro93] pp157-160 ISBN 0-89791-604-2 LCCN QA76.95 I59 1993 -\verb|www.acm.org/pubs/citations/proceedings/issac/164081/| +\bibitem[Lamport 02]{Lamp02} Lamport, Leslie +Specifying Systems'' +\verb|research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf| +Addison-Wesley ISBN 0-321-14306-X +%\verb|axiom-developer.org/axiom-website/papers/Lamp02.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel -A Unification of Liouvillian Extensions'' -%\verb|axiom-developer.org/axiom-website/papers/Bro90b.pdf| +\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D. +Investigating some Embedded Verification Techniques for Computer + Algebra Systems'' +\verb|www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz| +%\verb|axiom-developer.org/axiom-website/papers/Mart97.ps| abstract = " - We generalize Liouville's theory of elementary functions to a larger - class of differential extensions. Elementary, Liouvillian and - trigonometric extensions are all special cases of our extensions. In - the transcendental case, we show how the rational techniques of - integration theory can be applied to our extensions, and we give a - unified presentation which does not require separate cases for - different monomials." + This paper reports some preliminary ideas on a collaborative project + between St. Andrews University in the UK and NAG Ltd. The project aims + to use embedded verification techniques to improve the reliability and + mathematical soundness of computer algebra systems. We give some + history of attempts to integrate computer algebra systems and + automated theorem provers and discuss possible advantages and + disadvantages of these approaches. We also discuss some possible case + studies." \end{chunk} \begin{chunk}{axiom.bib} -@book{Bron97, - author = "Bronstein, Manuel", - title = "Symbolic Integration I--Transcendental Functions", - publisher = "Springer, Heidelberg", - year = "1997", - isbn = "3-540-21493-3", - url = "http://evil-wire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf", - paper = "Bron97.pdf" +@book{Maso86, + author = "Mason, Ian A.", + title = "The Semantics of Destructive Lisp", + publisher = "Center for the Study of Language and Information", + year = "1986", + isbn = "0-937073-06-7", + abstract = " + Our basic premise is that the ability to construct and modify programs + will not improve without a new and comprehensive look at the entire + programming process. Past theoretical research, say, in the logic of + programs, has tended to focus on methods for reasoning about + individual programs; little has been done, it seems to us, to develop + a sound understanding of the process of programming -- the process by + which programs evolve in concept and in practice. At present, we lack + the means to describe the techniques of program construction and + improvement in ways that properly link verification, documentation and + adaptability." } \end{chunk} \begin{chunk}{ignore} -\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel -The Poor Man's Integrator, a parallel integration heuristic'' -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt| -\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples| -%\verb|axiom-developer.org/axiom-website/papers/Bro05a.txt| - -\end{chunk} - -\begin{chunk}{axiom.bib} -@article{Bron06, - author = "Bronstein, M.", - title = "Parallel integration", - journal = "Programming and Computer Software", - year = "2006", - issn = "0361-7688", - volume = "32", - number = "1", - doi = "10.1134/S0361768806010075", - url = "http://dx.doi.org/10.1134/S0361768806010075", - publisher = "Nauka/Interperiodica", - pages = "59-60", - paper = "Bron06.pdf", +\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan; +Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael +Use of Formal Methods at Amazon Web Services'' +\verb|research.microsoft.com/en-us/um/people/lamport/tla/| +\verb|formal-methods-amazon.pdf| abstract = " - Parallel integration is an alternative method for symbolic - integration. While also based on Liouville's theorem, it handles all - the generators of the differential field containing the integrand in - parallel'', i.e. all at once rather than considering only the topmost - one in a recursive fasion. Although it still contains heuristic - aspects, its ease of implementation, speed, high rate of success, and - ability to integrate functions that cannot be handled by the Risch - algorithm make it an attractive alternative." -} + In order to find subtle bugs in a system design, it is necessary to + have a precise description of that design. There are at least two + major benefits to writing a precise design; the author is forced to + think more clearly, which helps eliminate plausible hand-waving'', + and tools can be applied to check for errors in the design, even while + it is being written. In contrast, conventional design documents + consist of prose, static diagrams, and perhaps pseudo-code in an ad + hoc untestable language. Such descriptions are far from precise; they + are often ambiguous, or omit critical aspects such as partial failure + or the granularity of concurrency (i.e. which constructs are assumed + to be atomic). At the other end of the spectrum, the final executable + code is unambiguous, but contains an overwhelming amount of detail. We + needed to be able to capture the essence of a design in a few hundred + lines of precise description. As our designs are unavoidably complex, + we need a highly-expressive language, far above the level of code, but + with precise semantics. That expressivity must cover real-world + concurrency and fault-tolerance. And, as we wish to build services + quickly, we wanted a language that is simple to learn and apply, + avoiding esoteric concepts. We also very much wanted an existing + ecosystem of tools. We found what we were looking for in TLA+, a + formal specification language." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Bron07, - author = "Bronstein, Manuel", - title = "Structure theorems for parallel integration", - journal = "Journal of Symbolic Computation", - volume = "42", - number = "7", - pages = "757-769", - year = "2007", - month = "July", - paper = "Bron07.pdf", +\begin{chunk}{ignore} +\bibitem[Poll 99a]{P99a} Poll, Erik +The Type System of Axiom'' +\verb|www.cs.ru.nl/E.Poll/talks/axiom.pdf| +%\verb|axiom-developer.org/axiom-website/papers/P99a.pdf| abstract = " - We introduce structure theorems that refine Liouville's Theorem on - integration in closed form for general derivations on multivariate - rational function fields. By predicting the arguments of the new - logarithms that an appear in integrals, as well as the denominator of - the rational part, those theorems provide theoretical backing for the - Risch-Norman integration method. They also generalize its applicability - to non-monomial extensions, for example the Lambert W function." -} + This is a slide deck from a talk on the correspondence between + Axiom/Aldor types and Logic." \end{chunk} \begin{chunk}{ignore} -\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin -Integration on Computer Algebra Systems'' -The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 1933-2823 -\verb|12000.org/my_notes/ten_hard_integrals/paper.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Charl07.pdf| +\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon +The Type System of Aldor'' +\verb|www.cs.kent.ac.uk/pubs/1999/874/content.ps| +%\verb|axiom-developer.org/axiom-website/papers/PT99.pdf| abstract = " - In this article, we consider ten indefinite integrals and the ability - of three computer algebra systems (CAS) to evaluate them in - closed-form, appealing only to the class of real, elementary - functions. Although these systems have been widely available for many - years and have undergone major enhancements in new versions, it is - interesting to note that there are still indefinite integrals that - escape the capacity of these systems to provide antiderivatves. When - this occurs, we consider what a user may do to find a solution with - the aid of a CAS." + This paper gives a formal description of -- at least a part of -- + the type system of Aldor, the extension language of the Axiom. + In the process of doing this a critique of the design of the system + emerges." \end{chunk} \begin{chunk}{ignore} -\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin -Symbolic Integration Problems'' -\verb|www.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Charl08.pdf| +\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon +Adding the axioms to Axiom. Toward a system of automated reasoning in +Aldor'' +\verb|citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps| +%\verb|axiom-developer.org/axiom-website/papers/PTxx.pdf| abstract = " - A list of the 50 example integration problems from Kevin Charlwood's 2008 - article Integration on Computer Algebra Systems''. Each integral along - with its optimal antiderivative (that is, the best antiderivative found - so far) is shown." + This paper examines the proposal of using the type system of Axiom to + represent a logic, and thus to use the constructions of Axiom to + handle the logic and represent proofs and propositions, in the same + way as is done in theorem provers based on type theory such as Nuprl + or Coq. + + The paper shows an interesting way to decorate Axiom with pre- and + post-conditions. + + The Curry-Howard correspondence used is + \begin{verbatim} + PROGRAMMING LOGIC + Type Formula + Program Proof + Product/record type (...,...) Conjunction + Sum/union type \/ Disjunction + Function type -> Implication + Dependent function type (x:A) -> B(x) Universal quantifier + Dependent product type (x:A,B(x)) Existential quantifier + Empty type Exit Contradictory proposition + One element type Triv True proposition + \end{verbatim}" \end{chunk} \begin{chunk}{ignore} -\bibitem[Cherry 84]{Che84} Cherry, G.W. -Integration in Finite Terms with Special Functions: The Error Function'' -J. Symbolic Computation (1985) Vol 1 pp283-302 -%\verb|axiom-developer.org/axiom-website/papers/Che84.pdf| +\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon +Integrating Computer Algebra and Reasoning through the Type System +of Aldor'' +%\verb|axiom-developer.org/axiom-website/papers/PT00.pdf| abstract = " - A decision procedure for integrating a class of transcendental - elementary functions in terms of elementary functions and error - functions is described. The procedure consists of three mutually - exclusive cases. In the first two cases a generalised procedure for - completing squares is used to limit the error functions which can - appear in the integral of a finite number. This reduces the problem - to the solution of a differential equation and we use a result of - Risch (1969) to solve it. The third case can be reduced to the - determination of what we have termed $\sum$-decompositions. The resutl - presented here is the key procuedure to a more general algorithm which - is described fully in Cherry (1983)." + A number of combinations of reasoning and computer algebra systems + have been proposed; in this paper we describe another, namely a way to + incorporate a logic in the computer algebra system Axiom. We examine + the type system of Aldor -- the Axiom Library Compiler -- and show + that with some modifications we can use the dependent types of the + system to model a logic, under the Curry-Howeard isomorphism. We give + a number of example applications of the logi we construct and explain + a prototype implementation of a modified type-checking system written + in Haskell." \end{chunk} +\section{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Cherry 86]{Che86} Cherry, G.W. -Integration in Finite Terms with Special Functions: -The Logarithmic Integral'' -SIAM J. Comput. Vol 15 pp1-21 February 1986 +\bibitem[Boehm 86]{Boe86} Boehm, Hans-J.; Cartwright, Robert; Riggle, Mark; +O'Donnell, Michael J. +Exact Real Arithmetic: A Case Study in Higher Order Programming'' +\verb|dev.acm.org/pubs/citations/proceedings/lfp/319838/p162-boehm| +%\verb|axiom-developer.org/axiom-website/papers/Boe86.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Cherry 89]{Che89} Cherry, G.W. -An Analysis of the Rational Exponential Integral'' -SIAM J. Computing Vol 18 pp 893-905 (1989) -%\verb|axiom-developer.org/axiom-website/papers/Che89.pdf| - abstract = " - In this paper an algorithm is presented for integrating expressions of - the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of - $x$, in terms of a class of special functions called the special - incomplete $\Gamma$ functions. This class of special functions - includes the exponential integral, the error functions, the sine and - cosing integrals, and the Fresnel integrals. The algorithm presented - here is an improvement over those published previously for integrating - with special functions in the following ways: (i) This algorithm - combines all the above special functions into one algorithm, whereas - previously they were treated separately, (ii) Previous algorithms - require that the underlying field of constants be algebraically - closed. This algorithm, however, works over any field of - characteristic zero in which the basic field operations can be carried - out. (iii) This algorithm does not rely on Risch's solution of the - differential equation $y^\prime + fy = g$. Instead, a more direct - method of undetermined coefficients is used." +\bibitem[Briggs 04]{Bri04} Briggs, Keith +Exact real arithmetic'' +\verb|keithbriggs.info/documents/xr-kent-talk-pp.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bri04.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Churchill 06]{Chur06} Churchill, R.C. -Liouville's Theorem on Integration Terms of Elementary Functions'' -\verb|www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Chur06.pdf| +\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W. +Computation with the Extended Rational Numbers and an Application to +Interval Arithmetic'' +\verb|www.cs.berkeley.edu/~fateman/papers/extrat.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Fat94.pdf| abstract = " - This talk should be regarded as an elementary introduction to - differential algebra. It culminates in a purely algebraic proof, due - to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of - elementary'' integrals of elementary'' functions. The precise - meaning of elementary will be specified. As an application of that - theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$ - cannot be expressed in terms of elementary functions. - \begin{itemize} - \item Preliminaries on Meromorphic Functions - \item Basic (Ordinary) Differential Algebra - \item Differential Ring Extensions with No New Constants - \item Extending Derivations - \item Integration in Finite Terms - \end{itemize}" + Programming languages such as Common Lisp, and virtually every + computer algebra system (CAS), support exact arbitrary-precision + integer arithmetic as well as exect rational number computation. + Several CAS include interval arithmetic directly, but not in the + extended form indicated here. We explain why changes to the usual + rational number system to include infinity and not-a-number'' may be + useful, especially to support robust interval computation. We describe + techniques for implementing these changes." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold -On the Integration of Algebraic Functions'' -Springer-Verlag Lecture Notes in Computer Science 102 -ISBN 0-387-10290-6 +\begin{chunk}{axiom.bib} +@incollection{Lamb06, + author = "Lambov, Branimir", + title = "Interval Arithmetic Using SSE-2", + booktitle = "Lecture Notes in Computer Science", + publisher = "Springer-Verlag", + year = "2006", + isbn = "978-3-540-85520-0", + pages = "102-113" +} \end{chunk} +\section{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Davenport 79c]{Dav79c} Davenport, J. H. -Algorithms for the Integration of Algebraic Functions'' -Lecture Notes in Computer Science V 72 pp415-425 (1979) -%\verb|axiom-developer.org/axiom-website/papers/Dav79c.pdf| +\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David +Numerical Solution of Ordinary Differential Equations'' +\verb|homepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Atk09.pdf| abstract = " - The problem of finding elementary integrals of algebraic functions has - long been recognized as difficult, and has sometimes been thought - insoluble. Risch stated a theorem characterising the integrands with - elementary integrals, and we can use the language of algebraic - geometry and the techniques of Davenport to yield an algorithm that will - always produce the integral if it exists. We explain the difficulty in - the way of extending this algorithm, and outline some ways of solving - it. Using work of Manin we are able to solve the problem in all cases - where the algebraic expressions depend on a parameter as well as on - the variable of integration." + This book is an expanded version of supplementary notes that we used + for a course on ordinary differential equations for upper-division + undergraduate students and beginning graduate students in mathematics, + engineering, and sciences. The book introduces the numerical analysis + of differential equations, describing the mathematical background for + understanding numerical methods and giving information on what to + expect when using them. As a reason for studying numerical methods as + a part of a more general course on differential equations, many of the + basic ideas of the numerical analysis of differential equations are + tied closely to theoretical behavior associated with the problem being + solved. For example, the criteria for the stability of a numerical + method is closely connected to the stability of the differential + equation problem being solved." \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport 82a]{Dav82a} Davenport, J.H. -The Parallel Risch Algorithm (I) -%\verb|axiom-developer.org/axiom-website/papers/Dav82a.pdf| - abstract = " - In this paper we review the so-called parallel Risch'' algorithm for - the integration of transcendental functions, and explain what the - problems with it are. We prove a positive result in the case of - logarithmic integrands." +\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P. +A practical method for numerical evaluations of solutions of partial + differential equations of heat-conduction type'' +Advances in Computational Mathematics Vol 6 pp207-226 (1996) +\verb|www.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/| +\verb|CNMethod-original.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Cran96.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport 82]{Dav82} Davenport, J.H. -On the Parallel Risch Algorithm (III): Use of Tangents'' -SIGSAM V16 no. 3 pp3-6 August 1982 +\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien; +Zimmermann, Paul +Worst Cases for the Exponential Function +in the IEEE-754r decimal64 Format'' +in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0 +(2006) pp114-125 + abstract = " + We searched for the worst cases for correct rounding of the + exponential function in the IEEE 754r decimal64 format, and computed + all the bad cases whose distance from a breakpoint (for all rounding + modes) is less than $10^{-15}$ ulp, and we give the worst ones. In + particular, the worst case for + $\vert{}x\vert{} \ge 3 x 10^{-11}$ is + $+ exp(9.407822313572878x10^{-2} = + 1.09864568206633850000000000000000278\ldots +$ + This work can be extended to other elementary functions in the decimal64 + format and allows the design of reasonably fast routines that will + evaluate these functions with correct rounding, at least in some + situations." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 03]{Dav03} Davenport, James H. -The Difficulties of Definite Integration'' -\verb|www.researchgate.net/publication/| -\verb|247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Dav03.pdf| - abstract = " - Indefinite integration is the inverse operation to differentiation, - and, before we can understand what we mean by indefinite integration, - we need to understand what we mean by differentiation." +\begin{chunk}{axiom.bib} +@book{Hamm62, + author = "Hamming R W.", + title = "Numerical Methods for Scientists and Engineers", + publisher = "Dover", + year = "1973", + isbn = "0-486-65241-6" +} \end{chunk} +\section{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Fateman 02]{Fat02} Fateman, Richard -Symbolic Integration'' -\verb|inst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Fat02.pdf| +\bibitem [Bostock 14]{Bos14} Bostock, Mike +Visualizing Algorithms'' +\verb|bost.ocks.org/mike/algorithms| + abstract = " + This website hosts various ways of visualizing algorithms. The hope is + that these kind of techniques can be applied to Axiom." \end{chunk} -\begin{chunk}{axiom.bib} -@inproceedings{Gedd89, - author = "Geddes, K. O. and Stefanus, L. Y.", - title = "On the Risch-norman Integration Method and Its Implementation - in MAPLE", - booktitle = "Proc. of the ACM-SIGSAM 1989 Int. Symp. on Symbolic and - Algebraic Computation", - series = "ISSAC '89", - year = "1989", - isbn = "0-89791-325-6", - location = "Portland, Oregon, USA", - pages = "212--217", - numpages = "6", - url = "http://doi.acm.org/10.1145/74540.74567", - doi = "10.1145/74540.74567", - acmid = "74567", - publisher = "ACM", - address = "New York, NY, USA", - paper = "Gedd89.pdf", +\begin{chunk}{ignore} +\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A. +Representation of mathematical object in interactive books'' +%\verb|axiom-developer.org/axiom-website/papers/Leexx.pdf| abstract = " - Unlike the Recursive Risch Algorithm for the integration of - transcendental elementary functions, the Risch-Norman Method processes - the tower of field extensions directly in one step. In addition to - logarithmic and exponential field extensions, this method can handle - extentions in terms of tangents. Consequently, it allows trigonometric - functions to be treated without converting them to complex exponential - form. We review this method and describe its implementation in - MAPLE. A heuristic enhancement to this method is also presented." -} + We present a model for the representation of mathematical objects in + structured electronic documents, in a way that allows for interaction + with applications such as computer algebra systems and proof checkers. + Using a representation that reflects only the intrinsic information of + an object, and storing application-dependent information in so-called + {\sl application descriptions}, it is shown how the translation from + the internal to an external representation and {\sl vice versa} can be + achieved. Hereby a formalisation of the concept of {\sl context} is + introduced. The proposed scheme allows for a high degree of + application integration, e.g., parallel evaluation of subexpressions + (by different computer algebra systems), or a proof checker using a + computer algebra system to verify an equation involving a symbolic + computation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G. -The Risch Integration Algorithm'' -Algorithms for Computer Algebra, Ch 12 pp511-573 (1992) -%\verb|axiom-developer.org/axiom-website/papers/GCL92a.pdf| +\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell +The Design of a User Interface for Computer Algebra Systems'' +\verb|www.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD-91-626.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Soif91.pdf| + abstract = " + This thesis discusses the design and implementation of natural user + interfaces for Computer Algebra Systems. Such an interface must not + only display expressions generated by the Computer Algebra System in + standard mathematical notation, but must also allow easy manipulation + and entry of expressions in that notation. The user interface should + also assist in understanding of large expressions that are generated + by Computer Algebra Systems and should be able to accommodate new + notational forms." \end{chunk} \begin{chunk}{ignore} -\bibitem[Hardy 1916]{Hard16} Hardy, G.H. -The Integration of Functions of a Single Variable'' -Cambridge Unversity Press, Cambridge, 1916 -% REF:00002 +\bibitem[Victor 11]{Vict11} Victor, Bret +Up and Down the Ladder of Abstraction'' +\verb|worrydream.com/LadderOfAbstraction| + abstract = " + This interactive essay presents the ladder of abstraction, a technique for + thinking explicitly about these levels, so a designer can move among + them consciously and confidently. " \end{chunk} \begin{chunk}{ignore} -\bibitem[Harrington 78]{Harr87} Harrington, S.J. -A new symbolic integration system in reduce'' -\verb|comjnl.oxfordjournals.or/content/22/2/127.full.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Harr87.pdf| +\bibitem[Victor 12]{Vict12} Victor, Bret +Inventing on Principle'' +\verb|www.youtube.com/watch?v=PUv66718DII| abstract = " - A new integration system, employing both algorithmic and pattern match - integration schemes is presented. The organization of the system - differs from that of earlier programs in its emphasis on the - algorithmic approach to integration, its modularity and its ease of - revision. The new Norman-Rish algorithm and its implementation at the - University of Cambridge are employed, supplemented by a powerful - collection of simplification and transformation rules. The facility - for user defined integrals and functions is also included. The program - is both fast and powerful, and can be easily modified to incorporate - anticipated developments in symbolic integration." + This video raises the level of discussion about human-computer + interaction from a technical question to a question of effectively + capturing ideas. In particular, this applies well to Axiom's focus on + literate programming." \end{chunk} +\section{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@misc{Herm1872, - author = "Hermite, E.", - title = "Sur l'int\'{e}gration des fractions rationelles", - journal = "Nouvelles Annales de Math\'{e}matiques", - volume = "11", - pages = "145-148", - year = "1872" +@InProceedings{Kalt84, + author = "Kaltofen, E.", + title = "A Note on the {Risch} Differential Equation", + booktitle = "Proc. EUROSAM '84", + pages = "359--366", + crossref = "EUROSAM84", + year = "1984", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz", + paper = "Kalt84.ps" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis -Algorithms for Partial Fraction Decomposition and Rational Function - Integration'' -SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971) -pp441-457 -%\verb|axiom-developer.org/axiom-website/papers/Horo71.pdf| REF:00018 - abstract = " - Algorithms for symbolic partial fraction decomposition and indefinite - integration of rational functions are described. Two types of - partial fraction decomposition are investigated, square-free and - complete square-free. A method is derived, based on the solution of - a linear system, which produces the square-free decomposition of any - rational function, say A/B. The computing time is show to be - $O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$ - is a number which is closely related to the size of the coefficients - which occur in A and B. The complete square-free partical fraction - decomposition can then be directly obtained and it is shown that the - computing time for this process is also bounded by $O(n^4(ln nf)^2)$." +\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel; +Petkovsek, Marko +On Polynomial Solutions of Linear Operator Equations'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Abra95.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D. -Recursive integration of piecewise-continuous functions'' -\verb|www.cybertester.com/data/recint.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Jeff97.pdf| +\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel +On Solutions of Linear Functional Systems'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Abra01.pdf| abstract = " - An algorithm is given for the integration of a class of - piecewise-continuous functions. The integration is with respect to a - real variable, because the functions considered do not in general - allow integration in the complex plane to be defined. The class of - integrands includes commonly occurring waveforms, such as square - waves, triangular waves, and the floor function; it also includes the - signum function. The algorithm can be implemented recursively, and it - has the property of ensuring that integrals are continuous on domains - of maximum extent." + We describe a new direct algorithm for transforming a linear system of + recurrences into an equivalent one with nonsingular leading or + trailing matrix. Our algorithm, which is an improvement to the EG + elimination method, uses only elementary linear algebra operations + (ranks, kernels, and determinants) to produce an equation satisfied by + the degress of the solutions with finite support. As a consequence, we + can boudn and compute the polynomial and rational solutions of very + general linear functional systems such as systems of differential or + ($q$)-difference equations." \end{chunk} \begin{chunk}{ignore} -\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.; -Rich, A.D. -Integration of the signum, piecewise and related functions'' -\verb|cs.uwaterloo.ca/~glabahn/Papers/issac99-2.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Jeff99.pdf| +\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel +On the Factorization of Linear Ordinary Differential Operators'' +Mathematics and Computers in Simulation 42 pp 387-389 (1996) +%\verb|axiom-developer.org/axiom-website/papers/Bro96b.pdf| abstract = " - When a computer algebra system has an assumption facility, it is - possible to distinguish between integration problems with respect to a - real variable, and those with respect to a complex variable. Here, a - class of integration problems is defined in which the integrand - consists of compositions of continuous functions and signum functions, - and integration is with respect to a real variable. Algorithms are - given for evaluating such integrals." + After reviewing the arithmetic of linear ordinary differential + operators, we describe the current status of the factorisation + algorithm, specially with respect to factoring over non-algebraically + closed constant fields. We also describe recent results from Singer + and Ulmer that reduce determining the differential Galois group of an + operator to factoring." \end{chunk} \begin{chunk}{ignore} -\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref -A new symbolic computation for formal integration with exact power series'' -%\verb|axiom-developer.org/axiom-website/Kiym04.pdf| +\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko +An introduction to pseudo-linear algebra'' +Theoretical Computer Science V157 pp3-33 (1966) +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro96a.pdf| abstract = " - This paper describes a new symbolic algorithm for formal integration - of a class of functions in the context of exact power series by using - generalized hypergeometric series and computer algebraic technique." + Pseudo-linear algebra is the study of common properties of linear + differential and difference operators. We introduce in this paper its + basic objects (pseudo-derivations, skew polynomials, and pseudo-linear + operators) and describe several recent algorithms on them, which, when + applied in the differential and difference cases, yield algorithms for + uncoupling and solving systems of linear differential and difference + equations in closed form." \end{chunk} \begin{chunk}{ignore} -\bibitem[Knowles 93]{Know93} Knowles, P. -Integration of a class of transcendental liouvillian -functions with error-functions i'' -Journal of Symbolic Computation Vol 13 pp525-543 (1993) +\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel +Computer Algebra Algorithms for Linear Ordinary Differential and +Difference equations'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Broxb.pdf| + abstract = " + Galois theory has now produced algorithms for solving linear ordinary + differential and difference equations in closed form. In addition, + recent algorithmic advances have made those algorithms effective and + implementable in computer algebra systems. After introducing the + relevant parts of the theory, we describe the latest algorithms for + solving such equations." \end{chunk} \begin{chunk}{ignore} -\bibitem[Knowles 95]{Know95} Knowles, P. -Integration of a class of transcendental liouvillian -functions with error-functions ii'' -Journal of Symbolic Computation Vol 16 pp227-241 (1995) +\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel +An improved algorithm for factoring linear ordinary differential +operators'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| + abstract = " + We describe an efficient algorithm for computing the associated + equations appearing in the Beke-Schlesinger factorisation method for + linear ordinary differential operators. This algorithm, which is based + on elementary operations with sets of integers, can be easily + implemented for operators of any order, produces several possible + associated equations, of which only the simplest can be selected for + solving, and often avoids the degenerate case, where the order of the + associated equation is less than in the generic case. We conclude with + some fast heuristics that can produce some factorizations while using + only linear computations." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Krag09, - author = "Kragler, R.", - title = "On Mathematica Program for Poor Man's Integrator Algorithm", - journal = "Programming and Computer Software", - volume = "35", - number = "2", - pages = "63-78", - year = "2009", - issn = "0361-7688", - paper = "Krag09.pdf", +\begin{chunk}{ignore} +\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel +On Solutions of Linear Ordinary Differential Equations in their +Coefficient Field'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro90.pdf| abstract = " - In this paper by means of computer experiment we study advantages and - disadvantages of the heuristical method of parallel integrator''. For - this purpose we describe and use implementation of the method in - Mathematica. In some cases we compare this implementation with the original - one in Maple." -} + We describe a rational algorithm for finding the denominator of any + solution of a linear ordinary differential equation in its coefficient + field. As a consequence, there is now a rational algorithm for finding + all such solutions when the coefficients can be built up from the + rational functions by finitely many algebraic and primitive + adjunctions. This also eliminates one of the computational bottlenecks + in algorithms that either factor or search for Liouvillian solutions + of such equations with Liouvillian coefficients." \end{chunk} \begin{chunk}{ignore} -\bibitem[Lang 93]{Lang93} Lang, S. -Algebra'' -Addison-Wesly, New York, 3rd edition 1993 +\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel +$\sum^{IT}$ -- A strongly-typed embeddable computer algebra library'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro96.pdf| + abstract = " + We describe the new computer algebra library $\sum^{IT}$ and its + underlying design. The development of $\sum^{IT}$ is motivated by the + need to provide highly efficient implementations of key algorithms for + linear ordinary differential and ($q$)-difference equations to + scientific programmers and to computer algebra users, regardless of + the programming language or interactive system they use. As such, + $\sum^{IT}$ is not a computer algebra system per se, but a library (or + substrate) which is designed to be plugged'' with minimal efforts + into different types of client applications." \end{chunk} \begin{chunk}{ignore} -\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian -A Generalization of Liouville's Theorem on Integration in Finite Terms'' -\verb|www.mathnet.or.kr/mathnet/kms_tex/113666.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Leer02.pdf| +\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel +Solving linear ordinary differential equations over +$C(x,e^{\int{f(x)dx}})$ +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro99a.pdf| abstract = " - A generalization of Liouville's theorem on integration in finite - terms, by enlarging the class of fields to an extension called - Ei-Gamma extension is established. This extension includes the - $\mathcal{E}\mathcal{L}$-elementary extensions of Singer, Saunders and - Caviness and contains the Gamma function." + We describe a new algorithm for computing the solutions in + $F=C(x,e^{\int{f(x)dx}})$ of linear ordinary differential equations + with coefficients in $F$. Compared to the general algorithm, our + algorithm avoids the computation of exponential solutions of equations + with coefficients in $C(x)$, as well as the solving of linear + differential systems over $C(x)$. Our method is effective and has been + implemented." \end{chunk} \begin{chunk}{ignore} -\bibitem[Leslie 09]{Lesl09} Leslie, Martin -Why you can't integrate exp($x^2$)'' -\verb|math.arizona.edu/~mleslie/files/integrationtalk.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Lesl09.pdf| +\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel +On Solutions of Linear Ordinary Differential Equations in their + Coefficient Field'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro00.pdf| + abstract = " + We extend the notion of monomial extensions of differential fields, + i.e. simple transcendental extensions in which the polynomials are + closed under differentiation, to difference fields. The structure of + such extensions provides an algebraic framework for solving + generalized linear difference equations with coefficients in such + fields. We then describe algorithms for finding the denominator of any + solution of those equations in an important subclass of monomial + extensions that includes transcendental indefinite sums and + products. This reduces the general problem of finding the solutions of + such equations in their coefficient fields to bounding their + degrees. In the base case, this yields in particular a new algorithm + for computing the rational solutions of $q$-difference equations with + polynomial coefficients." \end{chunk} \begin{chunk}{ignore} -\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel -Symbolic definite (and indefinite) integration: methods and open issues'' -ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011) -\verb|www.sigsam.org/bulletin/articles/175/issue175.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Lich11.pdf| +\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien +Solutions of linear ordinary differential equations in terms of +special functions'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bro02.pdf| abstract = " - The computation of definite integrals presents one with a variety of - choices. There are various methods such as Newton-Leibniz or Slater's - convolution method. There are questions such as whether to split or - merge sums, how to search for singularities on the path of - integration, when to issue conditional results, how to assess - (possibly conditional) convergence, and more. These various - considerations moreover interact with one another in a multitude of - ways. Herein we discuss these various issues and illustrate with examples." + We describe a new algorithm for computing special function solutions + of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary + differential equations, where $m(x)$ is an arbitrary Liouvillian + function, $\eta(x)$ is an arbitrary rational function, and $F$ + satisfies a given second order linear ordinary differential + equations. Our algorithm, which is base on finding an appropriate + point transformation between the equation defining $F$ and the one to + solve, is able to find all rational transformations for a large class + of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$ + special functions of mathematical physics, such as Airy, Bessel, + Kummer and Whittaker functions. It is also able to identify the values + of the parameters entering those special functions, and can be + generalized to equations of higher order." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Liou1833a, - author = "Liouville, Joseph", - title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales - dont la valeur est alg\'{e}brique", - journal = "Journal de l'Ecole Polytechnique", - volume = "14", - pages = "124-128", - year = "1833" -} +\begin{chunk}{ignore} +\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M. +A Reduction for Regular Differential Systems'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bro03.pdf| + abstract = " + We propose a definition of regularity of a linear differential system + with coefficients in a monomial extension of a differential field, as + well as a global and truly rational (i.e. factorisation-free) + iteration that transforms a system with regular finite singularites + into an equivalent one with simple finite poles. We then apply our + iteration to systems satisfied by bases of algebraic function fields, + obtaining algorithms for computing the number of irreducible + components and the genus of algebraic curves." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Liou1833b, - author = "Liouville, Joseph", - title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales - dont la valeur est alg\'{e}brique", - journal = "Journal de l'Ecole Polytechnique", - volume = "14", - pages = "149-193", - year = "1833" -} +\begin{chunk}{ignore} +\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick +Linear recurrences with polynomial coefficients'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html| +%\verb|axiom-developer.org/axiom-website/papers/Bro03a.pdf| + abstract = " + We relate sequences generated by recurrences with polynomial + coefficients to interleaving and multiplexing of sequences generated + by recurrences with constant coefficients. In the special case of + finite fields, we show that such sequences are periodic and provide + linear complexity estimates for all three constructions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph -Note sur la determination des int\'egrales dont la -valeur est alg\'ebrique'' -Journal f\"ur die Reine und Angewandte Mathematik, -Vol 10 pp 247-259, (1833) +\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min +Picard-Vessiot Extensions for Linear Functional Systems'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bro05.pdf| + abstract = " + Picard-Vessiot extensions for ordinary differential and difference + equations are well known and are at the core of the associated Galois + theories. In this paper, we construct fundamental matrices and + Picard-Vessiot extensions for systems of linear partial functional + equations having finite linear dimension. We then use those extensions + to show that all the solutions of a factor of such a system can be + completed to solutions of the original system." \end{chunk} \begin{chunk}{ignore} -\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph -Sur la determination des int\'egrales dont la valeur est -alg\'ebrique'' -{\sl Journal de l'Ecole Polytechnique}, 14:124-193, 1833 +\bibitem[Davenport 86]{Dav86} Davenport, J.H. +The Risch Differential Equation Problem'' +SIAM J. COMPUT. Vol 15, No. 4 1986 +%\verb|axiom-developer.org/axiom-website/papers/Dav86.pdf| + abstract = " + We propose a new algorithm, similar to Hermite's method for the + integration of rational functions, for the resolution of Risch + differential equations in closed form, or proving that they have no + resolution. By requiring more of the presentation of our differential + fields (in particular that the exponentials be weakly normalized), we + can avoid the introduction of arbitrary constants which have to be + solved for later. + + We also define a class of fields known as exponentially reduced, and + show that solutions of Risch differential equations which arise from + integrating in these fields satisfy the natural'' degree constraints + in their main variables, and we conjecture (after Risch and Norman) + that this is true in all variables." \end{chunk} \begin{chunk}{ignore} -\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph -M\'emoire sur l'int\'gration d'une classe de fonctions -transcendentes'' -Journal f\"ur die Reine und Angewandte Mathematik, -Vol 13(2) pp 93-118, (1835) +\bibitem[Singer 9]{Sing91.pdf} singer, Michael F. +Liouvillian Solutions of Linear Differential Equations with Liouvillian + Coefficients'' +J. Symbolic Computation V11 No 3 pp251-273 (1991) +\verb|www.sciencedirect.com/science/article/pii/S074771710880048X| +%\verb|axiom-developer.org/axiom-website/papers/Sing91.pdf| + abstract = " + Let $L(y)=b$ be a linear differential equation with coefficients in a + differential field $K$. We discuss the problem of deciding if such an + equation has a non-zero solution in $K$ and give a decision procedure + in case $K$ is an elementary extension of the field of rational + functions or is an algebraic extension of a transcendental liouvillian + extension of the field of rational functions We show how one can use + this result to give a procedure to find a basis for the space of + solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field + and $L(y)$ has coefficients in $K$." \end{chunk} \begin{chunk}{ignore} -\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, Gholem-All -An Invitation to Integration in Finite Terms'' -College Mathematics Journal Vol 25 No 4 (1994) pp295-308 -\verb|www.rangevoting.org/MarchisottoZint.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Marc94.pdf| +\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin +Symbolic Solutions of Discontinuous Differential Equations'' +\verb|e-collection.library.ethz.ch/eserv/eth:39463/eth-39463-01.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mohr94.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Marik 91]{Mari91} Marik, Jan -A note on integration of rational functions'' -\verb|dml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_116-1991-4_9.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mari91.pdf| +\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin +A Normal Form for Function Rings of Piecewise Functions'' +J. Symbolic Computation (1998) Vol 26 pp607-619 +\verb|www.cas.mcmaster.ca/~mohrens/JSC.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mohr98.pdf| abstract = " - Let $P$ and $Q$ be polynomials in one variable with complex coefficients - and let $n$ be a natural number. Suppose that $Q$ is not constant and - has only simple roots. Then there is a rational function $\varphi$ - with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the - functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is - divisible by $Q$." + Computer algebra systems often have to deal with piecewise continuous + functions. These are, for example, the absolute value function, + signum, piecewise defined functions but also functions that are the + supremum or infimum of two functions. We present a new algebraic + approach to these types of problems. This paper presents a normal form + for a function ring containing piecewise polynomial functions of an + expression. The main result is that this normal form can be used to + decide extensional equality of two piecewise functions. Also we define + supremum and infimum for piecewise functions; in fact, we show that + the function ring forms a lattice. Additionally, a method to solve + equalities and inequalities in this function ring is + presented. Finally, we give a user interface'' to the algebraic + representation of the piecewise functions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Moses 76]{Mos76} Moses, Joel -An introduction to the Risch Integration Algorithm'' -ACM Proc. 1976 annual conference pp425-428 -%\verb|axiom-developer.org/axiom-website/papers/Mos76.pdf| REF:00048 +\bibitem[Weber 06]{Webe06} Weber, Andreas +Quantifier Elimination on Real Closed Fields and Differential Equations'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber2006a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe06.pdf| + keywords = "survey", abstract = " - Risch's decision procedure for determining the integrability in closed - form of the elementary functions of the calculus is presented via - examples. The exponential and logarithmic cases of the algorithsm had - been implemented for the MACSYMA system several years ago. The - implementation of the algebraic case of the algorithm is the subject - of current research." + This paper surveys some recent applications of quantifier elimination + on real closed fields in the context of differential + equations. Although polynomial vector fields give rise to solutions + involving the exponential and other transcendental functions in + general, many questions can be settled within the real closed field + without referring to the real exponential field. The technique of + quantifier elimination on real closed fields is not only of + theoretical interest, but due to recent advances on the algorithmic + side including algorithms for the simplification of quantifier-free + formulae the method has gained practical applications, e.g. in the + context of computing threshold conditions in epidemic modeling." \end{chunk} \begin{chunk}{ignore} -\bibitem[Moses 71a]{Mos71a} Moses, Joel -Symbolic Integration: The Stormy Decade'' -CACM Aug 1971 Vol 14 No 8 pp548-560 -\verb|www-inst.eecs.berkeley.edu/~cs282/sp02/readings/moses-int.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mos71a.pdf| REF:00017 +\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix +Liouvillian solutions of third order differential equations'' +J. Symbolic COmputations 36 pp 855-889 (2003) +\verb|www.sciencedirect.com/science/article/pii/S0747717103000658| +%\verb|axiom-developer.org/axiom-website/papers/Ulm03.pdf| abstract = " - Three approaches to symbolic integration in the 1960's are - described. The first, from artificial intelligence, led to Slagle's - SAINT and to a large degree to Moses' SIN. The second, from algebraic - manipulation, led to Monove's implementation and to Horowitz' and - Tobey's reexamination of the Hermite algorithm for integrating - rational functions. The third, from mathematics, led to Richardson's - proof of the unsolvability of the problem for a class of functions and - for Risch's decision procedure for the elementary functions. - Generalizations of Risch's algorithm to a class of special - functions and programs for solving differential equations and for - finding the definite integral are also described." + The Kovacic algorithm and its improvements give explicit formulae for + the Liouvillian solutions of second order linear differential + equations. Algorithms for third order differential equations also + exist, but the tools they use are more sophisticated and the + computations more involved. In this paper we refine parts of the + algorithm to find Liouvillian solutions of third order equations. We + show that,except for four finite groups and a reduction to the second + order case, it is possible to give a formula in the imprimitve + case. We also give necessary conditions and several simplifications + for the computation of the minimal polynomial for the remaining finite + set of finite groups (or any known finite group) by extracting + ramification information from the character table. Several examples + have been constructed, illustrating the possibilities and limitations." \end{chunk} +\section{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H. -Symbolic Integration -- The Dust Settles?'' -%\verb|axiom-developer.org/axiom-website/papers/Nor79.pdf| +\bibitem[Carette 04]{Car04} Carette, Jacques +Understanding Expression Simplification'' +\verb|www.cas.mcmaster.ca/~carette/publications/simplification.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Car04.pdf| abstract = " - By the end of the 1960s it had been shown that a computer could find - indefinite integrals with a competence exceeding that of typical - undergraduates. This practical advance was backed up by algorithmic - interpretations of a number of clasical results on integration, and by - some significant mathematical extensions to these same results. At - that time it would have been possible to claim that all the major - barriers in the way of a complete system for automated analysis had - been breached. In this paper we survey the work that has grown out of - the above-mentioned early results, showing where the development has - been smooth and where it has spurred work in seemingly unrelated fields." + We give the first formal definition of the concept of {\sl + simplification} for general expressions in the context of Computer + Algebra Systems. The main mathematical tool is an adaptation of the + theory of Minimum Description Length, which is closely related to + various theories of complexity, such as Kolmogorov Complexity and + Algorithmic Information Theory. In particular, we show how this theory + can justify the use of various magic constants'' for deciding + between some equivalent representations of an expression, as found in + implementations of simplification routines." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Ostrowski 46]{Ost46} Ostrowski, A. -Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes -d'expressions'' -Comm. Math. Helv., Vol 18 pp 283-308, (1946) -% REF:00008 +\section{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@TechReport{Kalt84b, + author = "Kaltofen, E.", + title = "The Algebraic Theory of Integration", + institution = "RPI", + address = "Dept. Comput. Sci., Troy, New York", + year = "1984", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf", + paper = "Kalt84b.pdf" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Raab 12]{Raab12} Raab, Clemens G. -Definite Integration in Differential Fields'' -\verb|www.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Raab12.pdf| - abstract = " - The general goal of this thesis is to investigate and develop computer - algebra tools for the simplification resp. evaluation of definite - integrals. One way of finding the value of a def- inite integral is - via the evaluation of an antiderivative of the integrand. In the - nineteenth century Joseph Liouville was among the first who analyzed - the structure of elementary antiderivatives of elementary functions - systematically. In the early twentieth century the algebraic structure - of differential fields was introduced for modeling the differential - properties of functions. Using this framework Robert H. Risch - published a complete algorithm for transcendental elementary - integrands in 1969. Since then this result has been extended to - certain other classes of integrands as well by Michael F. Singer, - Manuel Bronstein, and several others. On the other hand, if no - antiderivative of suitable form is available, then linear relations - that are satisfied by the parameter integral of interest may be found - based on the principle of parametric integration (often called - differentiating under the integral sign or creative telescoping). - - The main result of this thesis extends the results mentioned above to - a complete algo- rithm for parametric elementary integration for a - certain class of integrands covering a majority of the special - functions appearing in practice such as orthogonal polynomials, - polylogarithms, Bessel functions, etc. A general framework is provided - to model those functions in terms of suitable differential fields. If - the integrand is Liouvillian, then the present algorithm considerably - improves the efficiency of the corresponding algorithm given by Singer - et al. in 1985. Additionally, a generalization of Czichowski’s - algorithm for computing the logarithmic part of the integral is - presented. Moreover, also partial generalizations to include other - types of integrands are treated. +\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor +Definite Integration'' +\verb|www.cs.cmu.edu/~adamchik/articles/integr/mj.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Adamxx.pdf| - As subproblems of the integration algorithm one also has to find - solutions of linear or- dinary differential equations of a certain - type. Some contributions are also made to solve those problems in our - setting, where the results directly dealing with systems of - differential equations have been joint work with Moulay A. Barkatou. +\end{chunk} - For the case of Liouvillian integrands we implemented the algorithm in - form of our Mathematica package Integrator. Parts of the - implementation also deal with more general functions. Our procedures - can be applied to a significant amount of the entries in integral - tables, both indefinite and definite integrals. In addition, our - procedures have been successfully applied to interesting examples of - integrals that do not appear in these tables or for which current - standard computer algebra systems like Mathematica or Maple do not - succeed. We also give examples of how parameter integrals coming from - the work of other researchers can be solved with the software, e.g., - an integral arising in analyzing the entropy of certain processes." +\begin{chunk}{ignore} +\bibitem[Adamchik 97]{Adam97} Adamchik, Victor +A Class of Logarithmic Integrals'' +\verb|www.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Adam97.pdf| + abstract = " + A class of definite integrals involving cyclotomic polynomials and + nested logarithms is considered. The results are given in terms of + derivatives of the Hurwitz Zeta function. Some special cases for which + such derivatives can be expressed in closed form are also considered." \end{chunk} \begin{chunk}{ignore} -\bibitem[Raab 13]{Raab13} Raab, Clemens G. -Generalization of Risch's Algorithm to Special Functions'' -\verb|arxiv.org/pdf/1305.1481| -%\verb|axiom-developer.org/axiom-website/papers/Raab13.pdf| +\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios +Definite Integration using the Generalized Hypergeometric Functions'' +\verb|dspace.mit.edu/handle/1721.1/16269| +%\verb|axiom-developer.org/axiom-websitep/papers/Avgo77.pdf| abstract = " - Symbolic integration deals with the evaluation of integrals in closed - form. We present an overview of Risch's algorithm including recent - developments. The algorithms discussed are suited for both indefinite - and definite integration. They can also be used to compute linear - relations among integrals and to find identities for special functions - given by parameter integrals. The aim of this presentation is twofold: - to introduce the reader to some basic idea of differential algebra in - the context of integration and to raise awareness in the physics - community of computer algebra algorithms for indefinite and definite - integration." + A design for the definite integration of approximately fifty Special + Functions is described. The Generalized Hypergeometric Functions are + utilized as a basis for the representation of the members of the above + set of Special Functions. Only a relatively small number of formulas + that generally involve Generalized Hypergeometric Functions are + utilized for the integration stage. A last and crucial stage is + required for the integration process: the reduction of the Generalized + Hypergeometric Function to Elementary and/or Special Functions. + + The result of an early implementation which involves Laplace + transforms are given and some actual examples with their corresponding + timing are provided." \end{chunk} \begin{chunk}{ignore} -\bibitem[Raab xx]{Raabxx} Raab, Clemens G. -Integration in finite terms for Liouvillian functions'' -\verb|www.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Raabxx.pdf| +\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil +A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite + Terms'' +\verb|www.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bad89.pdf| abstract = " - Computing integrals is a common task in many areas of science, - antiderivatives are one way to accomplish this. The problem of - integration in finite terms can be states as follows. Given a - differential field $(F,D)$ and $f \in F$, compute $g$ in some - elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$ - exists. - - This problem has been solved for various classes of fields $F$. For - rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and - algorithms to compute it are known already for a long time. In 1969 - Risch published an algorithm that solves this problem when $(F,D)$ is - a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later - this has been extended towards integrands being Liouvillian functions - by Singer et. al. via the use of regular log-explicit extensions of - $(C(x),\frac{d}{dx})$. Our algorithm extends this to handling - transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly - without the need to embed them into log-explicit extensions. For - example, this means that - $\int{(z-x)x^{z-1}e^{-x}dx} = x^ze^{-x}$ - can be computed without including log(x) in the differential field." + The result obtained generalizes Liouville's Theorem by allowing, in + addition to the elementary functions, dilogarithms to appear in the + integral of an elementary function. The basic conclusion is that an + associated function to the dilogarihm, if dilogarithms appear in the + integral, appears linearly, with logarithms appearing in a non-linear + way." \end{chunk} \begin{chunk}{ignore} -\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J. -A Knowledge Repository for Indefinite Integration Based on Transformation Rules'' -\verb|www.apmaths.uwo.ca/~arich/A%2520Rule-based%2520Knowedge%2520Repository.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Rich09.pdf| +\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil +Integration in Finite Terms with Elementary Functions and Dilogarithms'' +\verb|dspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Bad94.pdf| abstract = " - Taking the specific problem domain of indefinite integration, we - describe the on-going development of a repository of mathematical - knowledge based on transformation rules. It is important that the - repository be not confused with a look-up table. The database of - transformation rules is at present encoded in Mathematica, but this is - only one convenient form of the repository, and it could be readily - translated into other formats. The principles upon which the set of - rules is compiled is described. One important principle is - minimality. The benefits of the approach are illustrated with - examples, and with the results of comparisons with other approaches." + In this thesis, we report on a new theorem that generalizes + Liouville's theorem on integration in finite terms. The new theorem + allows dilogarithms to occur in the integral in addition to elementary + functions. The proof is base on two identities for the dilogarithm, + that characterize all the possible algebraic relations among + dilogarithms of functions that are built up from the rational + functions by taking transcendental exponentials, dilogarithms, and + logarithms." \end{chunk} -\begin{chunk}{axiom.bib} -@techreport{Risc68, - author = "Risch, Robert", - title = "On the integration of elementary functions which are built up - using algebraic operations", - type = "Research Report", - number = "SP-2801/002/00", - institution = "System Development Corporation, Santa Monica, CA, USA", - year = "1968" -} +\begin{chunk}{ignore} +\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil +A Note on Symbolic Integration with Polylogarithms'' +J. Math Vol 8 pp229-241 (2011) +%\verb|axiom-developer.org/axiom-website/papers/Bad10.pdf| + abstract = " + We generalize partially Liouville's theorem on integration in finite + terms to allow polylogarithms of any order to occur in the integral in + addition to elementary functions. The result is a partial + generalization of a theorem proved by the author for the + dilogarithm. It is also a partial proof of a conjecture postulated by + the author in 1994. The basic conclusion is that an associated + function to the nth polylogarithm appears linearly with logarithms + appearing possibly in a polynomial way with non-constant coefficients." \end{chunk} -\begin{chunk}{axiom.bib} -@techreport{Risc69a, - author = "Risch, Robert", - title = "Further results on elementary functions", - type = "Research Report", - number = "RC-2042", - institution = "IBM Research, Yorktown Heights, NY, USA", - year = "1969" - -} +\begin{chunk}{ignore} +\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D. +A contour integral involving legendre polynomial and Meijer's G-function'' +\verb|link.springer.com/article/10.1007/BF03049565| +%\verb|axiom-developer.org/axiom-website/papers/Bajp70.pdf| + abstract = " + In this paper a countour integral involving Legendre polynomial and + Meijer's G-function is evaluated. the integral is of general character + and it is a generalization of results recently given by Meijer, + MacRobert and others. An integral involving regular radial Coulomb + wave function is also obtained as a particular case." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Risc69b, - author = "Risch, Robert", - title = "The problem of integration in finite terms", - journal = "Transactions of the American Mathematical Society", - volume = "139", - year = "1969", - pages = "167-189", - paper = "Ris69b.pdf", - abstract = "This paper deals with the problem of telling whether a - given elementary function, in the sense of analysis, has an elementary - indefinite integral." -} +\begin{chunk}{ignore} +\bibitem[Bronstein 89]{Bro89a} Bronstein, M. +An Algorithm for the Integration of Elementary Functions'' +Lecture Notes in Computer Science Vol 378 pp491-497 (1989) +%\verb|axiom-developer.org/axiom-website/papers/Bro89a.pdf| + abstract = " + Trager (1984) recently gave a new algorithm for the indefinite + integration of algebraic functions. His approach was rational'' in + the sense that the only algebraic extension computed in the smallest + one necessary to express the answer. We outline a generalization of + this approach that allows us to integrate mixed elementary + functions. Using only rational techniques, we are able to normalize + the integrand, and to check a necessary condition for elementary + integrability." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Risc70, - author = "Risch, Robert", - title = "The Solution of the Problem of Integration in Finite Terms", - journal = "Bull. AMS", - year = "1970", - issn = "0002-9904", - volume = "76", - number = "3", - pages = "605-609", - paper = "Risc70.pdf", +\begin{chunk}{ignore} +\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel +Integration of Elementary Functions'' +J. Symbolic Computation (1990) 9, pp117-173 September 1988 +%\verb|axiom-developer.org/axiom-website/papers/Bro90a.pdf| abstract = " - The problem of integration in finite terms asks for an algorithm for - deciding whether an elementary function has an elementary indefinite - integral and for finding the integral if it does. Elementary'' is - used here to denote those functions build up from the rational - functions using only exponentiation, logarithms, trigonometric, - inverse trigonometric and algebraic operations. This vaguely worded - question has several precise, but inequivalent formulations. The - writer has devised an algorithm which solves the classical problem of - Liouville. A complete account is planned for a future publication. The - present note is intended to indiciate some of the ideas and techniques - involved." -} + We extend a recent algorithm of Trager to a decision procedure for the + indefinite integration of elementary functions. We can express the + integral as an elementary function or prove that it is not + elementary. We show that if the problem of integration in finite terms + is solvable on a given elementary function field $k$, then it is + solvable in any algebraic extension of $k(\theta)$, where $\theta$ is + a logarithm or exponential of an element of $k$. Our proof considers + an element of such an extension field to be an algebraic function of + one variable over $k$. + + In his algorithm for the integration of algebraic functions, Trager + describes a Hermite-type reduction to reduce the problem to an + integrand with only simple finite poles on the associated Riemann + surface. We generalize that technique to curves over liouvillian + ground fields, and use it to simplify our integrands. Once the + multipe finite poles have been removed, we use the Puiseux expansions + of the integrand at infinity and a generalization of the residues to + compute the integral. We also generalize a result of Rothstein that + gives us a necessary condition for elementary integrability, and + provide examples of its use." \end{chunk} \begin{chunk}{axiom.bib} -@article{Risc79, - author = "Risch, Robert", - title = "Algebraic properties of the elementary functions of analysis", - journal = "American Journal of Mathematics", - volume = "101", - pages = "743-759", - year = "1979" +@article{Bron90c, + author = "Bronstein, Manuel", + title = "On the integration of elementary functions", + journal = "Journal of Symbolic Computation", + volume = "9", + number = "2", + pages = "117-173", + year = "1990", + month = "February" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Ritt 48]{Ritt48} Ritt, J.F. -Integration in Finite Terms'' -Columbia University Press, New York 1948 -% REF:00046 +\bibitem[Bronstein 93]{REF-BS93} Bronstein, Manuel; Salvy, Bruno +Full partial fraction decomposition of rational functions'' +In Bronstein [Bro93] pp157-160 ISBN 0-89791-604-2 LCCN QA76.95 I59 1993 +\verb|www.acm.org/pubs/citations/proceedings/issac/164081/| \end{chunk} \begin{chunk}{ignore} -\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell -Liouville's Theorem on Functions with Elementary Integrals'' -Pacific Journal of Mathematics Vol 24 No 1 (1968) -\verb|msp.org/pjm/1968/24-1/pjm-v24-n1-p16-p.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Ro68.pdf| REF:00047 +\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel +A Unification of Liouvillian Extensions'' +%\verb|axiom-developer.org/axiom-website/papers/Bro90b.pdf| abstract = " - Defining a function with one variable to be elemetary if it has an - explicit representation in terms of a finite number of algebraic - operations, logarithms, and exponentials. Liouville's theorem in its - simplest case says that if an algebraic function has an elementary - integral then the latter is itself an algebraic function plus a sum of - constant multiples of logarithms of algebraic functions. Ostrowski has - generalized Liouville's results to wider classes of meromorphic - functions on regions of the complex plane and J.F. Ritt has given the - classical account of the entire subject in his Integraion in Finite - Terms, Columbia University Press, 1948. In spite of the essentially - algebraic nature of the problem, all proofs so far have been analytic. - This paper gives a self contained purely algebraic exposition of the - probelm, making a few new points in addition to the resulting - simplicity and generalization." + We generalize Liouville's theory of elementary functions to a larger + class of differential extensions. Elementary, Liouvillian and + trigonometric extensions are all special cases of our extensions. In + the transcendental case, we show how the rational techniques of + integration theory can be applied to our extensions, and we give a + unified presentation which does not require separate cases for + different monomials." \end{chunk} \begin{chunk}{axiom.bib} -@article{Rose72, - author = "Rosenlicht, Maxwell", - title = "Integration in finite terms", - journal = "American Mathematical Monthly", - year = "1972", - volume = "79", - pages = "963-972", - paper = "Rose72.pdf" +@book{Bron97, + author = "Bronstein, Manuel", + title = "Symbolic Integration I--Transcendental Functions", + publisher = "Springer, Heidelberg", + year = "1997", + isbn = "3-540-21493-3", + url = "http://evil-wire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf", + paper = "Bron97.pdf" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Rothstein 76]{Ro76} Rothstein, Michael -Aspects of symbolic integration and simplifcation of exponential -and primitive functions'' -PhD thesis, University of Wisconsin-Madison (1976) -\verb|www.cs.kent.edu/~rothstei/dis.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Ro76.pdf| REF:00051 - abstract = " - In this thesis we cover some aspects of the theory necessary to obtain - a canonical form for functions obtained by integration and - exponentiation from the set of rational functions. - - These aspects include a new algorithm for symbolic integration of - functions involving logarithms and exponentials which avoids - factorization of polynomials in those cases where algebraic extension - of the constant field is not required, avoids partial fraction - decompositions, and only solves linear systems with a small number of - unknowns. - - We have also found a theorem which states, roughly speaking, that if - integrals which can be represented as logarithms are represented as - such, the only algebraic dependence that a new exponential or - logarithm can satify is given by the law of exponents or the law of - logarithms." - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F. -A structure theorem for exponential and primitive functions: a preliminary - report'' -ACM Sigsam Bulletin Vol 10 Issue 4 (1976) -%\verb|axiom-developer.org/axiom-website/papers/Ro76a.pdf| - abstract = " - In this paper a generalization of the Risch Structure Theorem is reported. - The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$ - is a differential field (in our applications $F$ will be a finitely - generated extension of $Q$, the field of rational numbers) and each $t_i$ - is either algebraic over $F_{i-1}=F(t_1,\ldots,t_{i-1})$, is an - exponential of an element in $F_{i-1}$, or is an integral of an element - in $F_{i-1}$. If $t_i$ is an integral and can be expressed using - logarithms, it must be so expressed for the generalized structure - theorem to apply." +\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel +The Poor Man's Integrator, a parallel integration heuristic'' +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt| +\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples| +%\verb|axiom-developer.org/axiom-website/papers/Bro05a.txt| \end{chunk} -\begin{chunk}{ignore} -\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F. -A structure theorem for exponential and primitive functions'' -SIAM J. Computing Vol 8 No 3 (1979) -%\verb|axiom-developer.org/axiom-website/papers/Ro76b.pdf| REF:00104 +\begin{chunk}{axiom.bib} +@article{Bron06, + author = "Bronstein, M.", + title = "Parallel integration", + journal = "Programming and Computer Software", + year = "2006", + issn = "0361-7688", + volume = "32", + number = "1", + doi = "10.1134/S0361768806010075", + url = "http://dx.doi.org/10.1134/S0361768806010075", + publisher = "Nauka/Interperiodica", + pages = "59-60", + paper = "Bron06.pdf", abstract = " - In this paper a new theorem is proved that generalizes a result of - Risch. The new theorem gives all the possible algebraic relationships - among functions that can be built up from the rational functions by - algebraic operations, by taking exponentials, and by integration. The - functions so generated are called exponential and primitive functions. - From the theorem an algorithm for determining algebraic dependence - among a given set of exponential and primitive functions is derived. - The algorithm is then applied to a problem in computer algebra." + Parallel integration is an alternative method for symbolic + integration. While also based on Liouville's theorem, it handles all + the generators of the differential field containing the integrand in + parallel'', i.e. all at once rather than considering only the topmost + one in a recursive fasion. Although it still contains heuristic + aspects, its ease of implementation, speed, high rate of success, and + ability to integrate functions that cannot be handled by the Risch + algorithm make it an attractive alternative." +} \end{chunk} \begin{chunk}{axiom.bib} -@article{Roth77, - author = "Rothstein, Michael", - title = "A new algorithm for the integration of exponential and - logarithmic functions", - journal = "Proceedings of the 1977 MACSYMA Users Conference", - year = "1977", - pages = "263-274", - publisher = "NASA Pub CP-2012" +@article{Bron07, + author = "Bronstein, Manuel", + title = "Structure theorems for parallel integration", + journal = "Journal of Symbolic Computation", + volume = "42", + number = "7", + pages = "757-769", + year = "2007", + month = "July", + paper = "Bron07.pdf", + abstract = " + We introduce structure theorems that refine Liouville's Theorem on + integration in closed form for general derivations on multivariate + rational function fields. By predicting the arguments of the new + logarithms that an appear in integrals, as well as the denominator of + the rational part, those theorems provide theoretical backing for the + Risch-Norman integration method. They also generalize its applicability + to non-monomial extensions, for example the Lambert W function." } \end{chunk} \begin{chunk}{ignore} -\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham -Abstract differential algebra and the analytic case'' -Proc. Amer. Math. Soc. Vol 9 pp159-164 (1958) +\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin +Integration on Computer Algebra Systems'' +The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 1933-2823 +\verb|12000.org/my_notes/ten_hard_integrals/paper.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Charl07.pdf| + abstract = " + In this article, we consider ten indefinite integrals and the ability + of three computer algebra systems (CAS) to evaluate them in + closed-form, appealing only to the class of real, elementary + functions. Although these systems have been widely available for many + years and have undergone major enhancements in new versions, it is + interesting to note that there are still indefinite integrals that + escape the capacity of these systems to provide antiderivatves. When + this occurs, we consider what a user may do to find a solution with + the aid of a CAS." \end{chunk} \begin{chunk}{ignore} -\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham -Abstract differential algebra and the analytic case. II'' -Proc. Amer. Math. Soc. Vol 23 pp689-691 (1969) +\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin +Symbolic Integration Problems'' +\verb|www.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Charl08.pdf| + abstract = " + A list of the 50 example integration problems from Kevin Charlwood's 2008 + article Integration on Computer Algebra Systems''. Each integral along + with its optimal antiderivative (that is, the best antiderivative found + so far) is shown." \end{chunk} \begin{chunk}{ignore} -\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F. -An extension of Liouville's theorem on integration in finite terms'' -SIAM J. of Comp. Vol 14 pp965-990 (1985) -\verb|www4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Sing85.pdf| +\bibitem[Cherry 84]{Che84} Cherry, G.W. +Integration in Finite Terms with Special Functions: The Error Function'' +J. Symbolic Computation (1985) Vol 1 pp283-302 +%\verb|axiom-developer.org/axiom-website/papers/Che84.pdf| abstract = " - In Part 1 of this paper, we give an extension of Liouville's Theorem - and give a number of examples which show that integration with special - functions involves some phenomena that do not occur in integration - with the elementary functions alone. Our main result generalizes - Liouville's Theorem by allowing, in addition to the elementary - functions, special functions such as the error function, Fresnel - integrals and the logarithmic integral (but not the dilogarithm or - exponential integral) to appear in the integral of an elementary - function. The basic conclusion is that these functions, if they - appear, appear linearly. We give an algorithm which decides if an - elementary function, built up using only exponential functions and - rational operations has an integral which can be expressed in terms of - elementary functions and error functions." + A decision procedure for integrating a class of transcendental + elementary functions in terms of elementary functions and error + functions is described. The procedure consists of three mutually + exclusive cases. In the first two cases a generalised procedure for + completing squares is used to limit the error functions which can + appear in the integral of a finite number. This reduces the problem + to the solution of a differential equation and we use a result of + Risch (1969) to solve it. The third case can be reduced to the + determination of what we have termed $\sum$-decompositions. The resutl + presented here is the key procuedure to a more general algorithm which + is described fully in Cherry (1983)." \end{chunk} \begin{chunk}{ignore} -\bibitem[Slagle 61]{Slag61} Slagle, J. -A heuristic program that solves symbolic integration problems in - freshman calculus'' -Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman. -% REF:00014 +\bibitem[Cherry 86]{Che86} Cherry, G.W. +Integration in Finite Terms with Special Functions: +The Logarithmic Integral'' +SIAM J. Comput. Vol 15 pp1-21 February 1986 \end{chunk} \begin{chunk}{ignore} -\bibitem[Terelius 09]{Tere09} Terelius, Bjorn -Symbolic Integration'' -%\verb|axiom-developer.org/axiom-website/papers/Tere09.pdf| +\bibitem[Cherry 89]{Che89} Cherry, G.W. +An Analysis of the Rational Exponential Integral'' +SIAM J. Computing Vol 18 pp 893-905 (1989) +%\verb|axiom-developer.org/axiom-website/papers/Che89.pdf| abstract = " - Symbolic integration is the problem of expressing an indefinite integral - $\int{f}$ of a given function $f$ as a finite combination $g$ of elementary - functions, or more generally, to determine whether a certain class of - functions contains an element $g$ such that $g^\prime = f$. - - In the first part of this thesis, we compare different algorithms for - symbolic integration. Specifically, we review the integration rules - taught in calculus courses and how they can be used systematically to - create a reasonable, but somewhat limited, integration method. Then we - present the differential algebra required to prove the transcendental - cases of Risch's algorithm. Risch's algorithm decides if the integral - of an elementary function is elementary and if so computes it. The - presentation is mostly self-contained and, we hope, simpler than - previous descriptions of the algorithm. Finally, we describe - Risch-Norman's algorithm which, although it is not a decision - procedure, works well in practice and is considerably simpler than the - full Risch algorithm. - - In the second part of this thesis, we briefly discuss an - implementation of a computer algebra system and some of the - experiences it has given us. We also demonstrate an implementation of - the rule-based approach and how it can be used, not only to compute - integrals, but also to generate readable derivations of the results." + In this paper an algorithm is presented for integrating expressions of + the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of + $x$, in terms of a class of special functions called the special + incomplete $\Gamma$ functions. This class of special functions + includes the exponential integral, the error functions, the sine and + cosing integrals, and the Fresnel integrals. The algorithm presented + here is an improvement over those published previously for integrating + with special functions in the following ways: (i) This algorithm + combines all the above special functions into one algorithm, whereas + previously they were treated separately, (ii) Previous algorithms + require that the underlying field of constants be algebraically + closed. This algorithm, however, works over any field of + characteristic zero in which the basic field operations can be carried + out. (iii) This algorithm does not rely on Risch's solution of the + differential equation $y^\prime + fy = g$. Instead, a more direct + method of undetermined coefficients is used." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Trag76, - author = "Trager, Barry", - title = "Algebraic factoring and rational function integration", - journal = "Proceedings of SYMSAC'76", - year = "1976", - pages = "219-226", - paper = "Trag76.pdf", +\begin{chunk}{ignore} +\bibitem[Churchill 06]{Chur06} Churchill, R.C. +Liouville's Theorem on Integration Terms of Elementary Functions'' +\verb|www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Chur06.pdf| abstract = " - This paper presents a new, simple, and efficient algorithm for - factoring polynomials in several variables over an algebraic number - field. The algorithm is then used interatively to construct the - splitting field of a polynomial over the integers. Finally the - factorization and splitting field algorithms are applied to the - problem of determining the transcendental part of the integral of a - rational function. In particular, a constructive procedure is given - for finding a least degree extension field in which the integral can - be expressed." -} + This talk should be regarded as an elementary introduction to + differential algebra. It culminates in a purely algebraic proof, due + to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of + elementary'' integrals of elementary'' functions. The precise + meaning of elementary will be specified. As an application of that + theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$ + cannot be expressed in terms of elementary functions. + \begin{itemize} + \item Preliminaries on Meromorphic Functions + \item Basic (Ordinary) Differential Algebra + \item Differential Ring Extensions with No New Constants + \item Extending Derivations + \item Integration in Finite Terms + \end{itemize}" \end{chunk} \begin{chunk}{ignore} -\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall -Algorithms for Manipulating Algebraic Functions'' -MIT Master's Thesis. -\verb|www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/fattorizzazione-EA.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Tr76a.pdf| REF:00050 - abstract = " - Given a base field $k$, of characteristic zero, with effective - procedures for performing arithmetic and factoring polynomials, this - thesis presents algorithms for extending those capabilities to - elements of a finite algebraic symbolic manipulation system. An - algebraic factorization algorithm along with a constructive version of - the primitive element theorem is used to construct splitting fields of - polynomials. These fields provide a context in which we can operate - symbolically with all the roots of a set of polynomials. One - application for this capability is rational function integrations. - Previously presented symbolic algorithms concentrated on finding the - rational part and were only able to compute the complete - integral in special cases. This thesis presents an algorithm for - finding an algebraic extension field of least degreee in which the - integral can be expressed, and then constructs the integral in that - field. The problem of algebraic function integration is also - examined, and a highly efficient procedure is presented for generating - the algebraic part of integrals whose function fields are defined by a - single radical extension of the rational functions." +\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold +On the Integration of Algebraic Functions'' +Springer-Verlag Lecture Notes in Computer Science 102 +ISBN 0-387-10290-6 \end{chunk} -\begin{chunk}{axiom.bib} -@phdthesis{Trag84, - author = "Trager, Barry", - title = "On the integration of algebraic functions", - school = "MIT", - year = "1984", - url = "http://www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/thesis.pdf", - paper = "Trag76.pdf", +\begin{chunk}{ignore} +\bibitem[Davenport 79c]{Dav79c} Davenport, J. H. +Algorithms for the Integration of Algebraic Functions'' +Lecture Notes in Computer Science V 72 pp415-425 (1979) +%\verb|axiom-developer.org/axiom-website/papers/Dav79c.pdf| abstract = " - We show how the rational'' approach for integrating algebraic - functions can be extended to handle elementary functions. The - resulting algorithm is a practical decision procedure for determining - whether a given elementary function has an elementary antiderivative, - and for computing it if it exists." -} + The problem of finding elementary integrals of algebraic functions has + long been recognized as difficult, and has sometimes been thought + insoluble. Risch stated a theorem characterising the integrands with + elementary integrals, and we can use the language of algebraic + geometry and the techniques of Davenport to yield an algorithm that will + always produce the integral if it exists. We explain the difficulty in + the way of extending this algorithm, and outline some ways of solving + it. Using work of Manin we are able to solve the problem in all cases + where the algebraic expressions depend on a parameter as well as on + the variable of integration." \end{chunk} \begin{chunk}{ignore} -\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas -Basic Concepts of Differential Algebra'' -\verb|www14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Wurf07.pdf| +\bibitem[Davenport 82a]{Dav82a} Davenport, J.H. +The Parallel Risch Algorithm (I) +%\verb|axiom-developer.org/axiom-website/papers/Dav82a.pdf| abstract = " - Modern computer algebra systems symbolically integrate a vast variety - of functions. To reveal the underlying structure it is necessary to - understand infinite integration not only as an analytical problem but - as an algebraic one. Introducing the differential field of elementary - functions we sketch the mathematical tools like Liouville's Principle - used in modern algorithms. We present Hermite's method for integration - of rational functions as well as the Rothstein/Trager method for - rational and for elementary functions. Further applications of the - mentioned algorithms in the field of ODE's conclude this paper." + In this paper we review the so-called parallel Risch'' algorithm for + the integration of transcendental functions, and explain what the + problems with it are. We prove a positive result in the case of + logarithmic integrands." \end{chunk} -\subsection{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Angell]{Angell} Angell, Tom -Guidelines for Partial Fraction Decomposition'' -\verb|www.math.udel.edu/~angell/partfrac_I.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Angell.pdf| +\bibitem[Davenport 82]{Dav82} Davenport, J.H. +On the Parallel Risch Algorithm (III): Use of Tangents'' +SIGSAM V16 no. 3 pp3-6 August 1982 \end{chunk} \begin{chunk}{ignore} -\bibitem[Laval 08]{Lava08} Laval, Philippe B. -Partial Fractions Decomposition'' -\verb|www.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Lava08.pdf| +\bibitem[Davenport 03]{Dav03} Davenport, James H. +The Difficulties of Definite Integration'' +\verb|www.researchgate.net/publication/| +\verb|247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Dav03.pdf| + abstract = " + Indefinite integration is the inverse operation to differentiation, + and, before we can understand what we mean by indefinite integration, + we need to understand what we mean by differentiation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Mudd 14]{Mudd14} Harvey Mudd College -Partial Fractions'' -\verb|www.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mudd14.pdf| +\bibitem[Fateman 02]{Fat02} Fateman, Richard +Symbolic Integration'' +\verb|inst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Fat02.pdf| \end{chunk} -\begin{chunk}{ignore} -\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja -Partial Fraction Expansion'' -\verb|www.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/| -\verb|Partial%20Fractions.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Raja14.pdf| +\begin{chunk}{axiom.bib} +@inproceedings{Gedd89, + author = "Geddes, K. O. and Stefanus, L. Y.", + title = "On the Risch-norman Integration Method and Its Implementation + in MAPLE", + booktitle = "Proc. of the ACM-SIGSAM 1989 Int. Symp. on Symbolic and + Algebraic Computation", + series = "ISSAC '89", + year = "1989", + isbn = "0-89791-325-6", + location = "Portland, Oregon, USA", + pages = "212--217", + numpages = "6", + url = "http://doi.acm.org/10.1145/74540.74567", + doi = "10.1145/74540.74567", + acmid = "74567", + publisher = "ACM", + address = "New York, NY, USA", + paper = "Gedd89.pdf", + abstract = " + Unlike the Recursive Risch Algorithm for the integration of + transcendental elementary functions, the Risch-Norman Method processes + the tower of field extensions directly in one step. In addition to + logarithmic and exponential field extensions, this method can handle + extentions in terms of tangents. Consequently, it allows trigonometric + functions to be treated without converting them to complex exponential + form. We review this method and describe its implementation in + MAPLE. A heuristic enhancement to this method is also presented." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Wootton 14]{Woot14} Wootton, Aaron -Integration of Rational Functions by Partial Fractions'' -\verb|faculty.up.edu/wootton/calc2/section7.4.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Woot14.pdf| +\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G. +The Risch Integration Algorithm'' +Algorithms for Computer Algebra, Ch 12 pp511-573 (1992) +%\verb|axiom-developer.org/axiom-website/papers/GCL92a.pdf| \end{chunk} -\subsection{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -This is used as a reference for the LeftOreRing category, in particular, -the least left common multiple (lcmCoef) function. \begin{chunk}{ignore} -\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark -A method for the Integration of Solutions of Ore Equations'' -Proc ISSAC 97 pp172-175 (1997) -%\verb|axiom-developer.org/axiom-website/papers/Abra97.pdf| - abstract = " - We introduce the notion of the adjoint Ore ring and give a definition - of adjoint polynomial, operator and equation. We apply this for - integrating solutions of Ore equations." +\bibitem[Hardy 1916]{Hard16} Hardy, G.H. +The Integration of Functions of a Single Variable'' +Cambridge Unversity Press, Cambridge, 1916 +% REF:00002 \end{chunk} \begin{chunk}{ignore} -\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e -Noncommutative Symmetric functions and $W$-polynomials'' -\verb|arxiv.org/pdf/math/0606614.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DL06.pdf| +\bibitem[Harrington 78]{Harr87} Harrington, S.J. +A new symbolic integration system in reduce'' +\verb|comjnl.oxfordjournals.or/content/22/2/127.full.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Harr87.pdf| abstract = " - Let $K$, $S$, $D$ be a division ring an endomorphism and a - $S$-derivation of $K$, respectively. In this setting we introduce - generalized noncommutative symmetric functions and obtain Vi\'ete - formula and decompositions of different operators. $W$-polynomials - show up naturally, their connetions with $P$-independency. Vandermonde - and Wronskian matrices are briefly studied. The different linear - factorizations of $W$-polynomials are analysed. Connections between - the existence of LLCM (least left common multiples) of monic linear - polynomials with coefficients in a ring and the left duo property are - established at the end of the paper." + A new integration system, employing both algorithmic and pattern match + integration schemes is presented. The organization of the system + differs from that of earlier programs in its emphasis on the + algorithmic approach to integration, its modularity and its ease of + revision. The new Norman-Rish algorithm and its implementation at the + University of Cambridge are employed, supplemented by a powerful + collection of simplification and transformation rules. The facility + for user defined integrals and functions is also included. The program + is both fast and powerful, and can be easily modified to incorporate + anticipated developments in symbolic integration." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z. -Univariate Ore Polynomial Rings in Computer Algebra'' -\verb|www.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Abra05.pdf| - abstract = " - We present some algorithms related to rings of Ore polynomials (or, - briefly, Ore rings) and describe a computer algebra library for basic - operations in an arbitrary Ore ring. The library can be used as a - basis for various algorithms in Ore rings, in particular, in - differential, shift, and $q$-shift rings." +\begin{chunk}{axiom.bib} +@misc{Herm1872, + author = "Hermite, E.", + title = "Sur l'int\'{e}gration des fractions rationelles", + journal = "Nouvelles Annales de Math\'{e}matiques", + volume = "11", + pages = "145-148", + year = "1872" +} \end{chunk} -\subsection{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Shoup 08]{Sho08} Shoup, Victor -A Computational Introduction to Number Theory'' -\verb|shoup.net/ntb/ntb-v2.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Sho08.pdf| +\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis +Algorithms for Partial Fraction Decomposition and Rational Function + Integration'' +SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971) +pp441-457 +%\verb|axiom-developer.org/axiom-website/papers/Horo71.pdf| REF:00018 + abstract = " + Algorithms for symbolic partial fraction decomposition and indefinite + integration of rational functions are described. Two types of + partial fraction decomposition are investigated, square-free and + complete square-free. A method is derived, based on the solution of + a linear system, which produces the square-free decomposition of any + rational function, say A/B. The computing time is show to be + $O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$ + is a number which is closely related to the size of the coefficients + which occur in A and B. The complete square-free partical fraction + decomposition can then be directly obtained and it is shown that the + computing time for this process is also bounded by $O(n^4(ln nf)^2)$." \end{chunk} -\subsection{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{ignore} +\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D. +Recursive integration of piecewise-continuous functions'' +\verb|www.cybertester.com/data/recint.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Jeff97.pdf| + abstract = " + An algorithm is given for the integration of a class of + piecewise-continuous functions. The integration is with respect to a + real variable, because the functions considered do not in general + allow integration in the complex plane to be defined. The class of + integrands includes commonly occurring waveforms, such as square + waves, triangular waves, and the floor function; it also includes the + signum function. The algorithm can be implemented recursively, and it + has the property of ensuring that integrals are continuous on domains + of maximum extent." -\subsection{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\end{chunk} -\begin{chunk}{axiom.bib} -@article{Beau03, - author = "Beaumont, James and Bradford, Russell and Davenport, James H.", - title = "Better simplification of elementary functions through power series", - journal = "2003 International Symposium on Symbolic and Algebraic Computation", - series = "ISSAC'03", - year = "2003", - month = "August", - paper = "Beau03.pdf", +\begin{chunk}{ignore} +\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.; +Rich, A.D. +Integration of the signum, piecewise and related functions'' +\verb|cs.uwaterloo.ca/~glabahn/Papers/issac99-2.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Jeff99.pdf| + abstract = " + When a computer algebra system has an assumption facility, it is + possible to distinguish between integration problems with respect to a + real variable, and those with respect to a complex variable. Here, a + class of integration problems is defined in which the integrand + consists of compositions of continuous functions and signum functions, + and integration is with respect to a real variable. Algorithms are + given for evaluating such integrals." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref +A new symbolic computation for formal integration with exact power series'' +%\verb|axiom-developer.org/axiom-website/Kiym04.pdf| + abstract = " + This paper describes a new symbolic algorithm for formal integration + of a class of functions in the context of exact power series by using + generalized hypergeometric series and computer algebraic technique." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Knowles 93]{Know93} Knowles, P. +Integration of a class of transcendental liouvillian +functions with error-functions i'' +Journal of Symbolic Computation Vol 13 pp525-543 (1993) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Knowles 95]{Know95} Knowles, P. +Integration of a class of transcendental liouvillian +functions with error-functions ii'' +Journal of Symbolic Computation Vol 16 pp227-241 (1995) + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Krag09, + author = "Kragler, R.", + title = "On Mathematica Program for Poor Man's Integrator Algorithm", + journal = "Programming and Computer Software", + volume = "35", + number = "2", + pages = "63-78", + year = "2009", + issn = "0361-7688", + paper = "Krag09.pdf", + abstract = " + In this paper by means of computer experiment we study advantages and + disadvantages of the heuristical method of parallel integrator''. For + this purpose we describe and use implementation of the method in + Mathematica. In some cases we compare this implementation with the original + one in Maple." +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Lang 93]{Lang93} Lang, S. +Algebra'' +Addison-Wesly, New York, 3rd edition 1993 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian +A Generalization of Liouville's Theorem on Integration in Finite Terms'' +\verb|www.mathnet.or.kr/mathnet/kms_tex/113666.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Leer02.pdf| + abstract = " + A generalization of Liouville's theorem on integration in finite + terms, by enlarging the class of fields to an extension called + Ei-Gamma extension is established. This extension includes the + $\mathcal{E}\mathcal{L}$-elementary extensions of Singer, Saunders and + Caviness and contains the Gamma function." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Leslie 09]{Lesl09} Leslie, Martin +Why you can't integrate exp($x^2$)'' +\verb|math.arizona.edu/~mleslie/files/integrationtalk.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Lesl09.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel +Symbolic definite (and indefinite) integration: methods and open issues'' +ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011) +\verb|www.sigsam.org/bulletin/articles/175/issue175.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Lich11.pdf| + abstract = " + The computation of definite integrals presents one with a variety of + choices. There are various methods such as Newton-Leibniz or Slater's + convolution method. There are questions such as whether to split or + merge sums, how to search for singularities on the path of + integration, when to issue conditional results, how to assess + (possibly conditional) convergence, and more. These various + considerations moreover interact with one another in a multitude of + ways. Herein we discuss these various issues and illustrate with examples." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Liou1833a, + author = "Liouville, Joseph", + title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales + dont la valeur est alg\'{e}brique", + journal = "Journal de l'Ecole Polytechnique", + volume = "14", + pages = "124-128", + year = "1833" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Liou1833b, + author = "Liouville, Joseph", + title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales + dont la valeur est alg\'{e}brique", + journal = "Journal de l'Ecole Polytechnique", + volume = "14", + pages = "149-193", + year = "1833" +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph +Note sur la determination des int\'egrales dont la +valeur est alg\'ebrique'' +Journal f\"ur die Reine und Angewandte Mathematik, +Vol 10 pp 247-259, (1833) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph +Sur la determination des int\'egrales dont la valeur est +alg\'ebrique'' +{\sl Journal de l'Ecole Polytechnique}, 14:124-193, 1833 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph +M\'emoire sur l'int\'gration d'une classe de fonctions +transcendentes'' +Journal f\"ur die Reine und Angewandte Mathematik, +Vol 13(2) pp 93-118, (1835) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, Gholem-All +An Invitation to Integration in Finite Terms'' +College Mathematics Journal Vol 25 No 4 (1994) pp295-308 +\verb|www.rangevoting.org/MarchisottoZint.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Marc94.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Marik 91]{Mari91} Marik, Jan +A note on integration of rational functions'' +\verb|dml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_116-1991-4_9.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mari91.pdf| + abstract = " + Let $P$ and $Q$ be polynomials in one variable with complex coefficients + and let $n$ be a natural number. Suppose that $Q$ is not constant and + has only simple roots. Then there is a rational function $\varphi$ + with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the + functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is + divisible by $Q$." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Moses 76]{Mos76} Moses, Joel +An introduction to the Risch Integration Algorithm'' +ACM Proc. 1976 annual conference pp425-428 +%\verb|axiom-developer.org/axiom-website/papers/Mos76.pdf| REF:00048 + abstract = " + Risch's decision procedure for determining the integrability in closed + form of the elementary functions of the calculus is presented via + examples. The exponential and logarithmic cases of the algorithsm had + been implemented for the MACSYMA system several years ago. The + implementation of the algebraic case of the algorithm is the subject + of current research." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Moses 71a]{Mos71a} Moses, Joel +Symbolic Integration: The Stormy Decade'' +CACM Aug 1971 Vol 14 No 8 pp548-560 +\verb|www-inst.eecs.berkeley.edu/~cs282/sp02/readings/moses-int.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mos71a.pdf| REF:00017 + abstract = " + Three approaches to symbolic integration in the 1960's are + described. The first, from artificial intelligence, led to Slagle's + SAINT and to a large degree to Moses' SIN. The second, from algebraic + manipulation, led to Monove's implementation and to Horowitz' and + Tobey's reexamination of the Hermite algorithm for integrating + rational functions. The third, from mathematics, led to Richardson's + proof of the unsolvability of the problem for a class of functions and + for Risch's decision procedure for the elementary functions. + Generalizations of Risch's algorithm to a class of special + functions and programs for solving differential equations and for + finding the definite integral are also described." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H. +Symbolic Integration -- The Dust Settles?'' +%\verb|axiom-developer.org/axiom-website/papers/Nor79.pdf| + abstract = " + By the end of the 1960s it had been shown that a computer could find + indefinite integrals with a competence exceeding that of typical + undergraduates. This practical advance was backed up by algorithmic + interpretations of a number of clasical results on integration, and by + some significant mathematical extensions to these same results. At + that time it would have been possible to claim that all the major + barriers in the way of a complete system for automated analysis had + been breached. In this paper we survey the work that has grown out of + the above-mentioned early results, showing where the development has + been smooth and where it has spurred work in seemingly unrelated fields." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Ostrowski 46]{Ost46} Ostrowski, A. +Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes +d'expressions'' +Comm. Math. Helv., Vol 18 pp 283-308, (1946) +% REF:00008 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Raab 12]{Raab12} Raab, Clemens G. +Definite Integration in Differential Fields'' +\verb|www.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Raab12.pdf| + abstract = " + The general goal of this thesis is to investigate and develop computer + algebra tools for the simplification resp. evaluation of definite + integrals. One way of finding the value of a def- inite integral is + via the evaluation of an antiderivative of the integrand. In the + nineteenth century Joseph Liouville was among the first who analyzed + the structure of elementary antiderivatives of elementary functions + systematically. In the early twentieth century the algebraic structure + of differential fields was introduced for modeling the differential + properties of functions. Using this framework Robert H. Risch + published a complete algorithm for transcendental elementary + integrands in 1969. Since then this result has been extended to + certain other classes of integrands as well by Michael F. Singer, + Manuel Bronstein, and several others. On the other hand, if no + antiderivative of suitable form is available, then linear relations + that are satisfied by the parameter integral of interest may be found + based on the principle of parametric integration (often called + differentiating under the integral sign or creative telescoping). + + The main result of this thesis extends the results mentioned above to + a complete algo- rithm for parametric elementary integration for a + certain class of integrands covering a majority of the special + functions appearing in practice such as orthogonal polynomials, + polylogarithms, Bessel functions, etc. A general framework is provided + to model those functions in terms of suitable differential fields. If + the integrand is Liouvillian, then the present algorithm considerably + improves the efficiency of the corresponding algorithm given by Singer + et al. in 1985. Additionally, a generalization of Czichowski’s + algorithm for computing the logarithmic part of the integral is + presented. Moreover, also partial generalizations to include other + types of integrands are treated. + + As subproblems of the integration algorithm one also has to find + solutions of linear or- dinary differential equations of a certain + type. Some contributions are also made to solve those problems in our + setting, where the results directly dealing with systems of + differential equations have been joint work with Moulay A. Barkatou. + + For the case of Liouvillian integrands we implemented the algorithm in + form of our Mathematica package Integrator. Parts of the + implementation also deal with more general functions. Our procedures + can be applied to a significant amount of the entries in integral + tables, both indefinite and definite integrals. In addition, our + procedures have been successfully applied to interesting examples of + integrals that do not appear in these tables or for which current + standard computer algebra systems like Mathematica or Maple do not + succeed. We also give examples of how parameter integrals coming from + the work of other researchers can be solved with the software, e.g., + an integral arising in analyzing the entropy of certain processes." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Raab 13]{Raab13} Raab, Clemens G. +Generalization of Risch's Algorithm to Special Functions'' +\verb|arxiv.org/pdf/1305.1481| +%\verb|axiom-developer.org/axiom-website/papers/Raab13.pdf| + abstract = " + Symbolic integration deals with the evaluation of integrals in closed + form. We present an overview of Risch's algorithm including recent + developments. The algorithms discussed are suited for both indefinite + and definite integration. They can also be used to compute linear + relations among integrals and to find identities for special functions + given by parameter integrals. The aim of this presentation is twofold: + to introduce the reader to some basic idea of differential algebra in + the context of integration and to raise awareness in the physics + community of computer algebra algorithms for indefinite and definite + integration." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Raab xx]{Raabxx} Raab, Clemens G. +Integration in finite terms for Liouvillian functions'' +\verb|www.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Raabxx.pdf| + abstract = " + Computing integrals is a common task in many areas of science, + antiderivatives are one way to accomplish this. The problem of + integration in finite terms can be states as follows. Given a + differential field $(F,D)$ and $f \in F$, compute $g$ in some + elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$ + exists. + + This problem has been solved for various classes of fields $F$. For + rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and + algorithms to compute it are known already for a long time. In 1969 + Risch published an algorithm that solves this problem when $(F,D)$ is + a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later + this has been extended towards integrands being Liouvillian functions + by Singer et. al. via the use of regular log-explicit extensions of + $(C(x),\frac{d}{dx})$. Our algorithm extends this to handling + transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly + without the need to embed them into log-explicit extensions. For + example, this means that + $\int{(z-x)x^{z-1}e^{-x}dx} = x^ze^{-x}$ + can be computed without including log(x) in the differential field." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J. +A Knowledge Repository for Indefinite Integration Based on Transformation Rules'' +\verb|www.apmaths.uwo.ca/~arich/A%2520Rule-based%2520Knowedge%2520Repository.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Rich09.pdf| + abstract = " + Taking the specific problem domain of indefinite integration, we + describe the on-going development of a repository of mathematical + knowledge based on transformation rules. It is important that the + repository be not confused with a look-up table. The database of + transformation rules is at present encoded in Mathematica, but this is + only one convenient form of the repository, and it could be readily + translated into other formats. The principles upon which the set of + rules is compiled is described. One important principle is + minimality. The benefits of the approach are illustrated with + examples, and with the results of comparisons with other approaches." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@techreport{Risc68, + author = "Risch, Robert", + title = "On the integration of elementary functions which are built up + using algebraic operations", + type = "Research Report", + number = "SP-2801/002/00", + institution = "System Development Corporation, Santa Monica, CA, USA", + year = "1968" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@techreport{Risc69a, + author = "Risch, Robert", + title = "Further results on elementary functions", + type = "Research Report", + number = "RC-2042", + institution = "IBM Research, Yorktown Heights, NY, USA", + year = "1969" + +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Risc69b, + author = "Risch, Robert", + title = "The problem of integration in finite terms", + journal = "Transactions of the American Mathematical Society", + volume = "139", + year = "1969", + pages = "167-189", + paper = "Ris69b.pdf", + abstract = "This paper deals with the problem of telling whether a + given elementary function, in the sense of analysis, has an elementary + indefinite integral." +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Risc70, + author = "Risch, Robert", + title = "The Solution of the Problem of Integration in Finite Terms", + journal = "Bull. AMS", + year = "1970", + issn = "0002-9904", + volume = "76", + number = "3", + pages = "605-609", + paper = "Risc70.pdf", + abstract = " + The problem of integration in finite terms asks for an algorithm for + deciding whether an elementary function has an elementary indefinite + integral and for finding the integral if it does. Elementary'' is + used here to denote those functions build up from the rational + functions using only exponentiation, logarithms, trigonometric, + inverse trigonometric and algebraic operations. This vaguely worded + question has several precise, but inequivalent formulations. The + writer has devised an algorithm which solves the classical problem of + Liouville. A complete account is planned for a future publication. The + present note is intended to indiciate some of the ideas and techniques + involved." +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Risc79, + author = "Risch, Robert", + title = "Algebraic properties of the elementary functions of analysis", + journal = "American Journal of Mathematics", + volume = "101", + pages = "743-759", + year = "1979" +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Ritt 48]{Ritt48} Ritt, J.F. +Integration in Finite Terms'' +Columbia University Press, New York 1948 +% REF:00046 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell +Liouville's Theorem on Functions with Elementary Integrals'' +Pacific Journal of Mathematics Vol 24 No 1 (1968) +\verb|msp.org/pjm/1968/24-1/pjm-v24-n1-p16-p.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Ro68.pdf| REF:00047 + abstract = " + Defining a function with one variable to be elemetary if it has an + explicit representation in terms of a finite number of algebraic + operations, logarithms, and exponentials. Liouville's theorem in its + simplest case says that if an algebraic function has an elementary + integral then the latter is itself an algebraic function plus a sum of + constant multiples of logarithms of algebraic functions. Ostrowski has + generalized Liouville's results to wider classes of meromorphic + functions on regions of the complex plane and J.F. Ritt has given the + classical account of the entire subject in his Integraion in Finite + Terms, Columbia University Press, 1948. In spite of the essentially + algebraic nature of the problem, all proofs so far have been analytic. + This paper gives a self contained purely algebraic exposition of the + probelm, making a few new points in addition to the resulting + simplicity and generalization." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Rose72, + author = "Rosenlicht, Maxwell", + title = "Integration in finite terms", + journal = "American Mathematical Monthly", + year = "1972", + volume = "79", + pages = "963-972", + paper = "Rose72.pdf" +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rothstein 76]{Ro76} Rothstein, Michael +Aspects of symbolic integration and simplifcation of exponential +and primitive functions'' +PhD thesis, University of Wisconsin-Madison (1976) +\verb|www.cs.kent.edu/~rothstei/dis.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Ro76.pdf| REF:00051 + abstract = " + In this thesis we cover some aspects of the theory necessary to obtain + a canonical form for functions obtained by integration and + exponentiation from the set of rational functions. + + These aspects include a new algorithm for symbolic integration of + functions involving logarithms and exponentials which avoids + factorization of polynomials in those cases where algebraic extension + of the constant field is not required, avoids partial fraction + decompositions, and only solves linear systems with a small number of + unknowns. + + We have also found a theorem which states, roughly speaking, that if + integrals which can be represented as logarithms are represented as + such, the only algebraic dependence that a new exponential or + logarithm can satify is given by the law of exponents or the law of + logarithms." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F. +A structure theorem for exponential and primitive functions: a preliminary + report'' +ACM Sigsam Bulletin Vol 10 Issue 4 (1976) +%\verb|axiom-developer.org/axiom-website/papers/Ro76a.pdf| + abstract = " + In this paper a generalization of the Risch Structure Theorem is reported. + The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$ + is a differential field (in our applications $F$ will be a finitely + generated extension of $Q$, the field of rational numbers) and each $t_i$ + is either algebraic over $F_{i-1}=F(t_1,\ldots,t_{i-1})$, is an + exponential of an element in $F_{i-1}$, or is an integral of an element + in $F_{i-1}$. If $t_i$ is an integral and can be expressed using + logarithms, it must be so expressed for the generalized structure + theorem to apply." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F. +A structure theorem for exponential and primitive functions'' +SIAM J. Computing Vol 8 No 3 (1979) +%\verb|axiom-developer.org/axiom-website/papers/Ro76b.pdf| REF:00104 + abstract = " + In this paper a new theorem is proved that generalizes a result of + Risch. The new theorem gives all the possible algebraic relationships + among functions that can be built up from the rational functions by + algebraic operations, by taking exponentials, and by integration. The + functions so generated are called exponential and primitive functions. + From the theorem an algorithm for determining algebraic dependence + among a given set of exponential and primitive functions is derived. + The algorithm is then applied to a problem in computer algebra." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Roth77, + author = "Rothstein, Michael", + title = "A new algorithm for the integration of exponential and + logarithmic functions", + journal = "Proceedings of the 1977 MACSYMA Users Conference", + year = "1977", + pages = "263-274", + publisher = "NASA Pub CP-2012" +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham +Abstract differential algebra and the analytic case'' +Proc. Amer. Math. Soc. Vol 9 pp159-164 (1958) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham +Abstract differential algebra and the analytic case. II'' +Proc. Amer. Math. Soc. Vol 23 pp689-691 (1969) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F. +An extension of Liouville's theorem on integration in finite terms'' +SIAM J. of Comp. Vol 14 pp965-990 (1985) +\verb|www4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Sing85.pdf| + abstract = " + In Part 1 of this paper, we give an extension of Liouville's Theorem + and give a number of examples which show that integration with special + functions involves some phenomena that do not occur in integration + with the elementary functions alone. Our main result generalizes + Liouville's Theorem by allowing, in addition to the elementary + functions, special functions such as the error function, Fresnel + integrals and the logarithmic integral (but not the dilogarithm or + exponential integral) to appear in the integral of an elementary + function. The basic conclusion is that these functions, if they + appear, appear linearly. We give an algorithm which decides if an + elementary function, built up using only exponential functions and + rational operations has an integral which can be expressed in terms of + elementary functions and error functions." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Slagle 61]{Slag61} Slagle, J. +A heuristic program that solves symbolic integration problems in + freshman calculus'' +Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman. +% REF:00014 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Terelius 09]{Tere09} Terelius, Bjorn +Symbolic Integration'' +%\verb|axiom-developer.org/axiom-website/papers/Tere09.pdf| + abstract = " + Symbolic integration is the problem of expressing an indefinite integral + $\int{f}$ of a given function $f$ as a finite combination $g$ of elementary + functions, or more generally, to determine whether a certain class of + functions contains an element $g$ such that $g^\prime = f$. + + In the first part of this thesis, we compare different algorithms for + symbolic integration. Specifically, we review the integration rules + taught in calculus courses and how they can be used systematically to + create a reasonable, but somewhat limited, integration method. Then we + present the differential algebra required to prove the transcendental + cases of Risch's algorithm. Risch's algorithm decides if the integral + of an elementary function is elementary and if so computes it. The + presentation is mostly self-contained and, we hope, simpler than + previous descriptions of the algorithm. Finally, we describe + Risch-Norman's algorithm which, although it is not a decision + procedure, works well in practice and is considerably simpler than the + full Risch algorithm. + + In the second part of this thesis, we briefly discuss an + implementation of a computer algebra system and some of the + experiences it has given us. We also demonstrate an implementation of + the rule-based approach and how it can be used, not only to compute + integrals, but also to generate readable derivations of the results." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Trag76, + author = "Trager, Barry", + title = "Algebraic factoring and rational function integration", + journal = "Proceedings of SYMSAC'76", + year = "1976", + pages = "219-226", + paper = "Trag76.pdf", + abstract = " + This paper presents a new, simple, and efficient algorithm for + factoring polynomials in several variables over an algebraic number + field. The algorithm is then used interatively to construct the + splitting field of a polynomial over the integers. Finally the + factorization and splitting field algorithms are applied to the + problem of determining the transcendental part of the integral of a + rational function. In particular, a constructive procedure is given + for finding a least degree extension field in which the integral can + be expressed." +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall +Algorithms for Manipulating Algebraic Functions'' +MIT Master's Thesis. +\verb|www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/fattorizzazione-EA.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Tr76a.pdf| REF:00050 + abstract = " + Given a base field $k$, of characteristic zero, with effective + procedures for performing arithmetic and factoring polynomials, this + thesis presents algorithms for extending those capabilities to + elements of a finite algebraic symbolic manipulation system. An + algebraic factorization algorithm along with a constructive version of + the primitive element theorem is used to construct splitting fields of + polynomials. These fields provide a context in which we can operate + symbolically with all the roots of a set of polynomials. One + application for this capability is rational function integrations. + Previously presented symbolic algorithms concentrated on finding the + rational part and were only able to compute the complete + integral in special cases. This thesis presents an algorithm for + finding an algebraic extension field of least degreee in which the + integral can be expressed, and then constructs the integral in that + field. The problem of algebraic function integration is also + examined, and a highly efficient procedure is presented for generating + the algebraic part of integrals whose function fields are defined by a + single radical extension of the rational functions." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@phdthesis{Trag84, + author = "Trager, Barry", + title = "On the integration of algebraic functions", + school = "MIT", + year = "1984", + url = "http://www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/thesis.pdf", + paper = "Trag76.pdf", + abstract = " + We show how the rational'' approach for integrating algebraic + functions can be extended to handle elementary functions. The + resulting algorithm is a practical decision procedure for determining + whether a given elementary function has an elementary antiderivative, + and for computing it if it exists." +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas +Basic Concepts of Differential Algebra'' +\verb|www14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Wurf07.pdf| + abstract = " + Modern computer algebra systems symbolically integrate a vast variety + of functions. To reveal the underlying structure it is necessary to + understand infinite integration not only as an analytical problem but + as an algebraic one. Introducing the differential field of elementary + functions we sketch the mathematical tools like Liouville's Principle + used in modern algorithms. We present Hermite's method for integration + of rational functions as well as the Rothstein/Trager method for + rational and for elementary functions. Further applications of the + mentioned algorithms in the field of ODE's conclude this paper." + +\end{chunk} + +\section{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Angell]{Angell} Angell, Tom +Guidelines for Partial Fraction Decomposition'' +\verb|www.math.udel.edu/~angell/partfrac_I.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Angell.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Laval 08]{Lava08} Laval, Philippe B. +Partial Fractions Decomposition'' +\verb|www.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Lava08.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Mudd 14]{Mudd14} Harvey Mudd College +Partial Fractions'' +\verb|www.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mudd14.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja +Partial Fraction Expansion'' +\verb|www.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/| +\verb|Partial%20Fractions.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Raja14.pdf| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Wootton 14]{Woot14} Wootton, Aaron +Integration of Rational Functions by Partial Fractions'' +\verb|faculty.up.edu/wootton/calc2/section7.4.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Woot14.pdf| + +\end{chunk} +\section{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +This is used as a reference for the LeftOreRing category, in particular, +the least left common multiple (lcmCoef) function. + +\begin{chunk}{ignore} +\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark +A method for the Integration of Solutions of Ore Equations'' +Proc ISSAC 97 pp172-175 (1997) +%\verb|axiom-developer.org/axiom-website/papers/Abra97.pdf| abstract = " - In , we introduced an algorithm for deciding whether a proposed - simplification of elementary functions was correct in the presence of - branch cuts. This algorithm used multivalued function simplification - followed by verification that the branches were consistent. + We introduce the notion of the adjoint Ore ring and give a definition + of adjoint polynomial, operator and equation. We apply this for + integrating solutions of Ore equations." - In  an algorithm was presented for zero-testing functions defined - by ordinary differential equations, in terms of their power series. +\end{chunk} - The purpose of the current paper is to investigate merging the two - techniques. In particular, we will show an explicit reduction to the - constant problem ." +\begin{chunk}{ignore} +\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e +Noncommutative Symmetric functions and $W$-polynomials'' +\verb|arxiv.org/pdf/math/0606614.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DL06.pdf| + abstract = " + Let $K$, $S$, $D$ be a division ring an endomorphism and a + $S$-derivation of $K$, respectively. In this setting we introduce + generalized noncommutative symmetric functions and obtain Vi\'ete + formula and decompositions of different operators. $W$-polynomials + show up naturally, their connetions with $P$-independency. Vandermonde + and Wronskian matrices are briefly studied. The different linear + factorizations of $W$-polynomials are analysed. Connections between + the existence of LLCM (least left common multiples) of monic linear + polynomials with coefficients in a ring and the left duo property are + established at the end of the paper." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z. +Univariate Ore Polynomial Rings in Computer Algebra'' +\verb|www.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Abra05.pdf| + abstract = " + We present some algorithms related to rings of Ore polynomials (or, + briefly, Ore rings) and describe a computer algebra library for basic + operations in an arbitrary Ore ring. The library can be used as a + basis for various algorithms in Ore rings, in particular, in + differential, shift, and $q$-shift rings." + +\end{chunk} + +\section{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt89d, + author = "Kaltofen, E. and Valente, T. and Yui, N.", + title = "An improved {Las Vegas} primality test", + booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC89", + pages = "26--33", + year = "1989", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf", + paper = "Kalt89d.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Beau07, - author = "Beaumont, James C. and Bradford, Russell J. and - Davenport, James H. and Phisanbut, Nalina", - title = "Testing elementary function identities using CAD", - journal = "Applicable Algebra in Engineering, Communication and Computing", - year = "2007", - volume = "18", - number = "6", - issn = "0938-1279", +@InCollection{Kalt91b, + author = "Kaltofen, E. and Yui, N.", + editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and + M. B. Nathanson", + title = "Explicit construction of {Hilbert} class fields of imaginary + quadratic fields by integer lattice reduction", + booktitle = "Number Theory New York Seminar 1989--1990", + pages = "150--202", publisher = "Springer-Verlag", - pages = "513-543", - paper = "Beau07.pdf", - abstract = " - One of the problems with manipulating function identities in computer - algebra systems is that they often involve functions which are - multivalued, whilst most users tend to work with single-valued - functions. The problem is that many well-known identities may no - longer be true everywhere in the complex plane when working with their - single-valued counterparts. Conversely, we cannot ignore them, since - in particular contexts they may be valid. We investigate the - practicality of a method to verify such identities by means of an - experiment; this is based on a set of test examples which one might - realistically meet in practice. Essentially, the method works as - follows. We decompose the complex plane via means of cylindrical - algebraic decomposition into regions with respect to the branch cuts - of the functions. We then test the identity numerically at a sample - point in the region. The latter step is facilitated by the notion of - the {\sl adherence} of a branch cut, which was previously introduced - by the authors. In addition to presenting the results of the - experiment, we explain how adherence relates to the proposal of - {\sl signed zeros} by W. Kahan, and develop this idea further in order to - allow us to cover previously untreatable cases. Finally, we discuss - other ways to improve upon our general methodology as well as topics - for future research." + year = "1991", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf", + paper = "Kalt91b.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt84a, + author = "Kaltofen, E. and Yui, N.", + title = "Explicit construction of the {Hilbert} class field of imaginary + quadratic fields with class number 7 and 11", + booktitle = "Proc. EUROSAM '84", + pages = "310--320", + crossref = "EUROSAM84", + year = "1984", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz", + paper = "Kalt84a.ps" +} + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Shoup 08]{Sho08} Shoup, Victor +A Computational Introduction to Number Theory'' +\verb|shoup.net/ntb/ntb-v2.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Sho08.pdf| + +\end{chunk} + +\section{Sparse Polynomial Interpolation} %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt07a, + author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong", + title = "On probabilistic analysis of randomization in hybrid + symbolic-numeric algorithms", + year = "2007", + booktitle = "Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput.", + crossref = "SNC07", + pages = "11--17", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KYZ07.pdf", + paper = "Kalt07a.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt07b, + author = "Kaltofen, Erich and Yang, Zhengfeng", + title = "On Exact and Approximate Interpolation of Sparse + Rational Functions", + year = "2007", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'07", + crossref = "ISSAC07", + pages = "203--210", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KaYa07.pdf", + paper = "Kalt07b.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@Article{Gies03, + author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wen-shin", + title = "Algorithms for Computing Sparsest Shifts of Polynomials in + Power, {Chebychev}, and {Pochhammer} Bases", + year = "2003", + journal = "Journal of Symbolic Computation", + volume = "36", + number = "3--4", + pages = "401--424", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/GKL03.pdf", + paper = "Gies03.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Gies02, + author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wen-shin", + title = "Algorithms for Computing the Sparsest Shifts for Polynomials via the + {Berlekamp}/{Massey} Algorithm", + booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC02", + pages = "101--108", + year = "2002", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/GKL02.pdf", + paper = "Gies02.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@Article{Kalt03b, + author = "Kaltofen, Erich and Lee, Wen-shin", + title = "Early Termination in Sparse Interpolation Algorithms", + year = "2003", + journal = "Journal of Symbolic Computation", + volume = "36", + number = "3--4", + pages = "365--400", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KL03.pdf", + paper = "Kalt03b.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt00a, + author = "Kaltofen, E. and Lee, W.-s. and Lobo, A.A.", + title = "Early termination in {Ben-Or/Tiwari} sparse interpolation + and a hybrid of {Zippel}'s algorithm", + booktitle = "Proc. 2000 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC2K", + pages = "192--201", + year = "2000", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/KLL2K.pdf", + paper = "Kalt00a.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt10b, + author = "Kaltofen, Erich L.", + title = "Fifteen years after {DSC} and {WLSS2} {What} parallel + computations {I} do today [{Invited} Lecture at {PASCO} 2010]", + year = "2010", + booktitle = "Proc. 2010 Internat. Workshop on Parallel Symbolic Comput.", + crossref = "PASCO10", + pages = "10--17", + month = "July", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_pasco.pdf", + paper = "Kalt10b.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt90, + author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.", + editor = "S. Watanabe and M. Nagata", + title = "Modular rational sparse multivariate polynomial interpolation", + booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.", + pages = "135--139", + publisher = "ACM Press", + year = "1990", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf", + paper = "Kalt90.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt88a, + author = "Kaltofen, E. and Yagati, Lakshman", + title = "Improved sparse multivariate polynomial interpolation algorithms", + booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.", + crossref = "ISSAC88", + pages = "467--474", + year = "1988", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf", + paper = "Kalt88a.pdf" } - + \end{chunk} +\section{Divisions and Algebraic Complexity} %%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@article{Brad02, - author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and - Jeffrey, DavidJ. and Watt, StephenM.", - title="Reasoning about the Elementary Functions of Complex Analysis", - journal="Annals of Mathematics and Artificial Intelligence", - year="2002", - issn="1012-2443", - volume="36", - number="3", - doi="10.1023/A:1016007415899", - url="http://dx.doi.org/10.1023/A%3A1016007415899", - publisher="Kluwer Academic Publishers", - keywords="elementary functions; branch cuts; complex identities", - pages="303-318", - paper = "Brad02.pdf", - abstract = " - There are many problems with the simplification of elementary - functions, particularly over the complex plane, though not - exclusively. Systems tend to make howlers'' or not to simplify - enough. In this paper we outline the unwinding number'' approach to - such problems, and show how it can be used to prevent errors and to - systematise such simplification, even though we have not yet reduced - the simplification process to a complete algorithm. The unsolved - problems are probably more amenable to the techniques of artificial - intelligence and theorem proving than the original problem of complex - variable analysis." +@InCollection{Gren11, + author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal + and Portier, Natacha", + title = "Symmetric Determinantal Representation of Formulas and Weakly + Skew Circuits", + booktitle = "Randomization, Relaxation, and Complexity in Polynomial + Equation Solving", + year = "2011", + editor = "Leonid Gurvits and Philippe P\'{e}bay and J. Maurice Rojas + and David Thompson", + pages = "61--96", + publisher = "American Mathematical Society", + address = "Providence, Rhode Island, USA", + isbn = "978-0-8218-5228-6", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/GKKP10.pdf", + paper = "Gren11.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@inproceedings{Chyz11, - author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno", - title = "On Kahan's Rules for Determining Branch Cuts", - booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing", - year = "2011", - isbn = "978-1-4673-0207-4", - location = "Timisoara", - pages = "47-51", - doi = "10.1109/SYNASC.2011.51", - acmid = "258794", - publisher = "IEEE", - paper = "Chyz11.pdf", - abstract = " - In computer algebra there are different ways of approaching the - mathematical concept of functions, one of which is by defining them as - solutions of differential equations. We compare different such - appraoches and discuss the occurring problems. The main focus is on - the question of determining possible branch cuts. We explore the - extent to which the treatment of branch cuts can be rendered (more) - algorithmic, by adapting Kahan's rules to the differential equation - setting." +@InProceedings{Kalt08a, + author = "Kaltofen, Erich and Koiran, Pascal", + title = "Expressing a Fraction of Two Determinants as a Determinant", + year = "2008", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08", + crossref = "ISSAC08", + pages = "141--146", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaKoi08.pdf", + paper = "Kalt08a.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Dave10, - author = "Davenport, James", - title = {The Challenges of Multivalued "Functions"}, - journal = "Lecture Notes in Computer Science", - volume = "6167", - year = "2010", - pages = "1-12", - paper = "Dave10.pdf", - abstract = " - Although, formally, mathematics is clear that a function is a - single-valued object, mathematical practice is looser, particularly - with n-th roots and various inverse functions. In this paper, we point - out some of the looseness, and ask what the implications are, both for - Artificial Intelligence and Symbolic Computation, of these practices. - In doing so, we look at the steps necessary to convert existing tests - into - \begin{itemize} - \item (a) rigorous statements - \item (b) rigorously proved statements - \end{itemize} - In particular we ask whether there might be a constant de Bruij factor'' -  as we make these texts more formal, and conclude that the answer - depends greatly on the interpretation being placed on the symbols." +@Article{Hitz95, + author = "Kitz, M.A. and Kaltofen, E.", + title = "Integer division in residue number systems", + journal = "IEEE Trans. Computers", + year = "1995", + volume = "44", + number = "8", + pages = "983--989", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/HiKa95.pdf", + paper = "Hitz95.pdf" } - + \end{chunk} \begin{chunk}{axiom.bib} -@article{Dave12, - author = "Davenport, James H. and Bradford, Russell and England, Matthew - and Wilson, David", - title = "Program Verification in the presence of complex numbers, functions - with branch cuts etc", - journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for - Scientific Computing", - year = "2012", - series = "SYNASC'12", - pages = "83-88", - publisher = "IEEE", - paper = "Dave12.pdf", - abstract = " - In considering the reliability of numerical programs, it is normal to - limit our study to the semantics dealing with numerical precision''. - On the other hand, there is a great deal of work on the reliability of - programs that essentially ignores the numerics. The thesis of this - paper is that there is a class of problems that fall between the two, - which could be described as does the low-level arithmetic implement - the high-level mathematics''. Many of these problems arise because - mathematics, particularly the mathematics of the complex numbers, is - more difficult than expected; for example the complex function log is - not continuous, writing down a program to compute an inverse function - is more complicated than just solving an equation, and many algebraic - simplification rules are not universally valid. +@InProceedings{Kalt92a, + author = "Kaltofen, E.", + title = "On computing determinants of matrices without divisions", + booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC92", + pages = "342--349", + year = "1992", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf", + paper = "Kalt92a.pdf" +} - The good news is that these problems are theoretically capable of - being solved, and are practically close to being solved, but not yet - solved, in several real-world examples. However, there is still a long - way to go before implementations match the theoretical possibilities." +\end{chunk} + +\begin{chunk}{axiom.bib} +@Article{Cant91, + author = "Cantor, D.G. and Kaltofen, E.", + title = "On fast multiplication of polynomials over arbitrary algebras", + journal = "Acta Inform.", + year = "1991", + volume = "28", + number = "7", + pages = "693--701", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf", + paper = "Cant91.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Jeff04, - author = "Jeffrey, D. J. and Norman, A. C.", - title = "Not Seeing the Roots for the Branches: Multivalued Functions in - Computer Algebra", - journal = "SIGSAM Bull.", - issue_date = "September 2004", - volume = "38", - number = "3", - month = "September", - year = "2004", - issn = "0163-5824", - pages = "57--66", - numpages = "10", - url = "http://doi.acm.org/10.1145/1040034.1040036", - doi = "10.1145/1040034.1040036", - acmid = "1040036", - publisher = "ACM", - address = "New York, NY, USA", - paper = "Jeff04.pdf", - abstract = " - We discuss the multiple definitions of multivalued functions and their - suitability for computer algebra systems. We focus the discussion by - taking one specific problem and considering how it is solved using - different definitions. Our example problem is the classical one of - calculating the roots of a cubic polynomial from the Cardano formulae, - which contains fractional powers. We show that some definitions of - these functions result in formulae that are correct only in the sense - that they give candidates for solutions; these candidates must then be - tested. Formulae that are based on single-valued functions, in - contract, are efficient and direct." +@Article{Kalt88b, + author = "Kaltofen, E.", + title = "Greatest common divisors of polynomials given by + straight-line programs", + journal = "J. ACM", + year = "1988", + volume = "35", + number = "1", + pages = "231--264", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf", + paper = "Kalt88b.pdf" } \end{chunk} +\section{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@inproceedings{Kaha86, - author = "Kahan, W.", - title = "Branch cuts for complex elementary functions", - booktitle = "The State of the Art in Numerical Analysis", - year = "1986", - month = "April", - editor = "Powell, M.J.D and Iserles, A.", - publisher = "Oxford University Press" +@PhdThesis{Kalt82, + author = "Kaltofen, E.", + title = "On the complexity of factoring polynomials with integer + coefficients", + school = "RPI", + address = "Troy, N. Y.", + year = "1982", + month = "December", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf", + paper = "Kalt82.pdf" } -\end{chunk} +\end{chunk} \begin{chunk}{axiom.bib} -@article{Rich96, - author = "Rich, Albert D. and Jeffrey, David J.", - title = "Function Evaluation on Branch Cuts", - journal = "SIGSAM Bull.", - issue_date = "June 1996", - volume = "30", - number = "2", - month = "June", - year = "1996", - issn = "0163-5824", - pages = "25--27", - numpages = "3", - url = "http://doi.acm.org/10.1145/235699.235704", - doi = "10.1145/235699.235704", - acmid = "235704", - publisher = "ACM", - address = "New York, NY, USA", - abstract = " - Once it is decided that a CAS will evaluate multivalued functions on - their principal branches, questions arise concerning the branch - definitions. The first questions concern the standardization of the - positions of the branch cuts. These questions have largely been - resolved between the various algebra systems and the numerical - libraries, although not completely. In contrast to the computer - systems, many mathematical textbooks are much further behind: for - example, many popular textbooks still specify that the argument of a - complex number lies between 0 and $2\pi$. We do not intend to discuss - these first questions here, however. Once the positions of the branch - cuts have been fixed, a second set of questions arises concerning the - evaluation of functions on their branch cuts." +@Article{Gath85, + author = "Gathen, Joachim von zur; Kaltofen, E.", + title = "Factoring sparse multivariate polynomials", + journal = "J. Comput. System Sci.", + year = "1985", + volume = "31", + pages = "265--287", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz", + paper = "Gath85.ps" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InCollection{Kalt11c, + author = "Kaltofen, Erich and Lecerf, Gr{\'e}goire", + title = "Section 11.5. {Factorization} of multivariate polynomials", + booktitle = "Handbook of Finite Fields", + crossref = "HFF11", + pages = "382--392", + year = "2011", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KL11.pdf", + paper = "Kalt11c.pdf" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt05b, + author = "Kaltofen, Erich and Koiran, Pascal", + title = "On the complexity of factoring bivariate supersparse + (lacunary) polynomials", + year = "2005", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05", + crossref = "ISSAC05", + pages = "208--215", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KaKoi05.pdf", + paper = "Kalt05b.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Patt96, - author = "Patton, Charles M.", - title = "A Representation of Branch-cut Information", - journal = "SIGSAM Bull.", - issue_date = "June 1996", - volume = "30", - number = "2", - month = "June", - year = "1996", - issn = "0163-5824", - pages = "21--24", - numpages = "4", - url = "http://doi.acm.org/10.1145/235699.235703", - doi = "10.1145/235699.235703", - acmid = "235703", - publisher = "ACM", - address = "New York, NY, USA", - paper = "Patt96.pdf", - abstract = " - Handling (possibly) multi-valued functions is a problem in all current - computer algebra systems. The problem is not an issue of technology. - Its solution, however, is tied to a uniform handling of the issues by - the mathematics community." +@InProceedings{Kalt06a, + author = "Kaltofen, Erich and Koiran, Pascal", + title = "Finding Small Degree Factors of Multivariate Supersparse + (Lacunary) Polynomials Over Algebraic Number Fields", + year = "2006", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06", + crossref = "ISSAC06", + pages = "162--168", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaKoi06.pdf", + paper = "Kalt06a.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Squi91, - author = "Squire, Jon S.", - title = "Rationale for the Proposed Standard for a Generic Package of - Complex Elementary Functions", - journal = "Ada Lett.", - issue_date = "Fall 1991", - volume = "XI", - number = "7", - month = "September", - year = "1991", - issn = "1094-3641", - pages = "166--179", - numpages = "14", - url = "http://doi.acm.org/10.1145/123533.123545", - doi = "10.1145/123533.123545", - acmid = "123545", - publisher = "ACM", - address = "New York, NY, USA", - paper = "Squi91.pdf", - abstract = " - This document provides the background on decisions that were made - during the development of the specification for Generic Complex - Elementary fuctions. It also rovides some information that was used to - develop error bounds, range, domain and definitions of complex - elementary functions." +@InProceedings{Kalt97a, + author = "Kaltofen, E. and Shoup, V.", + title = "Fast polynomial factorization over high algebraic extensions of + finite fields", + booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC97", + year = "1997", + pages = "184--188", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf", + paper = "Kalt97a.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Squi91a, - editor = "Squire, Jon S.", - title = "Proposed Standard for a Generic Package of Complex - Elementary Functions", - journal = "Ada Lett.", - issue_date = "Fall 1991", - volume = "XI", - number = "7", - month = "September", - year = "1991", - issn = "1094-3641", - pages = "140--165", - numpages = "26", - url = "http://doi.acm.org/10.1145/123533.123544", - doi = "10.1145/123533.123544", - acmid = "123544", - publisher = "ACM", - address = "New York, NY, USA", - abstract = " - This document defines the specification of a generic package of - complex elementary functions called Generic Complex Elementary - Functions. It does not provide the body of the package." +@Article{Kalt98, + author = "Kaltofen, E. and Shoup, V.", + title = "Subquadratic-time factoring of polynomials over finite fields", + journal = "Math. Comput.", + month = "July", + year = "1998", + volume = "67", + number = "223", + pages = "1179--1197", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/KaSh98.pdf", + paper = "Kalt98.pdf" } \end{chunk} -\subsection{Square-free Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{axiom.bib} -@article{Bern97, - author = "Bernardin, Laurent", - title = "On square-free factorization of multivariate polynomials over a - finite field", - journal = "Theoretical Computer Science", - volume = "187", - number = "1-2", - year = "1997", - month = "November", - pages = "105-116", - keywords = "axiomref", - paper = "Bern97.pdf", - abstract = " - In this paper we present a new deterministic algorithm for computing - the square-free decomposition of multivariate polynomials with - coefficients from a finite field. - - Our algorithm is based on Yun's square-free factorization algorithm - for characteristic 0. The new algorithm is more efficient than - existing, deterministic algorithms based on Musser's squarefree - algorithm +@InProceedings{Kalt95a, + author = "Kaltofen, E. and Shoup, V.", + title = "Subquadratic-time factoring of polynomials over finite fields", + booktitle = "Proc. 27th Annual ACM Symp. Theory Comput.", + year = "1995", + publisher = "ACM Press", + address = "New York, N.Y.", + pages = "398--406", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/KaSh95.ps.gz", + paper = "Kalt95a.ps" +} - We will show that the modular approach presented by Yun has no - significant performance advantage over our algorithm. The new - algorithm is also simpler to implement and it can rely on any existing - GCD algorithm without having to worry about choosing good'' evaluation - points. +\end{chunk} - To demonstrate this, we present some timings using implementations in - Maple (Char et al. 1991), where the new algorithm is used for Release - 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system - known to the author to use and implementation of Yun's modular - algorithm mentioned above." +\begin{chunk}{axiom.bib} +@InProceedings{Diaz95, + author = "Diaz, A. and Kaltofen, E.", + title = "On computing greatest common divisors with polynomials given by + black boxes for their evaluation", + booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC95", + pages = "232--239", + year = "1995", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DiKa95.ps.gz", + paper = "Diaz95.ps" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Chez07, - author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire", - title = "Lifting and recombination techniques for absolute factorization", - journal = "Journal of Complexity", - volume = "23", - number = "3", - year = "2007", - month = "June", - pages = "380-420", - paper = "Chez07.pdf", - abstract = " - In the vein of recent algorithmic advances in polynomial factorization - based on lifting and recombination techniques, we present new faster - algorithms for computing the absolute factorization of a bivariate - polynomial. The running time of our probabilistic algorithm is less - than quadratic in the dense size of the polynomial to be factored." +@InProceedings{Kalt88, + author = "Kaltofen, E. and Trager, B.", + title = "Computing with polynomials given by black boxes for their + evaluations: Greatest common divisors, factorization, separation of + numerators and denominators", + booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.", + pages = "296--305", + year = "1988", + organization = "IEEE", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz", + paper = "Kalt88.ps" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Lece07, - author = "Lecerf, Gr\'egoire", - title = "Improved dense multivariate polynomial factorization algorithms", - journal = "Journal of Symbolic Computation", - volume = "42", - number = "4", - year = "2007", - month = "April", - pages = "477-494", - paper = "Lece07.pdf", - abstract = " - We present new deterministic and probabilistic algorithms that reduce - the factorization of dense polynomials from several variables to one - variable. The deterministic algorithm runs in sub-quadratic time in - the dense size of the input polynomial, and the probabilistic - algorithm is softly optimal when the number of variables is at least - three. We also investigate the reduction from several to two variables - and improve the quantitative versions of Bertini's irreducibility theorem." +@InProceedings{Kalt85b, + author = "Kaltofen, E.", + title = "Computing with polynomials given by straight-line programs {II}; + sparse factorization", + booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.", + year = "1985", + pages = "451--458", + organization = "IEEE", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz", + paper = "Kalt85b.ps" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Wang77, - author = "Wang, Paul S.", - title = "An efficient squarefree decomposition algorithm", - journal = "ACM SIGSAM Bulletin", - volume = "11", - number = "2", - year = "1977", - month = "May", - pages = "4-6", - paper = "Wang77.pdf", - abstract = " - The concept of polynomial squarefree decomposition is an important one - in algebraic computation. The squarefree decomposition process has - many uses in computer symbolic computation. A recent survey by D. Yun -  describes many useful algorithms for this purpose. All of these - methods depend on computing the greated common divisor (gcd) of the - polynomial to be decomposed and its first derivative (with repect to - some variable). In the multivariate case, this gcd computation is - non-trivial and dominates the cost for the squarefree decompostion." +@InProceedings{Kalt86, + author = "Kaltofen, E.", + title = "Uniform closure properties of p-computable functions", + booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.", + year = "1986", + pages = "330--337", + organization = "ACM", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf", + paper = "Kalt86.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@article{Wang79, - author = "Wang, Paul S. and Trager, Barry M.", - title = "New Algorithms for Polynomial Square-Free Decomposition - over the Integers", - journal = "SIAM Journal on Computing", - volume = "8", - number = "3", - year = "1979", - publisher = "Society for Industrial and Applied Mathematics", - issn = "00975397", - paper = "Wang79.pdf", - abstract = " - Previously known algorithms for polynomial square-free decomposition - rely on greatest common divisor (gcd) computations over the same - coefficient domain where the decomposition is to be performed. In - particular, gcd of the given polynomial and its first derivative (with - respect to some variable) is obtained to begin with. Application of - modular homomorphism and $p$-adic construction (multivariate case) or - the Chinese remainder algorithm (univariate case) results in new - square-free decomposition algorithms which, generally speaking, take - less time than a single gcd between the given polynomial and its first - derivative. The key idea is to obtain one or several correct'' - homomorphic images of the desired square-free decomposition - first. This provides information as to how many different square-free - factors there are, their multiplicities and their homomorphic - images. Since the multiplicities are known, only the square-free - factors need to be constructed. Thus, these new algorithms are - relatively insensitive to the multiplicities of the square-free factors." +@InProceedings{Kalt87b, + author = "Kaltofen, E.", + title = "Single-factor Hensel lifting and its application to the + straight-line complexity of certain polynomials", + booktitle = "Proc. 19th Annual ACM Symp. Theory Comput.", + year = "1987", + pages = "443--452", + organization = "ACM", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf", + paper = "Kalt87b.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@inproceedings{Yun76, - author = "Yun, D.Y.Y", - title = "On square-free decomposition algorithms", - booktitle = "Proceedings of SYMSAC'76", - year = "1976", - keywords = "survey", - pages = "26-35" +@InCollection{Kalt89, + author = "Kaltofen, E.", + editor = "S. Micali", + title = "Factorization of polynomials given by straight-line programs", + booktitle = "Randomness and Computation", + pages = "375--412", + publisher = "JAI Press Inc.", + year = "1989", + volume = "5", + series = "Advances in Computing Research", + address = "Greenwhich, Connecticut", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf", + paper = "Kalt89.pdf" } \end{chunk} -\section{Axiom Citations in the Literature} - -\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{axiom.bib} +@Article{Gao04, + author = "Gao, Shuhong and Kaltofen, E. and Lauder, A.", + title = "Deterministic distinct degree factorization for polynomials + over finite fields", + year = "2004", + journal = "Journal of Symbolic Computation", + volume = "38", + number = "6", + pages = "1461--1470", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/GKL01.pdf", + paper = "Gao04.pdf" +} -\begin{chunk}{ignore} -\bibitem[ACM 89]{ACM89} ACM, editor -Proceedings of the ACM-SIGSAM 1989 International -Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press, -New York, NY 10036, USA, 1989, , LCCN QA76.95.I59 - year = "1989", - isbn = "0-89791-325-6", - keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[ACM 94]{ACM94} ACM, editor -ISSAC '94. Proceedings of the International -Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY, -10036, USA, 1994, . LCCN QA76.95.I59 - year = "1994", - isbn = "0-89791-638-7", - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Kalt87c, + author = "Kaltofen, E.", + title = "Deterministic irreducibility testing of polynomials over + large finite fields", + journal = "Journal of Symbolic Computation", + year = "1987", + volume = "4", + pages = "77--82", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz", + paper = "Kalt87c.ps" +} \end{chunk} \begin{chunk}{axiom.bib} -@article{Augo91, - author = "Augot, D. and Charpin, P. and Sendrier, N.", - title = "The miniumum distance of some binary codes via the - Newton's identities", - journal = "Cohen and Charping [CC91]", - year = "1991", - pages = "65-73", - isbn = "0-387-54303-1", - misc = "3-540-54303-1 (Berlin). LCCN QA268.E95 1990", - keywords = "axiomref", - paper = "Augo91.pdf" +@Article{Kalt95b, + author = "Kaltofen, E.", + title = "Effective {Noether} irreducibility forms and applications", + journal = "J. Comput. System Sci.", + year = "1995", + volume = "50", + number = "2", + pages = "274--295", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_jcss.pdf", + paper = "Kalt95b.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Adams 94]{AL94} - author = "Adams, William W. and Loustaunau, Philippe", - title = "An Introduction to Gr\"obner Bases", - year = "1994", -American Mathematical Society (1994) - isbn = "0-8218-3804-0", - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Kalt85a, + author = "Kaltofen, E.", + title = "Fast parallel absolute irreducibility testing", + journal = "Journal of Symbolic Computation", + year = "1985", + volume = "1", + number = "1", + pages = "57--67", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf", + paper = "Kalt85a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Andrews 84]{And84} - author = "Andrews, George E.", - title = "Ramanujan and SCRATCHPAD", - year = "1984", - pages = "383-??", - keywords = "axiomref", -In Golden and Hussain [GH84] - -\end{chunk} +\begin{chunk}{axiom.bib} +@Article{Gath85a, + author = "{von zur Gathen}, Joachim and Kaltofen, E.", + title = "Factoring multivariate polynomials over finite fields", + journal = "Math. Comput.", + year = "1985", + volume = "45", + pages = "251--261", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz", + paper = "Gath85a.ps" +} -\begin{chunk}{ignore} -\bibitem[Andrews 88]{And88} - author = "Andrews, G. E.", - title = "Application of Scratchpad to problems in special functions and - combinatorics", - year = "1988" - pages = "158-??", - isbn = "3-540-18928-9", - keywords = "axiomref", -In Janssen [Jan88], pages 158-?? ISBN -0-387-18928-9 LCCN QA155.7.E4T74 +\begin{chunk}{axiom.bib} +@Article{Kalt85e, + author = "Kaltofen, E.", + title = "Polynomial-time reductions from multivariate to bi- and univariate + integral polynomial factorization", + journal = "{SIAM} J. Comput.", + year = "1985", + volume = "14", + number = "2", + pages = "469--489", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf", + paper = "Kalt85e.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Anon 91]{Ano91} - author = "Anonymous", - year = "1991, - keywords = "axiomref", -Proceedings 1991 Annual Conference, American Society for -Engineering Education. Challenges of a Changing World. ASEE, Washington, DC - 2 vol. +\begin{chunk}{axiom.bib} +@InProceedings{Kalt82a, + author = "Kaltofen, E.", + title = "A polynomial-time reduction from bivariate to univariate + integral polynomial factorization", + booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.", + year = "1982", + pages = "57--64", + organization = "IEEE", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf", + paper = "Kalt82a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Anon 92]{Ano92} - author = "Anonymous", - year = "1992", - keywords = "axiomref", -Programming environments for high-level scientific problem solving. -IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science -and Technology, A-2:??, CODEN ITATEC. ISSN 0926-5473 - -\end{chunk} +\begin{chunk}{axiom.bib} +@InProceedings{Kalt03, + author = "Kaltofen, Erich", + title = "Polynomial Factorization: a Success Story", + year = "2003", + booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.", + crossref = "ISSAC03", + pages = "3--4", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/Ka03.pdf", + keywords = "survey", + paper = "Kalt03.pdf" +} -\begin{chunk}{ignore} -\bibitem[Anono 95]{Ano95} - author =Anonymous - keywords = "axiomref", - year = "1995", -GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und -Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 0044-2267 +\begin{chunk}{axiom.bib} +@InProceedings{Kalt92b, + author = "Kaltofen, E.", + title = "Polynomial factorization 1987-1991", + booktitle = "Proc. LATIN '92", + editor = "I. Simon", + series = "Lect. Notes Comput. Sci.", + volume = "583", + pages = "294--313", + publisher = "Springer-Verlag", + year = "1992", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf", + keywords = "survey", + paper = "Kalt92b.pdf" +} \end{chunk} -\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{axiom.bib} -@article{Bacl14, - author = "Baclawski, Krystian", - title = "SPAD language type checker", - journal = "unknown", - year = "2014", - url = "http://github.com/cahirwpz/phd", - keywords = "axiomref", - abstract = " - The project aims to deliver a new type checker for SPAD language. - Several improvements over current type checker are planned. - \begin{itemize} - \item introduce better type inference - \item introduce modern language constructs - \item produce understandable diagnostic messages - \item eliminate well known bugs in the type system - \item find new type errors - \end{itemize}" +@InCollection{Kalt90c, + author = "Kaltofen, E.", + editor = "D. V. Chudnovsky and R. D. Jenks", + title = "Polynomial Factorization 1982-1986", + booktitle = "Computers in Mathematics", + pages = "285--309", + publisher = "Marcel Dekker, Inc.", + year = "1990", + volume = "125", + series = "Lecture Notes in Pure and Applied Mathematics", + address = "New York, N. Y.", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz", + keywords = "survey", + paper = "Kalt90c.ps" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Blair 70]{BGJ70} - author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.", - title = "An interactive facility for symbolic mathematics", - year = "1970", - pages = "394-419", - keywords = "axiomref", -Proc. International Computing Symposium, Bonn, Germany, +\begin{chunk}{axiom.bib} +@InCollection{Kalt82b, + author = "Kaltofen, E.", + title = "Polynomial factorization", + editor = "B. Buchberger and G. Collins and R. Loos", + booktitle = "Computer Algebra", + edition = "2", + pages = "95--113", + publisher = "Springer-Verlag", + year = "1982", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz", + keywords = "survey", + paper = "Kalt82b.ps" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Blair 70a]{BJ70} - author = "Blair, Fred W. and Jenks, Richard D.", - title = "LPL: LISP programming language", - year = "1970", - keywords = "axiomref", -IBM Research Report, RC3062 Sept - -\end{chunk} +\section{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{axiom.bib} -\bibitem[Broadbery 95]{BGDW95} - author = "Broadbery, P. A. and G{\'o}mez-D{\'\i}az, T. and Watt, S. M.", - title = "On the Implementation of Dynamic Evaluation", - year = "1995", - pages = "77-84", - keywords = "axiomref", - isbn = "0-89791-699-9", - url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf", - paper = "BGDW95.pdf", +@article{Beau03, + author = "Beaumont, James and Bradford, Russell and Davenport, James H.", + title = "Better simplification of elementary functions through power series", + journal = "2003 International Symposium on Symbolic and Algebraic Computation", + series = "ISSAC'03", + year = "2003", + month = "August", + paper = "Beau03.pdf", abstract = " - Dynamic evaluation is a technique for producing multiple results - according to a decision tree which evolves with program execution. - Sometimes it is desired to produce results for all possible branches - in the decision tree, while on other occasions, it may be sufficient - to compute a single result which satisfies certain properties. This - techinique finds use in computer algebra where computing the correct - result depends on recognizing and properly handling special cases of - parameters. In previous work, programs using dynamic evaluation have - explored all branches of decision trees by repeating the computations - prior to decision points. + In , we introduced an algorithm for deciding whether a proposed + simplification of elementary functions was correct in the presence of + branch cuts. This algorithm used multivalued function simplification + followed by verification that the branches were consistent. - This paper presents two new implementations of dynamic evaluation - which avoid recomputing intermediate results. The first approach uses - Scheme continuations'' to record state for resuming program - execution. The second implementation uses the Unix fork'' operation - to form new processes to explore alternative branches in parallel." + In  an algorithm was presented for zero-testing functions defined + by ordinary differential equations, in terms of their power series. + + The purpose of the current paper is to investigate merging the two + techniques. In particular, we will show an explicit reduction to the + constant problem ." } \end{chunk} \begin{chunk}{axiom.bib} -\bibitem[Boehm 89]{Boe89} -@inproceedings{Boe89, - author = "Boehm, Hans-J.", - title = "Type Inference in the Presence of Type Abstraction", - year = "1989", - pages = "192-206", - keywords = "axiomref", - url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192-boehm", - paper = "Boe89.pdf", - booktitle = "ACM SIGPLAN Notices", - volume = "24", - number = "7", - month = "July", +@article{Beau07, + author = "Beaumont, James C. and Bradford, Russell J. and + Davenport, James H. and Phisanbut, Nalina", + title = "Testing elementary function identities using CAD", + journal = "Applicable Algebra in Engineering, Communication and Computing", + year = "2007", + volume = "18", + number = "6", + issn = "0938-1279", + publisher = "Springer-Verlag", + pages = "513-543", + paper = "Beau07.pdf", abstract = " - A number of recent programming language designs incorporate a type - checking system based on the Girard-Reynolds polymorphic - $\lambda$-calculus. This allows the construction of general purpose, - reusable software without sacrificing compile-time type checking. A - major factor constraining the implementation of these languages is the - difficulty of automatically inferring the lengthy type information - that is otherwise required if full use is made of these - languages. There is no known algorithm to solve any natural and fully - general formulation of the type inference'' problem. One very - reasonable formulation of the problem is known to be undecidable. - - Here we define a restricted version of the type inference problem and - present an efficient algorithm for its solution. We argue that the - restriction is sufficiently weak to be unobtrusive in practice." + One of the problems with manipulating function identities in computer + algebra systems is that they often involve functions which are + multivalued, whilst most users tend to work with single-valued + functions. The problem is that many well-known identities may no + longer be true everywhere in the complex plane when working with their + single-valued counterparts. Conversely, we cannot ignore them, since + in particular contexts they may be valid. We investigate the + practicality of a method to verify such identities by means of an + experiment; this is based on a set of test examples which one might + realistically meet in practice. Essentially, the method works as + follows. We decompose the complex plane via means of cylindrical + algebraic decomposition into regions with respect to the branch cuts + of the functions. We then test the identity numerically at a sample + point in the region. The latter step is facilitated by the notion of + the {\sl adherence} of a branch cut, which was previously introduced + by the authors. In addition to presenting the results of the + experiment, we explain how adherence relates to the proposal of + {\sl signed zeros} by W. Kahan, and develop this idea further in order to + allow us to cover previously untreatable cases. Finally, we discuss + other ways to improve upon our general methodology as well as topics + for future research." } - + \end{chunk} \begin{chunk}{axiom.bib} -@inproceedings{BHGM04, - author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne - and Martin, Ursula", - title = "Design verification for control engineering", - year = "2004", - month = "April", - booktitle = "Proc 4th Int. Conf. on Integrated Formal Methods", - keywords = "axiomref", +@article{Brad02, + author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and + Jeffrey, DavidJ. and Watt, StephenM.", + title="Reasoning about the Elementary Functions of Complex Analysis", + journal="Annals of Mathematics and Artificial Intelligence", + year="2002", + issn="1012-2443", + volume="36", + number="3", + doi="10.1023/A:1016007415899", + url="http://dx.doi.org/10.1023/A%3A1016007415899", + publisher="Kluwer Academic Publishers", + keywords="elementary functions; branch cuts; complex identities", + pages="303-318", + paper = "Brad02.pdf", abstract = " - We introduce control engineering as a new domain of application for - formal methods. We discuss design verification, drawing attention to - the role played by diagrammatic evaluation criteria involving numeric - plots of a design, such as Nichols and Bode plots. We show that - symbolic computation and computational logic can be used to discharge - these criteria and provide symbolic, automated, and very general - alternatives to these standard numeric tests. We illustrate our work - with reference to a standard reference model drawn from military - avionics." + There are many problems with the simplification of elementary + functions, particularly over the complex plane, though not + exclusively. Systems tend to make howlers'' or not to simplify + enough. In this paper we outline the unwinding number'' approach to + such problems, and show how it can be used to prevent errors and to + systematise such simplification, even though we have not yet reduced + the simplification process to a complete algorithm. The unsolved + problems are probably more amenable to the techniques of artificial + intelligence and theorem proving than the original problem of complex + variable analysis." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Boulanger 91]{Bou91} - author = "Boulanger, Jean-Louis", - title = "Etude de la compilation de scratchpad 2", - year = "1991", - month = "September", -Rapport de DEA Universite dl lille 1 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@inproceedings{Chyz11, + author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno", + title = "On Kahan's Rules for Determining Branch Cuts", + booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing", + year = "2011", + isbn = "978-1-4673-0207-4", + location = "Timisoara", + pages = "47-51", + doi = "10.1109/SYNASC.2011.51", + acmid = "258794", + publisher = "IEEE", + paper = "Chyz11.pdf", + abstract = " + In computer algebra there are different ways of approaching the + mathematical concept of functions, one of which is by defining them as + solutions of differential equations. We compare different such + appraoches and discuss the occurring problems. The main focus is on + the question of determining possible branch cuts. We explore the + extent to which the treatment of branch cuts can be rendered (more) + algorithmic, by adapting Kahan's rules to the differential equation + setting." +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{Bou93a, - author = "Boulanger, Jean-Louis", - title = "Axiom, language fonctionnel \a d\'evelopement objet", - year = "1993", - month = "October", - paper = "Bou93a.pdf", - keywords = "axiomref" +@article{Dave10, + author = "Davenport, James", + title = {The Challenges of Multivalued "Functions"}, + journal = "Lecture Notes in Computer Science", + volume = "6167", + year = "2010", + pages = "1-12", + paper = "Dave10.pdf", + abstract = " + Although, formally, mathematics is clear that a function is a + single-valued object, mathematical practice is looser, particularly + with n-th roots and various inverse functions. In this paper, we point + out some of the looseness, and ask what the implications are, both for + Artificial Intelligence and Symbolic Computation, of these practices. + In doing so, we look at the steps necessary to convert existing tests + into + \begin{itemize} + \item (a) rigorous statements + \item (b) rigorously proved statements + \end{itemize} + In particular we ask whether there might be a constant de Bruij factor'' +  as we make these texts more formal, and conclude that the answer + depends greatly on the interpretation being placed on the symbols." } - + \end{chunk} \begin{chunk}{axiom.bib} -@misc{Bou93b, - author = "Boulanger, Jean-Louis", - title = "AXIOM, A Functional Language with Object Oriented Development", - year = "1993", - paper = "Bou93b.pdf", - keywords = "axiomref", +@article{Dave12, + author = "Davenport, James H. and Bradford, Russell and England, Matthew + and Wilson, David", + title = "Program Verification in the presence of complex numbers, functions + with branch cuts etc", + journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for + Scientific Computing", + year = "2012", + series = "SYNASC'12", + pages = "83-88", + publisher = "IEEE", + paper = "Dave12.pdf", abstract = " - We present in this paper, a study about the computer algebra system - Axiom, which gives us many very interesting Software engineering - concepts. This language is a functional language with an Object - Oriented Development. This feature is very important for modeling the - mathematical world (Hierarchy) and provides a running with - mathematical sense. (All objects are functions). We present many - problems of running and development in Axiom. We can note that Aiom is - the only system of this category." + In considering the reliability of numerical programs, it is normal to + limit our study to the semantics dealing with numerical precision''. + On the other hand, there is a great deal of work on the reliability of + programs that essentially ignores the numerics. The thesis of this + paper is that there is a class of problems that fall between the two, + which could be described as does the low-level arithmetic implement + the high-level mathematics''. Many of these problems arise because + mathematics, particularly the mathematics of the complex numbers, is + more difficult than expected; for example the complex function log is + not continuous, writing down a program to compute an inverse function + is more complicated than just solving an equation, and many algebraic + simplification rules are not universally valid. + + The good news is that these problems are theoretically capable of + being solved, and are practically close to being solved, but not yet + solved, in several real-world examples. However, there is still a long + way to go before implementations match the theoretical possibilities." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Boulanger 94]{Bou94} - author = "Boulanger, J.L.", - title = "Object Oriented Method for Axiom", - year = "1995", - month = "February", - pages = "33-41", - paper = "Bou94.pdf", -ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 0362-1340 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@article{Jeff04, + author = "Jeffrey, D. J. and Norman, A. C.", + title = "Not Seeing the Roots for the Branches: Multivalued Functions in + Computer Algebra", + journal = "SIGSAM Bull.", + issue_date = "September 2004", + volume = "38", + number = "3", + month = "September", + year = "2004", + issn = "0163-5824", + pages = "57--66", + numpages = "10", + url = "http://doi.acm.org/10.1145/1040034.1040036", + doi = "10.1145/1040034.1040036", + acmid = "1040036", + publisher = "ACM", + address = "New York, NY, USA", + paper = "Jeff04.pdf", abstract = " - Axiom is a very powerful computer algebra system which combines two - language paradigms (functional and OOP). Mathematical world is complex - and mathematicians use abstraction to design it. This paper presents - some aspects of the object oriented development in Axiom. The Axiom - programming is based on several new tools for object oriented - development, it uses two levels of class and some operations such that - {\sl coerce}, {\sl retract}, or {\sl convert} which permit the type - evolution. These notions introduce the concept of multi-view." + We discuss the multiple definitions of multivalued functions and their + suitability for computer algebra systems. We focus the discussion by + taking one specific problem and considering how it is solved using + different definitions. Our example problem is the classical one of + calculating the roots of a cubic polynomial from the Cardano formulae, + which contains fractional powers. We show that some definitions of + these functions result in formulae that are correct only in the sense + that they give candidates for solutions; these candidates must then be + tested. Formulae that are based on single-valued functions, in + contract, are efficient and direct." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 87]{Bro87} - author = "Bronstein, Manuel", - title = "Integration of Algebraic and Mixed Functions", - year = "1987", -in [Wit87], p18 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@inproceedings{Kaha86, + author = "Kahan, W.", + title = "Branch cuts for complex elementary functions", + booktitle = "The State of the Art in Numerical Analysis", + year = "1986", + month = "April", + editor = "Powell, M.J.D and Iserles, A.", + publisher = "Oxford University Press" +} -\end{chunk} +\end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 89]{Bro89} - author= "Bronstein, M.", - title = "Simplification of real elementary functions", - year = "1989", - pages = "207-211", - isbn = "0-89791-325-6", -ACM [ACM89] pages LCCN QA76.95.I59 1989 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@article{Rich96, + author = "Rich, Albert D. and Jeffrey, David J.", + title = "Function Evaluation on Branch Cuts", + journal = "SIGSAM Bull.", + issue_date = "June 1996", + volume = "30", + number = "2", + month = "June", + year = "1996", + issn = "0163-5824", + pages = "25--27", + numpages = "3", + url = "http://doi.acm.org/10.1145/235699.235704", + doi = "10.1145/235699.235704", + acmid = "235704", + publisher = "ACM", + address = "New York, NY, USA", abstract = " - We describe an algorithm, based on Risch's real structure theorem, that - determines explicitly all the algebraic relations among a given set of - real elementary functions. We also provide examples from its - implementation that illustrate the advantages over the use of complex - logarithms and exponentials." + Once it is decided that a CAS will evaluate multivalued functions on + their principal branches, questions arise concerning the branch + definitions. The first questions concern the standardization of the + positions of the branch cuts. These questions have largely been + resolved between the various algebra systems and the numerical + libraries, although not completely. In contrast to the computer + systems, many mathematical textbooks are much further behind: for + example, many popular textbooks still specify that the argument of a + complex number lies between 0 and $2\pi$. We do not intend to discuss + these first questions here, however. Once the positions of the branch + cuts have been fixed, a second set of questions arises concerning the + evaluation of functions on their branch cuts." } \end{chunk} \begin{chunk}{axiom.bib} -\bibitem[Bronstein 91a]{Bro91a} -@inproceedings{Bron91a, - author = "Bronstein, M.", - title = "The Risch Differential Equation on an Algebraic Curve", - booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation", - series = "ISSAC'91", - year = "1991", - pages = "241-246", - isbn = "0-89791-437-6", - publisher = "ACM, NY", - keywords = "axiomref", - paper = "Bro91a.pdf", +@article{Patt96, + author = "Patton, Charles M.", + title = "A Representation of Branch-cut Information", + journal = "SIGSAM Bull.", + issue_date = "June 1996", + volume = "30", + number = "2", + month = "June", + year = "1996", + issn = "0163-5824", + pages = "21--24", + numpages = "4", + url = "http://doi.acm.org/10.1145/235699.235703", + doi = "10.1145/235699.235703", + acmid = "235703", + publisher = "ACM", + address = "New York, NY, USA", + paper = "Patt96.pdf", abstract = " - We present a new rational algorithm for solving Risch differential - equations over algebraic curves. This algorithm can also be used to - solve $n^{th}$-order linear ordinary differential equations with - coefficients in an algebraic extension of the rational functions. In - the general (mixed function'') case, this algorithm finds the - denominator of any solution of the equation." + Handling (possibly) multi-valued functions is a problem in all current + computer algebra systems. The problem is not an issue of technology. + Its solution, however, is tied to a uniform handling of the issues by + the mathematics community." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 91c]{Bro91c} - author = "Bronstein, Manuel", - title = "Computer Algebra and Indefinite Integrals", +\begin{chunk}{axiom.bib} +@article{Squi91, + author = "Squire, Jon S.", + title = "Rationale for the Proposed Standard for a Generic Package of + Complex Elementary Functions", + journal = "Ada Lett.", + issue_date = "Fall 1991", + volume = "XI", + number = "7", + month = "September", year = "1991", - paper = "Bro91c.pdf", -in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds) -Springer-Verlag, NY (1991) - keywords = "axiomref", + issn = "1094-3641", + pages = "166--179", + numpages = "14", + url = "http://doi.acm.org/10.1145/123533.123545", + doi = "10.1145/123533.123545", + acmid = "123545", + publisher = "ACM", + address = "New York, NY, USA", + paper = "Squi91.pdf", abstract = " - We give an overview, from an analytical point of view, of decision - procedures for determining whether an elementary function has an - elementary function has an elementary antiderivative. We give examples - of algebraic functions which are integrable and non-integrable in - closed form, and mention the current implementation of various computer - algebra systems." + This document provides the background on decisions that were made + during the development of the specification for Generic Complex + Elementary fuctions. It also rovides some information that was used to + develop error bounds, range, domain and definitions of complex + elementary functions." } -\end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 92]{Bro92} - author = "Bronstein, M.", - title = "Linear Ordinary Differential Equations: Breaking Through the - Order 2 Barrier", - year = "1992", - url = - "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz", - paper = "Bro92.pdf", - keywords = "axiomref", - abstract = " - A major subproblem for algorithms that either factor ordinary linear - differential equations or compute their closed form solutions is to - find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$ - where $K$ is the constant field for the coefficients of the equation. - While a decision procedure for this subproblem was known in the - $19^{th}$ century, it requires factoring polynomials over - $\overline{K}$ and has not been implemented in full generality. We - present here an efficient algorithm for this subproblem, which has - been implemented in the AXIOM computer algebra system for equations of - arbitrary order over arbitrary fields of characteristic 0. This - algorithm never needs to compute with the individual complex - singularities of the equation, and algebraic numbers are added only - when they appear in the potential solutions. Implementation of the - complete Singer algorithm for $n=2,3$ based on this building block is - in progress." -} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 93]{Bro93} - author = "Bronstein, Manuel (ed)", - year = "1993", - month = "July" - isbn = "0-89791-604-2", -ISSAC'93: proceedings of the 1993 International Symposium on Symbolic -and Algebraic Computation, Kiev, Ukraine, -ACM Press New York, NY 10036, USA, ISBN -LCCN QA76.95 I59 1993 ACM order number 505930 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@article{Squi91a, + editor = "Squire, Jon S.", + title = "Proposed Standard for a Generic Package of Complex + Elementary Functions", + journal = "Ada Lett.", + issue_date = "Fall 1991", + volume = "XI", + number = "7", + month = "September", + year = "1991", + issn = "1094-3641", + pages = "140--165", + numpages = "26", + url = "http://doi.acm.org/10.1145/123533.123544", + doi = "10.1145/123533.123544", + acmid = "123544", + publisher = "ACM", + address = "New York, NY, USA", + abstract = " + This document defines the specification of a generic package of + complex elementary functions called Generic Complex Elementary + Functions. It does not provide the body of the package." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Brunelli 08]{Brun08} - author = "Brunelli, J.C.", - title = "Streams and Lazy Evaluation Applied to Integrable Models", - year = "2008", - url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf", - paper = "Brun08.pdf", +\section{Square-free Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@article{Bern97, + author = "Bernardin, Laurent", + title = "On square-free factorization of multivariate polynomials over a + finite field", + journal = "Theoretical Computer Science", + volume = "187", + number = "1-2", + year = "1997", + month = "November", + pages = "105-116", keywords = "axiomref", + paper = "Bern97.pdf", abstract = " - Computer algebra procedures to manipulate pseudo-differential - operators are implemented to perform calculations with integrable - models. We use lazy evaluation and streams to represent and operate - with pseudo-differential operators. No order of truncation is needed - since terms are produced on demand. We give a series of concrete - examples using the computer algebra language MAPLE." + In this paper we present a new deterministic algorithm for computing + the square-free decomposition of multivariate polynomials with + coefficients from a finite field. -\end{chunk} + Our algorithm is based on Yun's square-free factorization algorithm + for characteristic 0. The new algorithm is more efficient than + existing, deterministic algorithms based on Musser's squarefree + algorithm -\begin{chunk}{ignore} -\bibitem[Bronstein 93]{BS93} - author = "Bronstein, Manuel and Salvy, Bruno", - title = "Full Partial Fraction Decomposition of Rational Functions", - year = "1993", - pages = "157-160", - isbn = "0-89791-604-2", -In Bronstein [Bro93] LCCN QA76.95 I59 1993 - keywords = "axiomref", + We will show that the modular approach presented by Yun has no + significant performance advantage over our algorithm. The new + algorithm is also simpler to implement and it can rely on any existing + GCD algorithm without having to worry about choosing good'' evaluation + points. + + To demonstrate this, we present some timings using implementations in + Maple (Char et al. 1991), where the new algorithm is used for Release + 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system + known to the author to use and implementation of Yun's modular + algorithm mentioned above." +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{Bro92a, - author = "Bronstein, Manuel", - title = "Integration and Differential Equations in Computer Algebra", - year = "1992", - url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576", - paper = "Bro92a.pdf", - keywords = "axiomref", +@article{Chez07, + author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire", + title = "Lifting and recombination techniques for absolute factorization", + journal = "Journal of Complexity", + volume = "23", + number = "3", + year = "2007", + month = "June", + pages = "380-420", + paper = "Chez07.pdf", abstract = " - We describe in this paper how the problems of computing indefinite - integrals and solving linear ordinary differential equations in closed - form are now solved by computer algebra systems. After a brief review - of the mathematical history of those problems, we outline the two - major algorithms for them (respectively the Risch and Singer - algorithms) and the recent improvements on those algorithms which has - allowed them to be implemented." + In the vein of recent algorithmic advances in polynomial factorization + based on lifting and recombination techniques, we present new faster + algorithms for computing the absolute factorization of a bivariate + polynomial. The running time of our probabilistic algorithm is less + than quadratic in the dense size of the polynomial to be factored." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Beneke 94]{BS94} - author = "Beneke, T. and Schwippert, W.", - title = "Double-track into the future: MathCAD will gain new users with - Standard and Plus versions", - year = "1994", - month = "July", - pages = "107-110", - keywords = "axiomref", -Elektronik, 43(15) CODEN EKRKAR ISSN 0013-5658 - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Bronstein 97a]{Bro97a} - author = "Bronstein, Manuel and Weil, Jacques-Arthur", - title = "On Symmetric Powers of Differential Operators", - series = "ISSAC'97", - year = "1997", - pages = "156-163", - keywords = "axiomref", - url = - "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html" - paper = "Bro97a.pdf", - publisher = "ACM, NY", +\begin{chunk}{axiom.bib} +@article{Lece07, + author = "Lecerf, Gr\'egoire", + title = "Improved dense multivariate polynomial factorization algorithms", + journal = "Journal of Symbolic Computation", + volume = "42", + number = "4", + year = "2007", + month = "April", + pages = "477-494", + paper = "Lece07.pdf", abstract = " - We present alternative algorithms for computing symmetric powers of - linear ordinary differential operators. Our algorithms are applicable - to operators with coefficients in arbitrary integral domains and - become faster than the traditional methods for symmetric powers of - sufficiently large order, or over sufficiently complicated coefficient - domains. The basic ideas are also applicable to other computations - involving cyclic vector techniques, such as exterior powers of - differential or difference operators." + We present new deterministic and probabilistic algorithms that reduce + the factorization of dense polynomials from several variables to one + variable. The deterministic algorithm runs in sub-quadratic time in + the dense size of the input polynomial, and the probabilistic + algorithm is softly optimal when the number of variables is at least + three. We also investigate the reduction from several to two variables + and improve the quantitative versions of Bertini's irreducibility theorem." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Borwein 00]{Bor00} - author = "Borwein, Jonathan", - title = "Multimedia tools for communicating mathematics", - year = "2000", - pages = "58", - isbn = "3-540-42450-4", - publisher = "Springer-Verlag", - keywords = "axiomref" +\begin{chunk}{axiom.bib} +@article{Wang77, + author = "Wang, Paul S.", + title = "An efficient squarefree decomposition algorithm", + journal = "ACM SIGSAM Bulletin", + volume = "11", + number = "2", + year = "1977", + month = "May", + pages = "4-6", + paper = "Wang77.pdf", + abstract = " + The concept of polynomial squarefree decomposition is an important one + in algebraic computation. The squarefree decomposition process has + many uses in computer symbolic computation. A recent survey by D. Yun +  describes many useful algorithms for this purpose. All of these + methods depend on computing the greated common divisor (gcd) of the + polynomial to be decomposed and its first derivative (with repect to + some variable). In the multivariate case, this gcd computation is + non-trivial and dominates the cost for the squarefree decompostion." +} \end{chunk} \begin{chunk}{axiom.bib} -@article{BT94, - author = "Brown, R. and Tonks, A.", - title = "Calculations with simplicial and cubical groups in AXIOM", - journal = "Journal of Symbolic Computation", - volume = "17", - number = "2", - pages = "159-179", - year = "1994", - month = "February", - misc = "CODEN JSYCEH ISSN 0747-7171", - keywords = "axiomref" +@article{Wang79, + author = "Wang, Paul S. and Trager, Barry M.", + title = "New Algorithms for Polynomial Square-Free Decomposition + over the Integers", + journal = "SIAM Journal on Computing", + volume = "8", + number = "3", + year = "1979", + publisher = "Society for Industrial and Applied Mathematics", + issn = "00975397", + paper = "Wang79.pdf", + abstract = " + Previously known algorithms for polynomial square-free decomposition + rely on greatest common divisor (gcd) computations over the same + coefficient domain where the decomposition is to be performed. In + particular, gcd of the given polynomial and its first derivative (with + respect to some variable) is obtained to begin with. Application of + modular homomorphism and $p$-adic construction (multivariate case) or + the Chinese remainder algorithm (univariate case) results in new + square-free decomposition algorithms which, generally speaking, take + less time than a single gcd between the given polynomial and its first + derivative. The key idea is to obtain one or several correct'' + homomorphic images of the desired square-free decomposition + first. This provides information as to how many different square-free + factors there are, their multiplicities and their homomorphic + images. Since the multiplicities are known, only the square-free + factors need to be constructed. Thus, these new algorithms are + relatively insensitive to the multiplicities of the square-free factors." } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Brow95, - author = "Brown, Ronald and Dreckmann, Winfried", - title = "Domains of data and domains of terms in AXIOM", - year = "1995", - keywords = "axiomref", - paper = "DB95.pdf", - abstract = " - The main new concept we wish to illustrate in this paper is a - distinction between domains of data'' and domains of terms'', and - its use in the programming of certain mathematical structures. - Although this distinction is implicit in much of the programming work - that has gone into the construction of Axiom categories and domains, - we believe that a formalisation of this is new, that standards and - conventions are necessary and will be useful in various other - contexts. We shall show how this concept may be used for the coding of - free categories and groupoids on directed graphs." +@inproceedings{Yun76, + author = "Yun, D.Y.Y", + title = "On square-free decomposition algorithms", + booktitle = "Proceedings of SYMSAC'76", + year = "1976", + keywords = "survey", + pages = "26-35" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds) -EUROCAL '85: European Conference on Computer Algebra, Linz, Austria, -LLCN QA155.7.E4 E86 - isbn = "0-387-15983-5, 0-387-15984-3", - year = "1985", - month = "April", - publisher = "Springer-Verlag, Berlin, Germany", - keywords = "axiomref", - misc = "Lecture Notes in Computer Science, Vol 204", +\section{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@InProceedings{Kalt83, + author = "Kaltofen, E.", + title = "On the complexity of finding short vectors in integer lattices", + booktitle = "Proc. EUROCAL '83", + series = "Lect. Notes Comput. Sci.", + year = "1983", + volume = "162", + pages = "236--244", + publisher = "Springer-Verlag", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf", + paper = "Kalt83.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{Buh05, - author = "Buhl, Soren L.", - title = "Some Reflections on Integrating a Computer Algebra System in R", - year = "2005", - keywords = "axiomref" +@InProceedings{Kalt85, + author = "Kaltofen, E.", + title = "Effective {Hilbert} Irreducibility", + booktitle = "Proc. EUROSAM '84", + pages = "275--284", + crossref = "EUROSAM84", + year = "1985", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz", + paper = "Kalt85.ps" } - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Burge 91]{Burg91} - author = "Burge, W.H.", - title = "Scratchpad and the Rogers-Ramanujan identities", - year = "1991", - pages = "189-190", - isbn = "0-89791-437-6", - keywords = "axiomref", - abstract = " - This note sketches the part played by Scratchpad in obtaining new - proofs of Euler's theorem and the Rogers-Ramanujan Identities." - + \end{chunk} \begin{chunk}{axiom.bib} -@techreport{BW87, - author = "Burge, W. and Watt, S.", - title = "Infinite structures in SCRATCHPAD II", - year = "1987", - institution = "IBM Research", - type = "Technical Report", - number = "RC 12794", - keywords = "axiomref" +@TechReport{Kalt85c, + author = "Kaltofen, E.", + title = "Sparse Hensel lifting", + institution = "RPI", + address = "Dept. Comput. Sci., Troy, N. Y.", + year = "1985", + number = "85-12", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf", + paper = "Kalt85c.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Burge 87a]{BWM87} - author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.", - title = "Streams and Power Series", - year = "1987", - pages = "9-12", - keywords = "axiomref", -in [Wit87], pp9-12 - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Burge 89]{BW89} - author = "Burge, W. H. and Watt, S. M.", - title = "Infinite structures in Scratchpad II", - year = "1989", - pages = "138-148", - isbn = "3-540-51517-8", - keywords = "axiomref", -in Davenport [Dav89], LCCN QA155.7.E4E86 1987 +\begin{chunk}{axiom.bib} +@InProceedings{Kalt85d, + author = "Kaltofen, E.", + title = "Sparse Hensel lifting", + booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2", + crossref = "EUROCAL85", + pages = "4--17", + year = "1985", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf", + paper = "Kalt85d.pdf" +} \end{chunk} -\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Calmet 94]{Cal94} Calmet, J. (ed) -Rhine Workshop on Computer Algebra, Proceedings. -Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Mill88, + author = "Miller, G.L. and Ramachandran, V. and Kaltofen, E.", + title = "Efficient parallel evaluation of straight-line code and + arithmetic circuits", + journal = "SIAM J. Comput.", + year = "1988", + volume = "17", + number = "4", + pages = "687--695", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf", + paper = "Mill88.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Camion 92]{CCM92} - author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre", - title = "A combinatorial problem in Hamming Graphs and its solution - in Scratchpad", - year = "1992", - month = "January", - keywords = "axiomref", -Rapports de recherche 1586, Institut National de Recherche en -Informatique et en Automatique, Le Chesnay, France, 12pp +\begin{chunk}{axiom.bib} +@Article{Greg88, + author = "Gregory, B.; Kaltofen, E.", + title = "Analysis of the binary complexity of asymptotically fast + algorithms for linear system solving", + journal = "SIGSAM Bulletin", + year = "1988", + month = "April", + volume = "22", + number = "2", + pages = "41--49", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf", + paper = "Grey88.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Caprotti 00]{CCR00} - author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred", - title = "Java Phrasebooks for Computer Algebra and Automated Deduction", - url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf", - paper = "CCR00.pdf", - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Kalt89a, + author = "Kaltofen, E.; Rolletschek, H.", + title = "Computing greatest common divisors and factorizations in + quadratic number fields", + journal = "Math. Comput.", + year = "1989", + volume = "53", + number = "188", + pages = "697--720", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf", + paper = "Kalt89a.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{CC99, - author = "Capriotti, O. and Carlisle, D.", - title = "OpenMath and MathML: Semantic Mark Up for Mathematics", - year = "1999", - url = "http://www.acm.org/crossroads/xrds6-2/openmath.html", - keywords = "axiomref" +@Unpublished{Kalt89b, + author = "Kaltofen, E.", + title = "Processor efficient parallel computation of polynomial greatest + common divisors", + year = "1989", + month = "July", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_gcd.ps.gz", + paper = "Kalt89b.ps" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Capr99, - author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and - Sterk, Hans", - title = "OpenMath Technology for Interactive Mathematical Documents", - year = "2002", - pages = "51-66", - publisher = "Springer-Verlag, Berlin, Germany", - url = "http://www.win.tue.nl/~hansc/lisbon.pdf", - paper = "Capr99.pdf", - misc = "in Multimedia Tools for Communicating Mathematics", - keywords = "axiomref" +@TechReport{Kalt89c, + author = "Kaltofen, E.", + title = "Parallel Algebraic Algorithm Design", + institution = "RPI", + address = "Dept. Comput. Sci., Troy, New York", + year = "1989", + month = "July", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz", + paper = "Kalt89c.ps" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Carp04, - author = "Carpent, Quentin and Conil, Christophe", - title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26", - year = "2004", - paper = "Carp04.pdf", - keywords = "axiomref" +@InProceedings{Cann89, + author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman", + title = "Solving systems of non-linear polynomial equations faster", + booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC89", + pages = "121--128", + year = "1989", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf", + paper = "Cann89.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Chu85, - author = "Chudnovsky, D.V and Chudnovsky, G.V.", - title = "Elliptic Curve Calculations in Scratchpad II", - year = "1985", - institution = "Mathematics Dept., IBM Research", - type = "Scratchpad II Newsletter 1 (1)", - keywords = "axiomref" +@Article{Kalt90b, + author = "Kaltofen, E.", + title = "Computing the irreducible real factors and components of an + algebraic curve", + journal = "Applic. Algebra Engin. Commun. Comput.", + year = "1990", + volume = "1", + number = "2", + pages = "135--148", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf", + paper = "Kalt90b.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Chudnovsky 87]{Chu87} - author = "Chudnovsky, D.V and Chudnovsky, G.V.", - title = "New Analytic Methods of Polynomial Root Finding", - year = "1987", - pages = "2", - keywords = "axiomref", -in [Wit87] +\begin{chunk}{axiom.bib} +@Article{Kalt90d, + author = "Kaltofen, E.; Trager, B.", + title = "Computing with polynomials given by black boxes for their + evaluations: Greatest common divisors, factorization, separation of + numerators and denominators", + journal = "J. Symbolic Comput.", + year = "1990", + volume = "9", + number = "3", + pages = "301--320", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf", + paper = "Kalt90d.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Chudnovsky 89]{Chu89} - author = "Chudnovsky, D.V. and Chudnovsky, G.V.", - title = "The computation of classical constants", - year = "1989", - month = "November", - pages = "8178-8182", - keywords = "axiomref", -Proc. Natl. Acad. Sci. USA Vol 86 +\begin{chunk}{axiom.bib} +@InProceedings{Kalt91a, + author = "Kaltofen, E. and Singer, M.F.", + editor = "D. V. Shirkov and V. A. Rostovtsev and V. P. Gerdt", + title = "Size efficient parallel algebraic circuits for partial derivatives", + booktitle = + "IV International Conference on Computer Algebra in Physical Research", + pages = "133--145", + publisher = "World Scientific Publ. Co.", + year = "1991", + address = "Singapore", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf", + paper = "Kalt91a.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@proceedings{CJ86, - editor = "Chudnovsky, David and Jenks, Richard", - title = "Computers in Mathematics", - year = "1986", - month = "July", - isbn = "0-8247-8341-7", - note = "International Conference on Computers and Mathematics", - publisher = "Marcel Dekker, Inc", - keywords = "axiomref" +@InProceedings{Kalt93, + author = "Kaltofen, E.", + title = "Computational Differentiation and Algebraic Complexity Theory", + booktitle = "Workshop Report on First Theory Institute on Computational + Differentiation", + editor = "C. H. Bischof and A. Griewank and P. M. Khademi", + publisher = "Argonne National Laboratory", + address = "Argonne, Illinois", + series = "Tech. Rep.", + volume = "ANL/MCS-TM-183", + month = "December", + year = "1993", + pages = "28--30", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf", + paper = "Kalt93.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Cohe03, - author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and - Reinaldo, Ernesto and Sterk, Hans", - title = "Interactive Mathematical Documents on the Web", - year = "2003", - pages = "289-306", - editor = "Joswig, M. and Takayma, N.", - publisher = "Springer-Verlag, Berlin, Germany", - keywords = "axiomref", - misc = "in Algebra, Geometry and Software Systems" +@Article{Kalt93b, + author = "Kaltofen, E.", + title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor", + journal = "SIGSAM Bulletin", + year = "1993", + volume = "27", + number = "4", + pages = "2", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz", + paper = "Kalt93b.ps" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed) -EUROCODE '90 International Symposium on -Coding Theory and Applications Proceedings. Springer-Verlag, Berlin, Germany -/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0-387-54303-1 -(New York), 3-540-54303-1 (Berlin), LCCN QA268.E95 1990 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt94, + author = "Kaltofen, E. and Pan, V.", + title = "Parallel solution of Toeplitz and Toeplitz-like linear + systems over fields of small positive characteristic", + booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.", + crossref = "PASCO94", + pages = "225--233", + year = "1994", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf", + paper = "Kalt94.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Conrad (a)]{CFMPxxa} - author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra", - title = "Approaching Inheritance from a Natural Mathematical Perspective - and from a Java Driven Viewpoint: a Comparative Review", - keywords = "axiomref", - paper = "CFMPxxa.pdf", - abstract = " - It is well-known that few object-oriented programming languages allow - objects to change their nature at run-time. There have been a number - of reasons presented for this, but it appears that there is a real - need for matters to change. In this paper we discuss the need for - object-oriented programming languages to reflect the dynamic nature of - problems, particularly those arising in a mathematical context. It is - from this context that we present a framework that realistically - represents the dynamic and evolving characteristic of problems and - algorithms." +\begin{chunk}{axiom.bib} +@InProceedings{Sama95, + author = "Samadani, M. and Kaltofen, E.", + title = "Prediction based task scheduling in distributed computing", + booktitle = "Languages, Compilers and Run-Time Systems for Scalable + Computers", + editor = "B. K. Szymanski and B. Sinharoy", + publisher = "Kluwer Academic Publ.", + address = "Boston", + pages = "317--320", + year = "1996", + url = + "http://www.math.ncsu.edu/~kaltofen/bibliography/95/SaKa95_poster.ps.gz", + paper = "Sama95.ps" +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{CFMPxxb, - author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra", - title = "Mathematical Use Cases lead naturally to non-standard Inheritance - Relationships: How to make them accessible in a mainstream language?", - paper = "CFMPxxb.pdf", - keywords = "axiomref", - abstract = " - Conceptually there is a strong correspondence between Mathematical - Reasoning and Object-Oriented techniques. We investigate how the ideas - of Method Renaming, Dynamic Inheritance and Interclassing can be used - to strengthen this relationship. A discussion is initiated concerning - the feasibility of each of these features." +@InProceedings{Erli96, + author = "Erlingsson, U. and Kaltofen, E. and Musser, D.", + title = "Generic {Gram}-{Schmidt} Orthogonalization by Exact Division", + booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC96", + year = "1996", + pages = "275--282", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/EKM96.pdf", + paper = "Erli96.pdf" } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Cuyp10, - author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem", - title = "Interactive Geometry inside MathDox", - year = "2010", - url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf", - paper = "Cuyp10", - keywords = "axiomref" +@InProceedings{Kalt96, + author = "Kaltofen, E. and Lobo, A.", + title = "On rank properties of {Toeplitz} matrices over finite fields", + booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC96", + year = "1996", + pages = "241--249", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf", + paper = "Kalt96.pdf" } \end{chunk} -\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{axiom.bib} -@inproceedings{Dalm97, - author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen}, - title = "An OpenMath 1.0 Implementation", - booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation", - series = "ISSAC'97", +@Article{Kalt97, + author = "Kaltofen, E.", + title = "Teaching Computational Abstract Algebra", + journal = "Journal of Symbolic Computation", + volume = "23", + number = "5-6", + pages = "503--515", year = "1997", - isbn = "0-89791-875-4", - location = "Kihei, Maui, Hawaii, USA", - pages = "241-248", - numpages = "8", - url = "http://doi.acm.org/10.1145/258726.258794", - doi = "10.1145/258726.258794", - acmid = "258794", - publisher = "ACM, New York, NY USA", - keywords = "axiomref" + note = "Special issue on education, L. Lambe, editor.", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/Ka97_jsc.pdf", + keywords = "axiomref,read", + paper = "Kalt97.pdf", + abstract = " + We report on the contents and pedagogy of a course in abstract algebra + that was taught with the aid of educational software developed within + the Mathematica system. We describe the topics covered and the + didactical use of the corresponding Mathematica packages, as well as + draw conclusions for future such courses from the students' comments + and our own experience." } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dalmas 92]{Dal92} Dalmas, S. -A polymorphic functional language applied to symbolic computation'' -In Wang [Wan92] pp369-375 ISBN 0-89791-489-9 (soft cover) 0-89791-490-2 -(hard cover) LCCN QA76.95.I59 1992 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Unpublished{Hitz97, + author = "Hitz, M. A. and Kaltofen, E.", + title = "The {Kharitonov} theorem and its applications in symbolic + mathematical computation", + year = "1997", + month = "May", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/HiKa97_kharit.pdf", + paper = "Hitz97.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@misc{Daly88, - author = "Daly, Timothy", - title = "Axiom in an Educational Setting, Axiom course slide deck", - year = "1988", - month = "January", - keywords = "axiomref" +@InProceedings{Bern99, + author = "Bernardin, L. and Char, B. and Kaltofen, E.", + title = "Symbolic Computation in {Java}: an Appraisement", + booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.", + crossref = "ISSAC99", + year = "1999", + pages = "237--244", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/BCK99.pdf", + paper = "Bern99.pdf" } \end{chunk} -\begin{chunk}{ignore}TPDHERE -\bibitem[Daly 02]{Dal02} Daly, Timothy -Axiom as open source'' -SIGSAM Bulletin (ACM Special Interest Group -on Symbolic and Algebraic Manipulation) 36(1) pp28-?? March 2002 -CODEN SIGSBZ ISSN 0163-5824 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt02, + author = "Kaltofen, Erich and McLean, Michael and Norris, Larry", + title = "{Using} {Maple} to Grade {Maple}' Assessment Software from + {North Carolina State University}", + booktitle = "Proceedings 2002 Maple Workshop", + year = "2002", + publisher = "Waterloo Maple Inc.", + address = "Waterloo, Canada", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KMN02.pdf", + paper = "Kalt02.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Daly 03]{Dal03} Daly, Timothy -The Axiom Wiki Website'' -\verb|axiom.axiom-developer.org| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Book{Grab03, + editor = "Grabmeier, J. and Kaltofen, E. and Weispfenning, V.", + title = "Computer Algebra Handbook", + publisher = "Springer-Verlag", + year = "2003", + note = "637 + xx~pages + CD-ROM. Includes E. Kaltofen and V. Weispfenning + \S1.4 Computer algebra -- impact on research, pages 4--6; + E. Kaltofen + \S2.2.3 Absolute factorization of polynomials, page 26; + E. Kaltofen and B. D. Saunders + \S2.3.1 Linear systems, pages 36--38; + R. M. Corless, E. Kaltofen and S. M. Watt + \S2.12.3 Hybrid methods, pages 112--125; + E. Kaltofen + \S4.2.17 FoxBox and other blackbox systems, pages 383--385.", + isbn = "3-540-65466-6", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/symnum.pdf", + paper = "Grab03.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Daly 06]{Dal06} Daly, Timothy -Axiom Volume 1: Tutorial'' -Lulu, Inc. 860 Aviation Parkway, -Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp -\verb|www.lulu.com/content/190827| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt07, + author = "Kaltofen, Erich and Li, Bin and Sivaramakrishnan, Kartik and + Yang, Zhengfeng and Zhi, Lihong", + title = "Lower bounds for approximate factorizations via semidefinite + programming (extended abstract)", + year = "2007", + booktitle = + "SNC'07 Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput.", + crossref = "SNC07", + pages = "203--204", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KLSYZ07.pdf", + paper = "Kalt07.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Daly 09]{Dal09} Daly, Timothy -The Axiom Literate Documentation'' -\verb|axiom-developer.org/axiom-website/documentation.html| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Borw07, + author = "Borwein, Peter and Kaltofen, Erich and Mossinghoff, Michael J.", + title = "Irreducible Polynomials and {Barker} Sequences", + journal = "{ACM} Communications in Computer Algebra", + volume = "162", + number = "4", + year = "2007", + pages = "118--121", + month = "December", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/BKM07.pdf", + paper = "Borw07.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Daly 13]{Dal13} Daly, Timothy -Literate Programming in the Large'' -April 8-9, 2013 Portland Oregon -\verb|conf.writethedocs.org| -\verb|daly.axiom-developer.org| -\verb|www.youtube.com/watch?v=Av0PQDVTP4A| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@Article{Kalt10, + author = "Kaltofen, Erich and Lavin, Mark", + title = "Efficiently Certifying Non-Integer Powers", + journal = "Computational Complexity", + year = "2010", + volume = "19", + number = "3", + month = "September", + pages = "355--366", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KaLa09.pdf", + paper = "Kalt10.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 79a]{Dav79a} Davenport, J.H. -What can SCRATCHPAD/370 do?'' -VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt11, + author = "Kaltofen, Erich L. and Nehring, Michael", + title = "Supersparse black box rational function interpolation", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11", + crossref = "ISSAC11", + month = "June", + year = "2011", + pages = "177--185", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KaNe11.pdf", + paper = "Kalt11.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D. -MODLISP -- an Introduction'' -Proc LISP80, 1980, and IBM RC8357 Oct 1980 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Gren11a, + author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal + and Portier, Natacha", + title = "Symmetric Determinantal Representation of Weakly Skew Circuits", + booktitle = "Proc. 28th Internat. Symp. on Theoretical Aspects of Computer + Science, STACS 2011", + crossref = "STACS11", + pages = "543--554", + year = "2011", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/GKKP11.pdf", + paper = "Gren11a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.; -Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.; -Trager, B. -Scratchpad'' -Mathematical Sciences Department, IBM Thomas Watson Research Center 1984 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt11a, + author = "Kaltofen, Erich L. and Nehring, Michael and Saunders, David B.", + title = "Quadratic-Time Certificates in Linear Algebra", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11", + crossref = "ISSAC11", + month = "June", + year = "2011", + pages = "171--176", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KNS11.pdf", + paper = "Kalt11a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 84a]{Dav84a} Davenport, James H. -A New Algebra System'' -%\verb|axiom-developer.org/axiom-website/papers/Dav84a.pdf| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt11b, + author = "Kaltofen, Erich L. and Lee, Wen-shin and Yang, Zhengfeng", + title = "Fast estimates of {Hankel} matrix condition numbers + and numeric sparse interpolation", + booktitle = "Proc. 2011 Internat. Workshop on Symbolic-Numeric Comput.", + month = "June", + crossref = "SNC11", + year = "2011", + pages = "130--136", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KLY11.pdf", + paper = "Kalt11b.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 85]{Dav85} Davenport, James H. -The LISP/VM Foundation of Scratchpad II'' -The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985 -IBM Corporation, Yorktown Heights, NY - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Guo12, + author = "Guo, Feng and Kaltofen, Erich L. and Zhi, Lihong", + title = "Certificates of Impossibility of {Hilbert}-{Artin} Representations + of a Given Degree for Definite Polynomials and Functions", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12", + crossref = "ISSAC12", + month = "July", + year = "2012", + pages = "195--202", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/GKZ12.pdf", + paper = "Guo12.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E. -Computer Algebra: Systems and Algorithms for Algebraic Computation. -Academic Press, New York, NY, USA, 1988, ISBN 0-12-204232-9 -\verb|staff.bath.ac.uk/masjhd/masternew.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DST88.pdf| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Come12a, + author = "Comer, Matthew T. and Kaltofen, Erich L. and Pernet, Cl{\'e}ment", + title = "Sparse Polynomial Interpolation and {Berlekamp}/\allowbreak + {Massey} Algorithms That Correct Outlier Errors in Input Values", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12", + crossref = "ISSAC12", + month = "July", + year = "2012", + pages = "138--145", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/CKP12.pdf", + paper = "Come12a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 14]{Dav14} Davenport, James H. -Computer Algebra textbook'' -\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Dav14.pdf| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Boye13, + author = "Boyer, Brice and Comer, Matthew T. and Kaltofen, Erich L.", + title = "Sparse Polynomial Interpolation by Variable Shift in + the Presence of Noise and Outliers in the Evaluations", + booktitle = + "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)", + year = "2013", + month = "October", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/BCK13.pdf", + paper = "Boye13.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed) -EUROCAL '87 European Conference on Computer Algebra Proceedings -Springer-Verlag, Berlin, Germany / Heidelberg, Germany / London, -UK / etc., 1989 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt13b, + author = "Kaltofen, Erich and Yang, Zhengfeng", + title = "Sparse multivariate function recovery from values with noise and + outlier errors", + year = "2013", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'13", + crossref = "ISSAC13", + pages = "219--226", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/KaYa13.pdf", + paper = "Kalt13b.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M. -Scratchpad's view of algebra I: Basic commutative algebra'' -In Miola [Mio90], pp40-54. ISBN 0-387-52531-9 (New York), -3-540-52531-9 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical -Report, ATR/1, NAG Ltd., Oxford, 1992 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt13c, + author = "Kaltofen, Erich L.", + title = "Symbolic Computation and Complexity Theory Transcript of My Talk", + booktitle = + "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)", + year = "2013", + month = "October", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/Ka13.pdf", + paper = "Kalt13c.pdf" +} \end{chunk} \begin{chunk}{axiom.bib} -@inproceedings{Dave91, - author = "Davenport, J. H. and Gianni, P. and Trager, B. M.", - title = "Scratchpad's View of Algebra II: - A Categorical View of Factorization", - booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation", - series = "ISSAC '91", - year = "1991", - isbn = "0-89791-437-6", - location = "Bonn, West Germany", - pages = "32--38", - numpages = "7", - url = "http://doi.acm.org/10.1145/120694.120699", - doi = "10.1145/120694.120699", - acmid = "120699", - publisher = "ACM", - address = "New York, NY, USA", - keywords = "axiomref", - paper = "Dave91.pdf", - abstract = " - This paper explains how Scratchpad solves the problem of presenting a - categorical view of factorization in unique factorization domains, - i.e. a view which can be propagated by functors such as - SparseUnivariatePolynomial or Fraction. This is not easy, as the - constructive version of the classical concept of - UniqueFactorizationDomain cannot be so propagated. The solution - adopted is based largely on Seidenberg's conditions (F) and (P), but - there are several additional points that have to be borne in mind to - produce reasonably efficient algorithms in the required generality. - - The consequence of the algorithms and interfaces presented in this - paper is that Scratchpad can factorize in any extension of the - integers or finite fields by any combination of polynomial, fraction - and algebraic extensions: a capability far more general than any other - computer algebra system possesses. The solution is not perfect: for - example we cannot use these general constructions to factorize - polyinmoals in $\overline{Z[\sqrt{-5}]}[x]$ since the domain - $Z[\sqrt{-5}]$ is not a unique factorization domain, even though - $\overline{Z[\sqrt{-5}]}$ is, since it is a field. Of course, we can - factor polynomials in $\overline{Z}[\sqrt{-5}][x]$" +@InProceedings{Kalt14, + author = "Kaltofen, Erich L. and Yang, Zhengfeng", + title = "Sparse Multivariate Function Recovery With a High Error Rate + in Evaluations", + year = "2014", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14", + crossref = "ISSAC14", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaYa14.pdf", + paper = "Kalt14.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M. -Scratchpad's view of algebra II: A categorical view of factorization'' -Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc., -Downer's Grove, IL, USA and Oxford, UK, December 1992 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Kalt14a, + author = "Kaltofen, Erich L. and Pernet, Cl{\'e}ment", + title = "Sparse Polynomial Interpolation Codes and Their Decoding + Beyond Half the Minimal Distance", + year = "2014", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14", + crossref = "ISSAC14", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaPe14.pdf", + paper = "Kalt14a.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 92a]{Dav92a} Davenport, J. H. -The AXIOM system'' -AXIOM Technical Report TR5/92 (ATR/3) -(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and -Oxford, UK, December 1992 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@InProceedings{Duma14, + author = "Dumas, Jean-Guillaume and Kaltofen, Erich L.", + title = "Essentially Optimal Interactive Certificates In Linear Algebra", + year = "2014", + booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14", + crossref = "ISSAC14", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/DuKa14.pdf", + paper = "Duma14.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 92b]{Dav92b} Davenport, J. H. -How does one program in the AXIOM system?'' -AXIOM Technical Report TR6/92 (ATR/4)(NP2493) -Numerical Algorithms Group, Inc., Downer's -Grove, IL, USA and Oxford, UK December 1992 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| -%\verb|axiom-developer.org/axiom-website/papers/Dav92b.pdf| - keywords = "axiomref", - abstract = " - Axiom is a computer algebra system superficially like many others, but - fundamentally different in its internal construction, and therefore in - the possibilities it offers to its users and programmers. In these - lecture notes, we will explain, by example, the methodology that the - author uses for programming substantial bits of mathematics in Axiom." +\begin{chunk}{axiom.bib} +@InProceedings{Boye14, + author = "Boyer, Brice B. and Kaltofen, Erich L.", + title = "Numerical Linear System Solving With Parametric Entries By + Error Correction", + year = "2014", + booktitle = "SNC'14 Proc. 2014 Int. Workshop on Symbolic-Numeric Comput.", + crossref = "SNC14", + url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/BoKa14.pdf", + paper = "Boye14.pdf" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M. -Scratchpad's view of algebra I: Basic commutative algebra'' -DISCO 90 Capri, Italy April 1990 ISBN 0-387-52531-9 pp40-54 -Technical Report TR3/92 (ATR/1)(NP2490), Numerical -Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK, -December 1992. -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| - keywords = "axiomref", +\section{Axiom Citations in the Literature} -\end{chunk} +\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Davenport 93]{Dav93} Davenport, J. H. -Primality testing revisited'' -Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc., -Downer's Grove, IL, USA and Oxford, UK, August 1993 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +\bibitem[ACM 89]{ACM89} ACM, editor +Proceedings of the ACM-SIGSAM 1989 International +Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press, +New York, NY 10036, USA, 1989, , LCCN QA76.95.I59 + year = "1989", + isbn = "0-89791-325-6", keywords = "axiomref", - \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele -The Unknown in Computer Algebra'' -\verb|axiom-wiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DFxx.pdf| +\bibitem[ACM 94]{ACM94} ACM, editor +ISSAC '94. Proceedings of the International +Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY, +10036, USA, 1994, . LCCN QA76.95.I59 + year = "1994", + isbn = "0-89791-638-7", keywords = "axiomref", - abstract = " - Computer algebra systems have to deal with the confusion between - programming variables'' and mathematical symbols''. We claim that - they should also deal with unknowns'', i.e. elements whose values - are unknown, but whose type is known. For examples $x^p \ne x$ if $x$ - is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have - extended Axiom to deal with this concept." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Davenport 00]{Dav00} Davenport, James -13th OpenMath Meeting'' -James H. Davenport -A New Algebra System'' -May 1984 -\verb|xml.coverpages.org/openmath13.html| -%\verb|axiom-developer.org/axiom-website/papers/Dav00.pdf| +\begin{chunk}{axiom.bib} +@article{Augo91, + author = "Augot, D. and Charpin, P. and Sendrier, N.", + title = "The miniumum distance of some binary codes via the + Newton's identities", + journal = "Cohen and Charping [CC91]", + year = "1991", + pages = "65-73", + isbn = "0-387-54303-1", + misc = "3-540-54303-1 (Berlin). LCCN QA268.E95 1990", keywords = "axiomref", + paper = "Augo91.pdf" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport 12]{Dav12} Davenport, J.H. -Computer Algebra'' -\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf| +\bibitem[Adams 94]{AL94} + author = "Adams, William W. and Loustaunau, Philippe", + title = "An Introduction to Gr\"obner Bases", + year = "1994", +American Mathematical Society (1994) + isbn = "0-8218-3804-0", keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier -Computer Algebra'' \hfill -\verb|staff.bath.ac.uk/masjhd/masternew.pdf| +\bibitem[Andrews 84]{And84} + author = "Andrews, George E.", + title = "Ramanujan and SCRATCHPAD", + year = "1984", + pages = "383-??", keywords = "axiomref", +In Golden and Hussain [GH84] \end{chunk} \begin{chunk}{ignore} -\bibitem[Dewar 94]{Dew94} Dewar, M. C. -Manipulating Fortran Code in AXIOM and the AXIOM-NAG Link'' -Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H. -and Laine, M. and Valkeila, E. pp1-12 University of Helsinki, Finland (1994) +\bibitem[Andrews 88]{And88} + author = "Andrews, G. E.", + title = "Application of Scratchpad to problems in special functions and + combinatorics", + year = "1988" + pages = "158-??", + isbn = "3-540-18928-9", keywords = "axiomref", +In Janssen [Jan88], pages 158-?? ISBN +0-387-18928-9 LCCN QA155.7.E4T74 \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Dewa, - author = "Dewar, Mike", - title = "OpenMath: An Overview", - url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf", - paper = "Dewa.pdf", - keywords = "axiomref" -} +\begin{chunk}{ignore} +\bibitem[Anon 91]{Ano91} + author = "Anonymous", + year = "1991, + keywords = "axiomref", +Proceedings 1991 Annual Conference, American Society for +Engineering Education. Challenges of a Changing World. ASEE, Washington, DC + 2 vol. \end{chunk} \begin{chunk}{ignore} -\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D. -Algebraic extensions and algebraic closure in Scratchpad II'' -In Gianni [Gia89], pp440-446 ISBN 3-540-51084-2 -LCCN QA76.95.I57 1998 Conference held jointly with AAECC-6 +\bibitem[Anon 92]{Ano92} + author = "Anonymous", + year = "1992", keywords = "axiomref", +Programming environments for high-level scientific problem solving. +IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science +and Technology, A-2:??, CODEN ITATEC. ISSN 0926-5473 \end{chunk} \begin{chunk}{ignore} -\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard -Branch Cuts in Computer Algebra'' -1994 ISSAC, Oxford (UK), July 1994 -\verb|www.cs.berkeley.edu/~fateman/papers/ding.ps| -%\verb|axiom-developer.org/axiom-website/papers/Din94.pdf| +\bibitem[Anono 95]{Ano95} + author =Anonymous + keywords = "axiomref", + year = "1995", +GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und +Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 0044-2267 + +\end{chunk} + +\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@article{Bacl14, + author = "Baclawski, Krystian", + title = "SPAD language type checker", + journal = "unknown", + year = "2014", + url = "http://github.com/cahirwpz/phd", keywords = "axiomref", abstract = " - Many standard functions, such as the logarithms and square root - functions, cannot be defined continuously on the complex - plane. Mistaken assumptions about the properties of these functions - lead computer algebra systems into various conundrums. We discuss how - they can manipulate such functions in a useful fashion." + The project aims to deliver a new type checker for SPAD language. + Several improvements over current type checker are planned. + \begin{itemize} + \item introduce better type inference + \item introduce modern language constructs + \item produce understandable diagnostic messages + \item eliminate well known bugs in the type system + \item find new type errors + \end{itemize}" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[DLMF]{DLMF}. -Digital Library of Mathematical Functions'' -\verb|dlmf.nist.gov/software/#T1| +\bibitem[Blair 70]{BGJ70} + author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.", + title = "An interactive facility for symbolic mathematics", + year = "1970", + pages = "394-419", keywords = "axiomref", +Proc. International Computing Symposium, Bonn, Germany, \end{chunk} \begin{chunk}{ignore} -\bibitem[Dooley 99]{Doo99} Dooley, Sam editor. -ISSAC 99: July 29-31, 1999, Simon Fraser University, -Vancouver, BC, Canada: proceedings of the 1999 International Symposium on -Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999. -ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 +\bibitem[Blair 70a]{BJ70} + author = "Blair, Fred W. and Jenks, Richard D.", + title = "LPL: LISP programming language", + year = "1970", keywords = "axiomref", +IBM Research Report, RC3062 Sept \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel -A System for Axiomatic Programming'' -Proc. Conf. on Intelligent Computer Mathematics, Springer (2012) -\verb|www.axiomatics.org/~gdr/liz/cicm-2012.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DR12.pdf| +\begin{chunk}{axiom.bib} +\bibitem[Broadbery 95]{BGDW95} + author = "Broadbery, P. A. and G{\'o}mez-D{\'\i}az, T. and Watt, S. M.", + title = "On the Implementation of Dynamic Evaluation", + year = "1995", + pages = "77-84", keywords = "axiomref", + isbn = "0-89791-699-9", + url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf", + paper = "BGDW95.pdf", abstract = " - We present the design and implementation of a system for axiomatic - programming, and its application to mathematical software - construction. Key novelties include a direct support for user-defined - axioms establishing local equality between types, and overload - resolution based on equational theories and user-defined local - axioms. We illustrate uses of axioms, and their organization into - concepts, in structured generic programming as practiced in - computational mathematical systems." + Dynamic evaluation is a technique for producing multiple results + according to a decision tree which evolves with program execution. + Sometimes it is desired to produce results for all possible branches + in the decision tree, while on other occasions, it may be sufficient + to compute a single result which satisfies certain properties. This + techinique finds use in computer algebra where computing the correct + result depends on recognizing and properly handling special cases of + parameters. In previous work, programs using dynamic evaluation have + explored all branches of decision trees by repeating the computations + prior to decision points. + + This paper presents two new implementations of dynamic evaluation + which avoid recomputing intermediate results. The first approach uses + Scheme continuations'' to record state for resuming program + execution. The second implementation uses the Unix fork'' operation + to form new processes to explore alternative branches in parallel." } + \end{chunk} -\begin{chunk}{ignore} -\bibitem[Doye 97]{Doy97} Doye, Nicolas James -Order Sorted Computer Algebra and Coercions'' -Ph.D. Thesis University of Bath 1997 -%\verb|axiom-developer.org/axiom-website/papers/Doy97.pdf| +\begin{chunk}{axiom.bib} +\bibitem[Boehm 89]{Boe89} +@inproceedings{Boe89, + author = "Boehm, Hans-J.", + title = "Type Inference in the Presence of Type Abstraction", + year = "1989", + pages = "192-206", keywords = "axiomref", + url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192-boehm", + paper = "Boe89.pdf", + booktitle = "ACM SIGPLAN Notices", + volume = "24", + number = "7", + month = "July", abstract = " - Computer algebra systems are large collections of routines for solving - mathematical problems algorithmically, efficiently and above all, - symbolically. The more advanced and rigorous computer algebra systems - (for example, Axiom) use the concept of strong types based on - order-sorted algebra and category theory to ensure that operations are - only applied to expressions when they make sense''. - - In cases where Axiom uses notions which are not covered by current - mathematics we shall present new mathematics which will allow us to - prove that all such cases are reducible to cases covered by the - current theory. On the other hand, we shall also point out all the - cases where Axiom deviates undesirably from the mathematical ideal. - Furthermore we shall propose solutions to these deviations. + A number of recent programming language designs incorporate a type + checking system based on the Girard-Reynolds polymorphic + $\lambda$-calculus. This allows the construction of general purpose, + reusable software without sacrificing compile-time type checking. A + major factor constraining the implementation of these languages is the + difficulty of automatically inferring the lengthy type information + that is otherwise required if full use is made of these + languages. There is no known algorithm to solve any natural and fully + general formulation of the type inference'' problem. One very + reasonable formulation of the problem is known to be undecidable. - Strongly typed systems (especially of mathematics) become unusable - unless the system can change the type in a way a user expects. We wish - any change expected by a user to be automated, natural'', and - unique. Coercions'' are normally viewed as natural type changing - maps''. This thesis shall rigorously define the word coercion'' in - the context of computer algebra systems. + Here we define a restricted version of the type inference problem and + present an efficient algorithm for its solution. We argue that the + restriction is sufficiently weak to be unobtrusive in practice." +} - We shall list some assumptions so that we may prove new results so - that all coercions are unique. This concept is called coherence''. +\end{chunk} - We shall give an algorithm for automatically creating all coercions in - type system which adheres to a set of assumptions. We shall prove that - this is an algorithm and that it always returns a coercion when one - exists. Finally, we present a demonstration implementation of this - automated coerion algorithm in Axiom." +\begin{chunk}{axiom.bib} +@inproceedings{BHGM04, + author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne + and Martin, Ursula", + title = "Design verification for control engineering", + year = "2004", + month = "April", + booktitle = "Proc 4th Int. Conf. on Integrated Formal Methods", + keywords = "axiomref", + abstract = " + We introduce control engineering as a new domain of application for + formal methods. We discuss design verification, drawing attention to + the role played by diagrammatic evaluation criteria involving numeric + plots of a design, such as Nichols and Bode plots. We show that + symbolic computation and computational logic can be used to discharge + these criteria and provide symbolic, automated, and very general + alternatives to these standard numeric tests. We illustrate our work + with reference to a standard reference model drawn from military + avionics." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Doye 99]{Doy99} Doye, Nicolas J. -Automated coercion for Axiom'' -In Dooley [Doo99], pp229-235 -ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 ACM Press -\verb|www.acm.org/citation.cfm?id=309944| +\bibitem[Boulanger 91]{Bou91} + author = "Boulanger, Jean-Louis", + title = "Etude de la compilation de scratchpad 2", + year = "1991", + month = "September", +Rapport de DEA Universite dl lille 1 keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio -Modeling Inheritance as Coercion in a Symbolic Computation System'' -ISSAC 2001 ACM 1-58113-417-7/01/0007 -%\verb|axiom-developer.org/axiom-website/papers/DR01.pdf| - keywords = "axiomref", - abstract = " - In this paper the analysis of the data structures used in a symbolic - computation system, called Kenzo, is undertaken. We deal with the - specification of the inheritance relationship since Kenzo is an - object-oriented system, written in CLOS, the Common Lisp Object - System. We focus on a particular case, namely the relationship between - simplicial sets and chain complexes, showing how the order-sorted - algebraic specifications formalisms can be adapted, through the - inheritance as coercion'' metaphor, in order to model this Kenzo - fragment." +\begin{chunk}{axiom.bib} +@misc{Bou93a, + author = "Boulanger, Jean-Louis", + title = "Axiom, language fonctionnel \a d\'evelopement objet", + year = "1993", + month = "October", + paper = "Bou93a.pdf", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and - Linton, Steve -Embedded Verification Techniques for Computer Algebra Systems'' -Grant citation GR/L48256 Nov 1, 1997-Feb 28, 2001 -\verb|www.cs.st-andrews.ac.uk/research/output/detail?output=ML97.php| +\begin{chunk}{axiom.bib} +@misc{Bou93b, + author = "Boulanger, Jean-Louis", + title = "AXIOM, A Functional Language with Object Oriented Development", + year = "1993", + paper = "Bou93b.pdf", keywords = "axiomref", + abstract = " + We present in this paper, a study about the computer algebra system + Axiom, which gives us many very interesting Software engineering + concepts. This language is a functional language with an Object + Oriented Development. This feature is very important for modeling the + mathematical world (Hierarchy) and provides a running with + mathematical sense. (All objects are functions). We present many + problems of running and development in Axiom. We can note that Aiom is + the only system of this category." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne; -Kelsey, Tom; Martin, Ursula; Owre, Sam -Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS'' -TPHOLS 2001, Edinburgh -\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf| -%\verb|axiom-developer.org/axiom-website/papers/DGKM01.pdf| +\bibitem[Boulanger 94]{Bou94} + author = "Boulanger, J.L.", + title = "Object Oriented Method for Axiom", + year = "1995", + month = "February", + pages = "33-41", + paper = "Bou94.pdf", +ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 0362-1340 keywords = "axiomref", abstract = " - We describe an interface between version 6 of the Maple computer - algebra system with the PVS automated theorem prover. The interface is - designed to allow Maple users access to the robust and checkable proof - environment of PVS. We also extend this environment by the provision - of a library of proof strategies for use in real analysis. We - demonstrate examples using the interface and the real analysis - library. These examples provide proofs which are both illustrative and - applicable to genuine symbolic computation problems." + Axiom is a very powerful computer algebra system which combines two + language paradigms (functional and OOP). Mathematical world is complex + and mathematicians use abstraction to design it. This paper presents + some aspects of the object oriented development in Axiom. The Axiom + programming is based on several new tools for object oriented + development, it uses two levels of class and some operations such that + {\sl coerce}, {\sl retract}, or {\sl convert} which permit the type + evolution. These notions introduce the concept of multi-view." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 92]{DJ92} Duval D.; Jung, F. -Examples of problem solving using computer algebra'' -IFIP Transactions. A. Computer Science and Technology, A-2 pp133-141, 143 1992 -CODEN ITATEC. ISSN 0926-5473 +\bibitem[Bronstein 87]{Bro87} + author = "Bronstein, Manuel", + title = "Integration of Algebraic and Mixed Functions", + year = "1987", +in [Wit87], p18 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 94]{Duv94} Duval, Dominique -Symbolic or algebraic computation?'' -Madrid Spain, NAG conference (private copy of paper) +\bibitem[Bronstein 89]{Bro89} + author= "Bronstein, M.", + title = "Simplification of real elementary functions", + year = "1989", + pages = "207-211", + isbn = "0-89791-325-6", +ACM [ACM89] pages LCCN QA76.95.I59 1989 keywords = "axiomref", + abstract = " + We describe an algorithm, based on Risch's real structure theorem, that + determines explicitly all the algebraic relations among a given set of + real elementary functions. We also provide examples from its + implementation that illustrate the advantages over the use of complex + logarithms and exponentials." +} \end{chunk} \begin{chunk}{axiom.bib} -@article{Duva95, - author = "Duval, D.", - title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom", - journal = "Journal of Pure and Applied Algebra", - volume = "99", - year = "1995", - pages = "267--295.", - keywords = "axiomref" +\bibitem[Bronstein 91a]{Bro91a} +@inproceedings{Bron91a, + author = "Bronstein, M.", + title = "The Risch Differential Equation on an Algebraic Curve", + booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation", + series = "ISSAC'91", + year = "1991", + pages = "241-246", + isbn = "0-89791-437-6", + publisher = "ACM, NY", + keywords = "axiomref", + paper = "Bro91a.pdf", + abstract = " + We present a new rational algorithm for solving Risch differential + equations over algebraic curves. This algorithm can also be used to + solve $n^{th}$-order linear ordinary differential equations with + coefficients in an algebraic extension of the rational functions. In + the general (mixed function'') case, this algorithm finds the + denominator of any solution of the equation." } \end{chunk} -\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William -The Sage Project'' -\verb|wstein.org/papers/icms/icms_2010.pdf| -%\verb|axiom-developer.org/axiom-website/papers/ES10.pdf| +\bibitem[Bronstein 91c]{Bro91c} + author = "Bronstein, Manuel", + title = "Computer Algebra and Indefinite Integrals", + year = "1991", + paper = "Bro91c.pdf", +in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds) +Springer-Verlag, NY (1991) keywords = "axiomref", abstract = " - Sage is a free, open source, self-contained distribution of - mathematical software, including a large library that provides a - unified interface to the components of this distribution. This library - also builds on the components of Sage to implement novel algorithms - covering a broad range of mathematical functionality from algebraic - combinatorics to number theory and arithmetic geometry." - + We give an overview, from an analytical point of view, of decision + procedures for determining whether an elementary function has an + elementary function has an elementary antiderivative. We give examples + of algebraic functions which are integrable and non-integrable in + closed form, and mention the current implementation of various computer + algebra systems." +} \end{chunk} -\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Fateman 90]{Fat90} Fateman, R. J. -Advances and trends in the design and construction of algebraic -manipulation systems'' -In Watanabe and Nagata [WN90], pp60-67 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990 +\bibitem[Bronstein 92]{Bro92} + author = "Bronstein, M.", + title = "Linear Ordinary Differential Equations: Breaking Through the + Order 2 Barrier", + year = "1992", + url = + "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz", + paper = "Bro92.pdf", keywords = "axiomref", - + abstract = " + A major subproblem for algorithms that either factor ordinary linear + differential equations or compute their closed form solutions is to + find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$ + where $K$ is the constant field for the coefficients of the equation. + While a decision procedure for this subproblem was known in the + $19^{th}$ century, it requires factoring polynomials over + $\overline{K}$ and has not been implemented in full generality. We + present here an efficient algorithm for this subproblem, which has + been implemented in the AXIOM computer algebra system for equations of + arbitrary order over arbitrary fields of characteristic 0. This + algorithm never needs to compute with the individual complex + singularities of the equation, and algebraic numbers are added only + when they appear in the potential solutions. Implementation of the + complete Singer algorithm for $n=2,3$ based on this building block is + in progress." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Fateman 05]{Fat05} Fateman, R. J. -An incremental approach to building a mathematical expert out of software'' -4/19/2005\hfill -\verb|www.cs.berkeley.edu/~fateman/papers/axiom.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Fat05.pdf| +\bibitem[Bronstein 93]{Bro93} + author = "Bronstein, Manuel (ed)", + year = "1993", + month = "July" + isbn = "0-89791-604-2", +ISSAC'93: proceedings of the 1993 International Symposium on Symbolic +and Algebraic Computation, Kiev, Ukraine, +ACM Press New York, NY 10036, USA, ISBN +LCCN QA76.95 I59 1993 ACM order number 505930 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Fateman 06]{Fat06} Fateman, R. J. -Building Algebra Systems by Overloading Lisp'' -\verb|www.cs.berkeley.edu/~fateman/generic/overload-small.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Fat06.pdf| +\bibitem[Brunelli 08]{Brun08} + author = "Brunelli, J.C.", + title = "Streams and Lazy Evaluation Applied to Integrable Models", + year = "2008", + url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf", + paper = "Brun08.pdf", keywords = "axiomref", abstract = " - Some of the earliest computer algebra systems (CAS) looked like - overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula - Algol, and others each took advantage of a pre-existing language base - and expanded the notion of a numeric value to include mathematical - expressions. Much more recently, perhaps encouraged by the growth in - popularity of C++, we have seen a renewal of the use of overloading to - implement a CAS. - - This paper makes three points. 1. It is easy to do overloading in - Common Lisp, and show how to do it in detail. 2. Overloading per se - provides an easy solution to some simple programming problems. We show - how it can be used for a demonstration'' CAS. Other simple and - plausible overloadings interact nicely with this basic system. 3. Not - all goes so smoothly: we can view overloading as a case study and - perhaps an object lesson since it fails to solve a number of - fairly-well articulated and difficult design issues in CAS for which - other approaches are preferable." + Computer algebra procedures to manipulate pseudo-differential + operators are implemented to perform calculations with integrable + models. We use lazy evaluation and streams to represent and operate + with pseudo-differential operators. No order of truncation is needed + since terms are produced on demand. We give a series of concrete + examples using the computer algebra language MAPLE." \end{chunk} \begin{chunk}{ignore} -\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James -Parameters in Computer Algebra'' +\bibitem[Bronstein 93]{BS93} + author = "Bronstein, Manuel and Salvy, Bruno", + title = "Full Partial Fraction Decomposition of Rational Functions", + year = "1993", + pages = "157-160", + isbn = "0-89791-604-2", +In Bronstein [Bro93] LCCN QA76.95 I59 1993 keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James; -Naciri, Hanane -Multi-values Computer Algebra'' -ISSN 0249-6399 Institut National De Recherche en Informatique et en -Automatique Sept. 2000 No. 4001 -\verb|hal.inria.fr/inria-00072643/PDF/RR-4401.pdf| -%\verb|axiom-developer.org/axiom-website/papers/FDN00b.pdf| +\begin{chunk}{axiom.bib} +@misc{Bro92a, + author = "Bronstein, Manuel", + title = "Integration and Differential Equations in Computer Algebra", + year = "1992", + url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576", + paper = "Bro92a.pdf", keywords = "axiomref", abstract = " - One of the main strengths of computer algebra is being able to solve a - family of problems with one computation. In order to express not only - one problem but a family of problems, one introduces some symbols - which are in fact the parameters common to all the problems of the - family. - - The user must be able to understand in which way these parameters - affect the result when he looks at the answer. Otherwise it may lead - to completely wrong calculations, which when used for numerical - applications bring nonsensical answers. This is the case in most - current Computer Algebra Systems we know because the form of the - answer is never explicitly conditioned by the values of the - parameters. The user is not even informed that the given answer may be - wrong in some cases then computer algebra systems can not be entirely - trustworthy. We have introduced multi-valued expressions called {\sl - conditional} expressions, in which each potential value is associated - with a condition on some parameters. This is used, in particular, to - capture the situation in integration, where the form of the answer can - depend on whether certain quantities are positive, negative or - zero. We show that it is also necessary when solving modular linear - equations or deducing congruence conditions from complex expressions." - -\end{chunk} - -\begin{chunk}{ignore} -\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed) -EUROSAM '84: International Symposium on Symbolic and -Algebraic Computation, Cambridge, England, July 9-11, 1984, volume 174 of -Lecture Notes in Computer Science. Springer-Verlag, Berlin, Germany / -Heildelberg, Germany / London, UK / etc., 1984 ISBN 0-387-13350-X -LCCN QA155.7.E4 I57 1984 - keywords = "axiomref", + We describe in this paper how the problems of computing indefinite + integrals and solving linear ordinary differential equations in closed + form are now solved by computer algebra systems. After a brief review + of the mathematical history of those problems, we outline the two + major algorithms for them (respectively the Risch and Singer + algorithms) and the recent improvements on those algorithms which has + allowed them to be implemented." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Fitch 93]{Fit93} Fitch, J. (ed) -Design and Implementation of Symbolic Computation Systems -International Symposium DISCO '92 Proceedings. Springer-Verlag, Berlin, -Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0-387-57272-4 -(New York), 3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992 +\bibitem[Beneke 94]{BS94} + author = "Beneke, T. and Schwippert, W.", + title = "Double-track into the future: MathCAD will gain new users with + Standard and Plus versions", + year = "1994", + month = "July", + pages = "107-110", keywords = "axiomref", +Elektronik, 43(15) CODEN EKRKAR ISSN 0013-5658 \end{chunk} \begin{chunk}{ignore} -\bibitem[Fogus 11]{Fog11} Fogus, Michael -UnConj'' -\verb|clojure.com/blog/2011/11/22/unconj.html| +\bibitem[Bronstein 97a]{Bro97a} + author = "Bronstein, Manuel and Weil, Jacques-Arthur", + title = "On Symmetric Powers of Differential Operators", + series = "ISSAC'97", + year = "1997", + pages = "156-163", keywords = "axiomref", + url = + "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html" + paper = "Bro97a.pdf", + publisher = "ACM, NY", + abstract = " + We present alternative algorithms for computing symmetric powers of + linear ordinary differential operators. Our algorithms are applicable + to operators with coefficients in arbitrary integral domains and + become faster than the traditional methods for symmetric powers of + sufficiently large order, or over sufficiently complicated coefficient + domains. The basic ideas are also applicable to other computations + involving cyclic vector techniques, such as exterior powers of + differential or difference operators." \end{chunk} \begin{chunk}{ignore} -\bibitem[Fortenbacher 90]{For90} Fortenbacher, A. -Efficient type inference and coercion in computer algebra'' -In Miola [Mio90], pp56-60. ISBN 0-387-52531-9 (New York), 3-540-52531-9 -(Berlin). LCCN QA76.9.S88I576 1990 - keywords = "axiomref", +\bibitem[Borwein 00]{Bor00} + author = "Borwein, Jonathan", + title = "Multimedia tools for communicating mathematics", + year = "2000", + pages = "58", + isbn = "3-540-42450-4", + publisher = "Springer-Verlag", + keywords = "axiomref" \end{chunk} -\begin{chunk}{ignore} -\bibitem[Fouche 90]{Fou90} Fouche, Francois -Une implantation de l'algorithme de Kovacic en Scratchpad'' -Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e'' -Strasbourg, France, 1990 31pp - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@article{BT94, + author = "Brown, R. and Tonks, A.", + title = "Calculations with simplicial and cubical groups in AXIOM", + journal = "Journal of Symbolic Computation", + volume = "17", + number = "2", + pages = "159-179", + year = "1994", + month = "February", + misc = "CODEN JSYCEH ISSN 0747-7171", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[FSF 14]{FSF14} FSF -Free Software Directory'' -\verb|directory.fsf.org/wiki/Axiom| +\begin{chunk}{axiom.bib} +@misc{Brow95, + author = "Brown, Ronald and Dreckmann, Winfried", + title = "Domains of data and domains of terms in AXIOM", + year = "1995", keywords = "axiomref", + paper = "DB95.pdf", + abstract = " + The main new concept we wish to illustrate in this paper is a + distinction between domains of data'' and domains of terms'', and + its use in the programming of certain mathematical structures. + Although this distinction is implicit in much of the programming work + that has gone into the construction of Axiom categories and domains, + we believe that a formalisation of this is new, that standards and + conventions are necessary and will be useful in various other + contexts. We shall show how this concept may be used for the coding of + free categories and groupoids on directed graphs." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Frisco ]{Fris} Frisco -Objectives and Results'' -\verb|www.nag.co.uk/projects/frisco/frisco/node3.htm| +\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds) +EUROCAL '85: European Conference on Computer Algebra, Linz, Austria, +LLCN QA155.7.E4 E86 + isbn = "0-387-15983-5, 0-387-15984-3", + year = "1985", + month = "April", + publisher = "Springer-Verlag, Berlin, Germany", keywords = "axiomref", + misc = "Lecture Notes in Computer Science, Vol 204", \end{chunk} -\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael -Buchberger's algorithm and staggered linear bases'' -In Bruce W. Char, editor. Proceedings of the 1986 -Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, 1986 -Waterloo, Ontario, pp218-221 ACM Press, New York, NY 10036, USA, 1986. -ISBN 0-89791-199-7 LCCN QA155.7.E4 A281 1986 ACM order number 505860 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Buh05, + author = "Buhl, Soren L.", + title = "Some Reflections on Integrating a Computer Algebra System in R", + year = "2005", + keywords = "axiomref" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M. -On an installation of Buchberger's algorithm'' -Journal of Symbolic Computation, 6(2-3) pp275-286 1988 -CODEN JSYCEH ISSN 0747-7171 -\verb|www.sciencedirect.com/science/article/pii/S0747717188800488/pdf| -\verb|?md5=f6ccf63002ef3bc58aaa92e12ef18980&| -\verb|pid=1-s2.0-S0747717188800488-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/GM88.pdf| +\bibitem[Burge 91]{Burg91} + author = "Burge, W.H.", + title = "Scratchpad and the Rogers-Ramanujan identities", + year = "1991", + pages = "189-190", + isbn = "0-89791-437-6", keywords = "axiomref", abstract = " - Buchberger's algorithm calculates Groebner bases of polynomial - ideals. Its efficiency depends strongly on practical criteria for - detecting superfluous reductions. Buchberger recommends two - criteria. The more important one is interpreted in this paper as a - criterion for detecting redundant elements in a basis of a module of - syzygies. We present a method for obtaining a reduced, nearly minimal - basis of that module. The simple procedure for detecting (redundant - syzygies and )superfluous reductions is incorporated now in our - installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE - 3.3. The paper concludes with statistics stressing the good - computational properties of these installations." + This note sketches the part played by Scratchpad in obtaining new + proofs of Euler's theorem and the Rogers-Ramanujan Identities." \end{chunk} \begin{chunk}{axiom.bib} -@book{Gedd92, - author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George", - title = "Algorithms For Computer Algebra", - publisher = "Kluwer Academic Publishers", - isbn = "0-7923-9259-0", - month = "September", - year = "1992", +@techreport{BW87, + author = "Burge, W. and Watt, S.", + title = "Infinite structures in SCRATCHPAD II", + year = "1987", + institution = "IBM Research", + type = "Technical Report", + number = "RC 12794", keywords = "axiomref" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Gianni 87]{Gia87} Gianni, Patrizia -Primary Decomposition of Ideals'' -in [Wit87], pp12-13 +\bibitem[Burge 87a]{BWM87} + author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.", + title = "Streams and Power Series", + year = "1987", + pages = "9-12", keywords = "axiomref", +in [Wit87], pp9-12 \end{chunk} \begin{chunk}{ignore} -\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.; -Zacharias, Gail. -Gr\"obner Bases and Primary Decomposition of Polynomial Ideals'' -J. Symbolic Computation 6, 149-167 (1988) -\verb|www.sciencedirect.com/science/article/pii/S0747717188800403/pdf| -\verb|?md5=40c29b67947035884904fd4597ddf710&| -\verb|pid=1-s2.0-S0747717188800403-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Gia88.pdf| +\bibitem[Burge 89]{BW89} + author = "Burge, W. H. and Watt, S. M.", + title = "Infinite structures in Scratchpad II", + year = "1989", + pages = "138-148", + isbn = "3-540-51517-8", keywords = "axiomref", +in Davenport [Dav89], LCCN QA155.7.E4E86 1987 \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed) -Symbolic and Algebraic Computation. -International Symposium ISSAC '88, Rome, Italy, July 4-8, 1988. Proceedings, -volume 358 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, -Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3-540-51084-2 -LCCN QA76.95.I57 1988 Conference held jointly with AAECC-6 - keywords = "axiomref", - -\end{chunk} +\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T. -Algebraic solution of systems of polynomial equations using -Gr{\"o}bner bases.'' -In Huguet and Poli [HP89], pp247-257 ISBN 3-540-51082-6 LCCN QA268.A35 1987 +\bibitem[Calmet 94]{Cal94} Calmet, J. (ed) +Rhine Workshop on Computer Algebra, Proceedings. +Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Gil 92]{Gil92} Gil, I. -Computation of the Jordan canonical form of a square matrix (using -the Axiom programming language)'' -In Wang [Wan92], pp138-145. -ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover) -LCCN QA76.95.I59 1992 +\bibitem[Camion 92]{CCM92} + author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre", + title = "A combinatorial problem in Hamming Graphs and its solution + in Scratchpad", + year = "1992", + month = "January", keywords = "axiomref", +Rapports de recherche 1586, Institut National de Recherche en +Informatique et en Automatique, Le Chesnay, France, 12pp \end{chunk} \begin{chunk}{ignore} -\bibitem[Gomez-Diaz 92]{Gom92} G\'omez-D'iaz, Teresa -Quelques applications de l\'evaluation dynamique'' -Ph.D. Thesis L'Universite De Limoges March 1992 +\bibitem[Caprotti 00]{CCR00} + author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred", + title = "Java Phrasebooks for Computer Algebra and Automated Deduction", + url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf", + paper = "CCR00.pdf", keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gomez-Diaz 93]{Gom93} G\'omez-D\'iaz, Teresa -Examples of using Dynamic Constructible Closure'' -IMACS Symposium SC-1993 -%\verb|axiom-developer.org/axiom-website/papers/Gom93.pdf| - keywords = "axiomref", - abstract = " - We present here some examples of using the Dynamic Constructible - Closure'' program, which performs automatic case distinction in - computations involving parameters over a base field $K$. This program - is an application of the Dynamic Evaluation'' principle, which - generalizes traditional evaluation and was first used to deal with - algebraic numbers." +\begin{chunk}{axiom.bib} +@misc{CC99, + author = "Capriotti, O. and Carlisle, D.", + title = "OpenMath and MathML: Semantic Mark Up for Mathematics", + year = "1999", + url = "http://www.acm.org/crossroads/xrds6-2/openmath.html", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A. -Using MathCAD in teaching material and energy balance concepts'' -In Anonymous [Ano91], pp345-349 (vol. 1) 2 vols. - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Capr99, + author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and + Sterk, Hans", + title = "OpenMath Technology for Interactive Mathematical Documents", + year = "2002", + pages = "51-66", + publisher = "Springer-Verlag, Berlin, Germany", + url = "http://www.win.tue.nl/~hansc/lisbon.pdf", + paper = "Capr99.pdf", + misc = "in Multimedia Tools for Communicating Mathematics", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds) -Proceedings of the 1984 MACSYMA Users' Conference: -Schenectady, New York, July 23-25, 1984, General Electric, -Schenectady, NY, USA, 1984 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Carp04, + author = "Carpent, Quentin and Conil, Christophe", + title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26", + year = "2004", + paper = "Carp04.pdf", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H. -Official verion 1.0 of the Meta Content Dictionary'' -\verb|www.inf.ethz.ch/personal/gonnet/ContDict/Meta| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Chu85, + author = "Chudnovsky, D.V and Chudnovsky, G.V.", + title = "Elliptic Curve Calculations in Scratchpad II", + year = "1985", + institution = "Mathematics Dept., IBM Research", + type = "Scratchpad II Newsletter 1 (1)", + keywords = "axiomref" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P. -An abstract data type development of graded rings'' -In Fitch [Fit93], pp193-202. ISBN 0-387-57272-4 (New York), -3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992 +\bibitem[Chudnovsky 87]{Chu87} + author = "Chudnovsky, D.V and Chudnovsky, G.V.", + title = "New Analytic Methods of Polynomial Root Finding", + year = "1987", + pages = "2", keywords = "axiomref", +in [Wit87] \end{chunk} \begin{chunk}{ignore} -\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom; -Martin, Ursula -Hidden verification for computational mathematics'' -Journal of Symbolic Computation, Vol39, Num 5, pp539-567 (2005) -\verb|www.sciencedirect.com/science/article/pii/S0747717105000295| -%\verb|axiom-developer.org/axiom-website/papers/GKM05.pdf| +\bibitem[Chudnovsky 89]{Chu89} + author = "Chudnovsky, D.V. and Chudnovsky, G.V.", + title = "The computation of classical constants", + year = "1989", + month = "November", + pages = "8178-8182", keywords = "axiomref", - abstract = " - We present hidden verification as a means to make the power of - computational logic available to users of computer algebra systems - while shielding them from its complexity. We have implemented in PVS a - library of facts about elementary and transcendental function, and - automatic procedures to attempt proofs of continuity, convergence and - differentiability for functions in this class. These are called - directly from Maple by a simple pipe-lined interface. Hence we are - able to support the analysis of differential equations in Maple by - direct calls to PVS for: result refinement and verification, discharge - of verification conditions, harnesses to ensure more reliable - differential equation solvers, and verifiable look-up tables." +Proc. Natl. Acad. Sci. USA Vol 86 \end{chunk} -\begin{chunk}{ignore} -\bibitem[Grabe 98]{Gra98} Gr\"abe, Hans-Gert -About the Polynomial System Solve Facility of Axiom, Macyma, Maple -Mathematica, MuPAD, and Reduce'' -%\verb|axiom-developer.org/axiom-website/papers/Gra98.pdf| - keywords = "axiomref", - abstract = " - We report on some experiences with the general purpose Computer - Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and - Reduce solving systems of polynomial equations and the way they - present their solutions. This snapshot (taken in the spring of 1996) - of the current power of the different systems in a special area - concentrates on both CPU-times and the quality of the output." +\begin{chunk}{axiom.bib} +@proceedings{CJ86, + editor = "Chudnovsky, David and Jenks, Richard", + title = "Computers in Mathematics", + year = "1986", + month = "July", + isbn = "0-8247-8341-7", + note = "International Conference on Computers and Mathematics", + publisher = "Marcel Dekker, Inc", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U. -Das ComputeralgebraSystem AXIOM bei kryptologischen und -verkehrstheoretischen Untersuchungen des -Forschunginstituts der Deutschen Bundespost TELEKOM'' -Technischer Report TR 75.91.20, IBM Wissenschaftliches -Zentrum, Heidelberg, Germany, 1991 +\begin{chunk}{axiom.bib} +@misc{Cohe03, + author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and + Reinaldo, Ernesto and Sterk, Hans", + title = "Interactive Mathematical Documents on the Web", + year = "2003", + pages = "289-306", + editor = "Joswig, M. and Takayma, N.", + publisher = "Springer-Verlag, Berlin, Germany", keywords = "axiomref", + misc = "in Algebra, Geometry and Software Systems" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A. -Finite fields in Axiom'' -AXIOM Technical Report TR7/92 (ATR/5)(NP2522), -Numerical Algorithms Group, Inc., Downer's -Grove, IL, USA and Oxford, UK, 1992 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| -and Technical Report, IBM Heidelberg Scientific Center, 1992 +\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed) +EUROCODE '90 International Symposium on +Coding Theory and Applications Proceedings. Springer-Verlag, Berlin, Germany +/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0-387-54303-1 +(New York), 3-540-54303-1 (Berlin), LCCN QA268.E95 1990 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich; -Weispfenning, Volker (eds) -Computer algebra handbook: foundations, applications, systems. -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -2003. ISBN 3-540-65466-6 637pp Includes CDROM -\verb|www.springer.com/sgw/cda/frontpage/| -\verb|0,11855,1-102-22-1477871-0,00.html| +\bibitem[Conrad (a)]{CFMPxxa} + author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra", + title = "Approaching Inheritance from a Natural Mathematical Perspective + and from a Java Driven Viewpoint: a Comparative Review", keywords = "axiomref", + paper = "CFMPxxa.pdf", + abstract = " + It is well-known that few object-oriented programming languages allow + objects to change their nature at run-time. There have been a number + of reasons presented for this, but it appears that there is a real + need for matters to change. In this paper we discuss the need for + object-oriented programming languages to reflect the dynamic nature of + problems, particularly those arising in a mathematical context. It is + from this context that we present a framework that realistically + represents the dynamic and evolving characteristic of problems and + algorithms." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D. -SCRATCHPAD/1 -- an interactive facility for symbolic mathematics'' -In Petrick [Pet71], pp42-58. LCCN QA76.5.S94 1971 -\verb|delivery.acm.org/10.1145/810000/806266/p42-griesmer.pdf| -SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic -Manipulation pp45-48 -%\verb|axiom-developer.org/axiom-website/papers/GJ71.pdf| REF:00027 +\begin{chunk}{axiom.bib} +@misc{CFMPxxb, + author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra", + title = "Mathematical Use Cases lead naturally to non-standard Inheritance + Relationships: How to make them accessible in a mainstream language?", + paper = "CFMPxxb.pdf", keywords = "axiomref", abstract = " - The SCRATCHPAD/1 system is designed to provide an interactive symbolic - computational facility for the mathematician user. The system features - a user language designed to capture the style and succinctness of - mathematical notation, together with a facility for conveniently - introducing new notations into the language. A comprehensive system - library incorporates symbolic capabilities provided by such systems as - SIN, MATHLAB, and REDUCE." + Conceptually there is a strong correspondence between Mathematical + Reasoning and Object-Oriented techniques. We investigate how the ideas + of Method Renaming, Dynamic Inheritance and Interclassing can be used + to strengthen this relationship. A discussion is initiated concerning + the feasibility of each of these features." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R. -Experience with an online symbolic math system SCRATCHPAD'' -in Online'72 [Onl72] ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Cuyp10, + author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem", + title = "Interactive Geometry inside MathDox", + year = "2010", + url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf", + paper = "Cuyp10", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D. -SCRATCHPAD: A capsule view'' -ACM SIGPLAN Notices, 7(10) pp93-102, 1972. Proceedings of the symposium -on Two-dimensional man-machine communications. Mark B. Wells and -James B. Morris (eds.). - keywords = "axiomref", - -\end{chunk} +\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{chunk}{ignore} -\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y -SCRATCHPAD User's Manual'' -IBM Research Publication RA70 June 1975 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@inproceedings{Dalm97, + author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen}, + title = "An OpenMath 1.0 Implementation", + booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation", + series = "ISSAC'97", + year = "1997", + isbn = "0-89791-875-4", + location = "Kihei, Maui, Hawaii, USA", + pages = "241-248", + numpages = "8", + url = "http://doi.acm.org/10.1145/258726.258794", + doi = "10.1145/258726.258794", + acmid = "258794", + publisher = "ACM, New York, NY USA", + keywords = "axiomref" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y -A Set of SCRATCHPAD Examples'' -April 1976 (private copy) +\bibitem[Dalmas 92]{Dal92} Dalmas, S. +A polymorphic functional language applied to symbolic computation'' +In Wang [Wan92] pp369-375 ISBN 0-89791-489-9 (soft cover) 0-89791-490-2 +(hard cover) LCCN QA76.95.I59 1992 keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M. -Introduction to Gauss'' -SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic -Manipulation), 28(3) pp3-19 August 1994 CODEN SIGSBZ ISSN 0163-5824 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@misc{Daly88, + author = "Daly, Timothy", + title = "Axiom in an Educational Setting, Axiom course slide deck", + year = "1988", + month = "January", + keywords = "axiomref" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik -On Computing Limits in a Symbolic Manipulation System'' -Thesis, Swiss Federal Institute of Technology Z\"urich 1996 -Diss. ETH No. 11432 -\verb|www.cybertester.com/data/gruntz.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Gru96.pdf| - keywords = "axiomref", - abstract = " - This thesis presents an algorithm for computing (one-sided) limits - within a symbolic manipulation system. Computing limtis is an - important facility, as limits are used both by other functions such as - the definite integrator and to get directly some qualitative - information about a given function. - - The algorithm we present is very compact, easy to understand and easy - to implement. It overcomes the cancellation problem other algorithms - suffer from. These goals were achieved using a uniform method, namely - by expanding the whole function into a series in terms of its most - rapidly varying subexpression instead of a recursive bottom up - expansion of the function. In the latter approach exact error terms - have to be kept with each approximation in order to resolve the - cancellation problem, and this may lead to an intermediate expression - swell. Our algorithm avoids this problem and is thus suited to be - implemented in a symbolic manipulation system." +\begin{chunk}{ignore}TPDHERE +\bibitem[Daly 02]{Dal02} Daly, Timothy +Axiom as open source'' +SIGSAM Bulletin (ACM Special Interest Group +on Symbolic and Algebraic Manipulation) 36(1) pp28-?? March 2002 +CODEN SIGSBZ ISSN 0163-5824 + keywords = "axiomref", \end{chunk} -\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Boyle 88]{Boyl88} Boyle, Ann -Future Directions for Research in Symbolic Computation'' -Soc. for Industrial and Applied Mathematics, Philadelphia (1990) -\verb|www.eecis.udel.edu/~caviness/wsreport.pdf| -%\verb|axiom-developer.org/axiom-website/Boyl88.pdf| +\bibitem[Daly 03]{Dal03} Daly, Timothy +The Axiom Wiki Website'' +\verb|axiom.axiom-developer.org| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.; -Watt, Stephen M. -Construction of Algebraic Error Control Codes (ECC) on the Elliptic -Riemann Surface'' -in [Wit87], pp5-8 +\bibitem[Daly 06]{Dal06} Daly, Timothy +Axiom Volume 1: Tutorial'' +Lulu, Inc. 860 Aviation Parkway, +Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp +\verb|www.lulu.com/content/190827| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Heck 01]{Hec01} Heck, A. -Variables in computer algebra, mathematics and science'' -The International Journal of Computer Algebra in Mathematics Education -Vol. 8 No. 3 pp195-210 (2001) +\bibitem[Daly 09]{Dal09} Daly, Timothy +The Axiom Literate Documentation'' +\verb|axiom-developer.org/axiom-website/documentation.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds). -Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. -5th International Conference AAECC-5 Proceedings. -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -1989. ISBN 3-540-51082-6. LCCN QA268.A35 1987 +\bibitem[Daly 13]{Dal13} Daly, Timothy +Literate Programming in the Large'' +April 8-9, 2013 Portland Oregon +\verb|conf.writethedocs.org| +\verb|daly.axiom-developer.org| +\verb|www.youtube.com/watch?v=Av0PQDVTP4A| keywords = "axiomref", \end{chunk} -\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds) -Proceedings SC 93 -International IMACS Symposium on Symbolic Computation. New Trends and -Developments. LIFL Univ. Lille, Lille France, 1993 +\bibitem[Davenport 79a]{Dav79a} Davenport, J.H. +What can SCRATCHPAD/370 do?'' +VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed) -Trends in Computer Algebra, International Symposium -Bad Neuenahr, May 19-21, 1987, Proceedings, volume 296 of Lecture Notes in -Computer Science. -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -1988 ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 +\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D. +MODLISP -- an Introduction'' +Proc LISP80, 1980, and IBM RC8357 Oct 1980 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 69]{Jen69} Jenks, R. D. -META/LISP: An interactive translator writing system'' -Research Report International Business Machines, Inc., Thomas J. -Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970 +\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.; +Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.; +Trager, B. +Scratchpad'' +Mathematical Sciences Department, IBM Thomas Watson Research Center 1984 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 71]{Jen71} Jenks, R. D. -META/PLUS: The syntax extension facility for SCRATCHPAD'' -Research Report RC 3259, International Business Machines, Inc., Thomas J. -Watson Research Center, Yorktown Heights, NY, USA, 1971 -% REF:00040 +\bibitem[Davenport 84a]{Dav84a} Davenport, James H. +A New Algebra System'' +%\verb|axiom-developer.org/axiom-website/papers/Dav84a.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 74]{Jen74} Jenks, R. D. -The SCRATCHPAD language'' -ACM SIGPLAN Notices, 9(4) pp101-111 1974 CODEN SINODQ. ISSN 0362-1340 +\bibitem[Davenport 85]{Dav85} Davenport, James H. +The LISP/VM Foundation of Scratchpad II'' +The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985 +IBM Corporation, Yorktown Heights, NY keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jen76]{Jen76} Jenks, Richard D. -A pattern compiler'' -In Richard D. Jenks, editor, -SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic -Computation, August 10-12, 1976, Yorktown Heights, New York, pp60-65, -ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976 -QA9.58.A11 1976 +\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E. +Computer Algebra: Systems and Algorithms for Algebraic Computation. +Academic Press, New York, NY, USA, 1988, ISBN 0-12-204232-9 +\verb|staff.bath.ac.uk/masjhd/masternew.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DST88.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 79]{Jen79} Jenks, R. D. -MODLISP: An Introduction'' -Proc EUROSAM 79, pp466-480, 1979 and IBMRC8073 Jan 1980 +\bibitem[Davenport 14]{Dav14} Davenport, James H. +Computer Algebra textbook'' +\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Dav14.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M. -A Language for Computational Algebra'' -Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation, -Snowbird, Utah August, 1981 +\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed) +EUROCAL '87 European Conference on Computer Algebra Proceedings +Springer-Verlag, Berlin, Germany / Heidelberg, Germany / London, +UK / etc., 1989 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M. -A Language for Computational Algebra'' -SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981 +\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M. +Scratchpad's view of algebra I: Basic commutative algebra'' +In Miola [Mio90], pp40-54. ISBN 0-387-52531-9 (New York), +3-540-52531-9 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical +Report, ATR/1, NAG Ltd., Oxford, 1992 keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M. -A Language for Computational Algebra'' -IBM Research Report RC8930 IBM Yorktown Heights, NY +\begin{chunk}{axiom.bib} +@inproceedings{Dave91, + author = "Davenport, J. H. and Gianni, P. and Trager, B. M.", + title = "Scratchpad's View of Algebra II: + A Categorical View of Factorization", + booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation", + series = "ISSAC '91", + year = "1991", + isbn = "0-89791-437-6", + location = "Bonn, West Germany", + pages = "32--38", + numpages = "7", + url = "http://doi.acm.org/10.1145/120694.120699", + doi = "10.1145/120694.120699", + acmid = "120699", + publisher = "ACM", + address = "New York, NY, USA", keywords = "axiomref", + paper = "Dave91.pdf", + abstract = " + This paper explains how Scratchpad solves the problem of presenting a + categorical view of factorization in unique factorization domains, + i.e. a view which can be propagated by functors such as + SparseUnivariatePolynomial or Fraction. This is not easy, as the + constructive version of the classical concept of + UniqueFactorizationDomain cannot be so propagated. The solution + adopted is based largely on Seidenberg's conditions (F) and (P), but + there are several additional points that have to be borne in mind to + produce reasonably efficient algorithms in the required generality. + + The consequence of the algorithms and interfaces presented in this + paper is that Scratchpad can factorize in any extension of the + integers or finite fields by any combination of polynomial, fraction + and algebraic extensions: a capability far more general than any other + computer algebra system possesses. The solution is not perfect: for + example we cannot use these general constructions to factorize + polyinmoals in $\overline{Z[\sqrt{-5}]}[x]$ since the domain + $Z[\sqrt{-5}]$ is not a unique factorization domain, even though + $\overline{Z[\sqrt{-5}]}$ is, since it is a field. Of course, we can + factor polynomials in $\overline{Z}[\sqrt{-5}][x]$" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D. -The new SCRATCHPAD language and system for computer algebra'' -In Golden and Hussain [GH84], pp409-?? +\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M. +Scratchpad's view of algebra II: A categorical view of factorization'' +Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc., +Downer's Grove, IL, USA and Oxford, UK, December 1992 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D. -A primer: 11 keys to New Scratchpad'' -In Fitch [Fit84], pp123-147. ISBN 0-387-13350-X LCCN QA155.7.E4 I57 1984 +\bibitem[Davenport 92a]{Dav92a} Davenport, J. H. +The AXIOM system'' +AXIOM Technical Report TR5/92 (ATR/3) +(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and +Oxford, UK, December 1992 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.; -Watt, Stephen M. -Scratchpad II: An Abstract Datatype System for Mathematical Computation'' -Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc., -Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp -\verb|www.csd.uwo.ca/~watt/pub/reprints/1987-ima-spadadt.pdf| -%\verb|axiom-developer.org/axiom-website/papers/JWS86.pdf| +\bibitem[Davenport 92b]{Dav92b} Davenport, J. H. +How does one program in the AXIOM system?'' +AXIOM Technical Report TR6/92 (ATR/4)(NP2493) +Numerical Algorithms Group, Inc., Downer's +Grove, IL, USA and Oxford, UK December 1992 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +%\verb|axiom-developer.org/axiom-website/papers/Dav92b.pdf| keywords = "axiomref", abstract = " - Scratchpad II is an abstract datatype language and system that is - under development in the Computer Algebra Group, Mathematical Sciences - Department, at the IBM Thomas J. Watson Research Center. Some features - of APL that made computation particularly elegant have been borrowed. - Many different kinds of computational objects and data structures are - provided. Facilities for computation include symbolic integration, - differentiation, factorization, solution of equations and linear - algebra. Code economy and modularity is achieved by having - polymorphic packages of functions that may create datatypes. The use - of categories makes these facilities as general as possible." + Axiom is a computer algebra system superficially like many others, but + fundamentally different in its internal construction, and therefore in + the possibilities it offers to its users and programmers. In these + lecture notes, we will explain, by example, the methodology that the + author uses for programming substantial bits of mathematics in Axiom." \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.; -Watt, Stephen M. -Scratchpad II: an Abstract Datatype System for Mathematical Computation'' -Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296, -Springer Verlag, (1987) +\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M. +Scratchpad's view of algebra I: Basic commutative algebra'' +DISCO 90 Capri, Italy April 1990 ISBN 0-387-52531-9 pp40-54 +Technical Report TR3/92 (ATR/1)(NP2490), Numerical +Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK, +December 1992. +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M. -Scratchpad II: An abstract datatype system for mathematical computation'' -In Jan{\ss}en [Jan88], -pp12-?? ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 +\bibitem[Davenport 93]{Dav93} Davenport, J. H. +Primality testing revisited'' +Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc., +Downer's Grove, IL, USA and Oxford, UK, August 1993 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 88a]{Jen88a} Jenks, R. D. -A Guide to Programming in BOOT'' -Computer Algebra Group, Mathematical Sciences Department, IBM Research -Draft September 5, 1988 +\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele +The Unknown in Computer Algebra'' +\verb|axiom-wiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DFxx.pdf| + keywords = "axiomref", + abstract = " + Computer algebra systems have to deal with the confusion between + programming variables'' and mathematical symbols''. We claim that + they should also deal with unknowns'', i.e. elements whose values + are unknown, but whose type is known. For examples $x^p \ne x$ if $x$ + is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have + extended Axiom to deal with this concept." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Davenport 00]{Dav00} Davenport, James +13th OpenMath Meeting'' +James H. Davenport +A New Algebra System'' +May 1984 +\verb|xml.coverpages.org/openmath13.html| +%\verb|axiom-developer.org/axiom-website/papers/Dav00.pdf| + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Davenport 12]{Dav12} Davenport, J.H. +Computer Algebra'' +\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 88b]{Jen88b} Jenks, Richard -The Scratchpad II Computer Algebra System Interactive Environment Users -Guide'' - Spring 1988 +\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier +Computer Algebra'' \hfill +\verb|staff.bath.ac.uk/masjhd/masternew.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M. -Scratchpad II: an abstract datatype system for mathematical computation'' -In Jan{\ss}en -[Jan88], pp12-37. ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 +\bibitem[Dewar 94]{Dew94} Dewar, M. C. +Manipulating Fortran Code in AXIOM and the AXIOM-NAG Link'' +Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H. +and Laine, M. and Valkeila, E. pp1-12 University of Helsinki, Finland (1994) keywords = "axiomref", \end{chunk} \begin{chunk}{axiom.bib} -@book{Jenk92, - author = "Jenks, Richard D. and Sutor, Robert S.", - title = "AXIOM: The Scientific Computation System", - publisher = "Springer-Verlag, Berlin, Germany", - year = "1992", - isbn = "0-387-97855-0", - keywords = "axiomref" +@misc{Dewa, + author = "Dewar, Mike", + title = "OpenMath: An Overview", + url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf", + paper = "Dewa.pdf", + keywords = "axiomref" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M. -How to make AXIOM into a Scratchpad'' -In ACM [ACM94], pp32-40 ISBN 0-89791-638-7 LCCN QA76.95.I59 1994 -%\verb|axiom-developer.org/axiom-website/papers/JT94.pdf| +\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D. +Algebraic extensions and algebraic closure in Scratchpad II'' +In Gianni [Gia89], pp440-446 ISBN 3-540-51084-2 +LCCN QA76.95.I57 1998 Conference held jointly with AAECC-6 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki -Algebra, geometry, and software systems'' -Springer-Verlag ISBN 3-540-00256-1 p291 +\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard +Branch Cuts in Computer Algebra'' +1994 ISSAC, Oxford (UK), July 1994 +\verb|www.cs.berkeley.edu/~fateman/papers/ding.ps| +%\verb|axiom-developer.org/axiom-website/papers/Din94.pdf| keywords = "axiomref", + abstract = " + Many standard functions, such as the logarithms and square root + functions, cannot be defined continuously on the complex + plane. Mistaken assumptions about the properties of these functions + lead computer algebra systems into various conundrums. We discuss how + they can manipulate such functions in a useful fashion." \end{chunk} \begin{chunk}{ignore} -\bibitem[Joyner 06]{J006} Joyner, David -OSCAS - Maxima'' -SIGSAM Communications in Computer Algebra, 157 2006 -\verb|sage.math.washington.edu/home/wdj/sigsam/oscas-cca1.pdf| +\bibitem[DLMF]{DLMF}. +Digital Library of Mathematical Functions'' +\verb|dlmf.nist.gov/software/#T1| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Joyner 14]{JO14} Joyner, David -Links to some open source mathematical programs'' -\verb|www.opensourcemath.org/opensource_math.html| +\bibitem[Dooley 99]{Doo99} Dooley, Sam editor. +ISSAC 99: July 29-31, 1999, Simon Fraser University, +Vancouver, BC, Canada: proceedings of the 1999 International Symposium on +Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999. +ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 keywords = "axiomref", \end{chunk} -\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Kauers 08]{Kau08} Kauers, Manuel -Integration of Algebraic Functions: A Simple Heuristic for Finding -the Logarithmic Part'' -ISSAC July 2008 ACM 978-1-59593-904 pp133-140 -\verb|www.risc.jku.at/publications/download/risc_3427/Ka01.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Kau08.pdf| +\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel +A System for Axiomatic Programming'' +Proc. Conf. on Intelligent Computer Mathematics, Springer (2012) +\verb|www.axiomatics.org/~gdr/liz/cicm-2012.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DR12.pdf| keywords = "axiomref", abstract = " - A new method is proposed for finding the logarithmic part of an - integral over an algebraic function. The method uses Gr{\"o}bner bases - and is easy to implement. It does not have the feature of finding a - closed form of an integral whenever there is one. But it very often - does, as we will show by a comparison with the built-in integrators of - some computer algebra systems." - + We present the design and implementation of a system for axiomatic + programming, and its application to mathematical software + construction. Key novelties include a direct support for user-defined + axioms establishing local equality between types, and overload + resolution based on equational theories and user-defined local + axioms. We illustrate uses of axioms, and their organization into + concepts, in structured generic programming as practiced in + computational mathematical systems." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G. -Production of Argument SubPrograms in the AXIOM -- NAG -link: examples involving nonleanr systems'' -Technical Report TR1/94 -ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and -Oxford, UK, 1994 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +\bibitem[Doye 97]{Doy97} Doye, Nicolas James +Order Sorted Computer Algebra and Coercions'' +Ph.D. Thesis University of Bath 1997 +%\verb|axiom-developer.org/axiom-website/papers/Doy97.pdf| keywords = "axiomref", + abstract = " + Computer algebra systems are large collections of routines for solving + mathematical problems algorithmically, efficiently and above all, + symbolically. The more advanced and rigorous computer algebra systems + (for example, Axiom) use the concept of strong types based on + order-sorted algebra and category theory to ensure that operations are + only applied to expressions when they make sense''. + + In cases where Axiom uses notions which are not covered by current + mathematics we shall present new mathematics which will allow us to + prove that all such cases are reducible to cases covered by the + current theory. On the other hand, we shall also point out all the + cases where Axiom deviates undesirably from the mathematical ideal. + Furthermore we shall propose solutions to these deviations. + + Strongly typed systems (especially of mathematics) become unusable + unless the system can change the type in a way a user expects. We wish + any change expected by a user to be automated, natural'', and + unique. Coercions'' are normally viewed as natural type changing + maps''. This thesis shall rigorously define the word coercion'' in + the context of computer algebra systems. + + We shall list some assumptions so that we may prove new results so + that all coercions are unique. This concept is called coherence''. + + We shall give an algorithm for automatically creating all coercions in + type system which adheres to a set of assumptions. We shall prove that + this is an algorithm and that it always returns a coercion when one + exists. Finally, we present a demonstration implementation of this + automated coerion algorithm in Axiom." \end{chunk} \begin{chunk}{ignore} -\bibitem[Kelsey 99]{Kel99} Kelsey, Tom -Formal Methods and Computer Algebra: A Larch Specification of AXIOM -Categories and Functors'' -Ph.D. Thesis, University of St Andrews, 1999 +\bibitem[Doye 99]{Doy99} Doye, Nicolas J. +Automated coercion for Axiom'' +In Dooley [Doo99], pp229-235 +ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 ACM Press +\verb|www.acm.org/citation.cfm?id=309944| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom -Formal specification of computer algebra'' -University of St Andrews, 6th April 2000 -\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbs.ps| -%\verb|axiom-developer.org/axiom-website/papers/Kel00a.pdf| +\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio +Modeling Inheritance as Coercion in a Symbolic Computation System'' +ISSAC 2001 ACM 1-58113-417-7/01/0007 +%\verb|axiom-developer.org/axiom-website/papers/DR01.pdf| keywords = "axiomref", abstract = " - We investigate the use of formal methods languages and tools in the - design and development of computer algebra systems (henceforth CAS). - We demonstrate that errors in CAS design can be identified and - corrected by the use of (i) abstract specifications of types and - procedures, (ii) automated proofs of properties of the specifications, - and (iii) interface specifications which assist the verification of - pre- and post conditions of implemented code." + In this paper the analysis of the data structures used in a symbolic + computation system, called Kenzo, is undertaken. We deal with the + specification of the inheritance relationship since Kenzo is an + object-oriented system, written in CLOS, the Common Lisp Object + System. We focus on a particular case, namely the relationship between + simplicial sets and chain complexes, showing how the order-sorted + algebraic specifications formalisms can be adapted, through the + inheritance as coercion'' metaphor, in order to model this Kenzo + fragment." \end{chunk} \begin{chunk}{ignore} -\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom -Formal specification of computer algebra'' -(slides) University of St Andrews, Sept 21, 2000 -\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbstalk.ps| +\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and + Linton, Steve +Embedded Verification Techniques for Computer Algebra Systems'' +Grant citation GR/L48256 Nov 1, 1997-Feb 28, 2001 +\verb|www.cs.st-andrews.ac.uk/research/output/detail?output=ML97.php| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Kendall 99a]{Ken99a} Kendall, W.S. -Itovsn3 in AXIOM: modules, algebras and stochastic differentials'' -\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/| -\verb|kendall/personal/ppt/328.ps.gz| +\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne; +Kelsey, Tom; Martin, Ursula; Owre, Sam +Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS'' +TPHOLS 2001, Edinburgh +\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf| +%\verb|axiom-developer.org/axiom-website/papers/DGKM01.pdf| keywords = "axiomref", + abstract = " + We describe an interface between version 6 of the Maple computer + algebra system with the PVS automated theorem prover. The interface is + designed to allow Maple users access to the robust and checkable proof + environment of PVS. We also extend this environment by the provision + of a library of proof strategies for use in real analysis. We + demonstrate examples using the interface and the real analysis + library. These examples provide proofs which are both illustrative and + applicable to genuine symbolic computation problems." \end{chunk} \begin{chunk}{ignore} -\bibitem[Kendall 99b]{Ken99b} Kendall, W.S. -Symbolic It\^o calculus in AXIOM: an ongoing story -\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/| -\verb|kendall/personal/ppt/327.ps.gz| +\bibitem[Duval 92]{DJ92} Duval D.; Jung, F. +Examples of problem solving using computer algebra'' +IFIP Transactions. A. Computer Science and Technology, A-2 pp133-141, 143 1992 +CODEN ITATEC. ISSN 0926-5473 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Kosleff 91]{Kos91} P.-V. Koseleff -Word games in free Lie algebras: several bases and formulas'' -Theoretical Computer Science 79(1) pp241-256 Feb. 1991 CODEN TCSCDI -ISSN 0304-3975 +\bibitem[Duval 94]{Duv94} Duval, Dominique +Symbolic or algebraic computation?'' +Madrid Spain, NAG conference (private copy of paper) keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H. -Implementation of a geometry theorem proving package in SCRATCHPAD II'' -In Davenport [Dav89] pp246-257 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987 - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@article{Duva95, + author = "Duval, D.", + title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom", + journal = "Journal of Pure and Applied Algebra", + volume = "99", + year = "1995", + pages = "267--295.", + keywords = "axiomref" +} \end{chunk} -\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Lahey 08]{Lah08} Lahey, Tim -Sage Integration Testing'' -\verb|github.com/tjl/sage_int_testing| Dec. 2008 +\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William +The Sage Project'' +\verb|wstein.org/papers/icms/icms_2010.pdf| +%\verb|axiom-developer.org/axiom-website/papers/ES10.pdf| keywords = "axiomref", + abstract = " + Sage is a free, open source, self-contained distribution of + mathematical software, including a large library that provides a + unified interface to the components of this distribution. This library + also builds on the components of Sage to implement novel algorithms + covering a broad range of mathematical functionality from algebraic + combinatorics to number theory and arithmetic geometry." \end{chunk} +\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Lambe 89]{Lam89} Lambe, L. A. -Scratchpad II as a tool for mathematical research'' -Notices of the AMS, February 1928 pp143-147 +\bibitem[Fateman 90]{Fat90} Fateman, R. J. +Advances and trends in the design and construction of algebraic +manipulation systems'' +In Watanabe and Nagata [WN90], pp60-67 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lambe 91]{Lam91} Lambe, L. A. -Resolutions via homological perturbation'' -Journal of Symbolic Computation 12(1) pp71-87 July 1991 -CODEN JSYCEH ISSN 0747-7171 +\bibitem[Fateman 05]{Fat05} Fateman, R. J. +An incremental approach to building a mathematical expert out of software'' +4/19/2005\hfill +\verb|www.cs.berkeley.edu/~fateman/papers/axiom.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Fat05.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lambe 92]{Lam92} Lambe, Larry -Next Generation Computer Algebra Systems AXIOM and the Scratchpad -Concept: Applications to Research in Algebra'' -$21^{st}$ Nordic Congress of Mathematicians 1992 -%\verb|axiom-developer.org/axiom-website/papers/Lam92.pdf| +\bibitem[Fateman 06]{Fat06} Fateman, R. J. +Building Algebra Systems by Overloading Lisp'' +\verb|www.cs.berkeley.edu/~fateman/generic/overload-small.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Fat06.pdf| keywords = "axiomref", abstract = " - One way in which mathematicians deal with infinite amounts of data is - symbolic representation. A simple example is the quadratic equation - $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ - a formula which uses symbolic representation to describe the solutions - to an infinite class of equations. Most computer algebra systems can - deal with polynomials with symbolic coefficients, but what if symbolic - exponents are called for (e.g. $1+t^i$)? What if symbolic limits on - summations are also called for, for example - $1+t+\ldots+t^i=\sum_j{t^j}$ + Some of the earliest computer algebra systems (CAS) looked like + overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula + Algol, and others each took advantage of a pre-existing language base + and expanded the notion of a numeric value to include mathematical + expressions. Much more recently, perhaps encouraged by the growth in + popularity of C++, we have seen a renewal of the use of overloading to + implement a CAS. - The Scratchpad Concept'' is a theoretical ideal which allows the - implementation of objects at this level of abstraction and beyond in a - mathematically consistent way. The Axiom computer algebra system is an - implementation of a major part of the Scratchpad Concept. Axiom - (formerly called Scratchpad) is a language with extensible - parameterized types and generic operators which is based on the - notions of domains and categories. By examining some aspects of the - Axiom system, the Scratchpad Concept will be illustrated. It will be - shown how some complex problems in homologicial algebra were solved - through the use of this system." + This paper makes three points. 1. It is easy to do overloading in + Common Lisp, and show how to do it in detail. 2. Overloading per se + provides an easy solution to some simple programming problems. We show + how it can be used for a demonstration'' CAS. Other simple and + plausible overloadings interact nicely with this basic system. 3. Not + all goes so smoothly: we can view overloading as a case study and + perhaps an object lesson since it fails to solve a number of + fairly-well articulated and difficult design issues in CAS for which + other approaches are preferable." \end{chunk} \begin{chunk}{ignore} -\bibitem[Lambe 93]{Lam93} Lambe, Larry -On Using Axiom to Generate Code'' -(preprint) 1993 +\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James +Parameters in Computer Algebra'' keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard -Object-Oriented Mathematical Programming and Symbolic/Numeric Interface'' -$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993 -%\verb|axiom-developer.org/axiom-website/papers/LL93.pdf| +\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James; +Naciri, Hanane +Multi-values Computer Algebra'' +ISSN 0249-6399 Institut National De Recherche en Informatique et en +Automatique Sept. 2000 No. 4001 +\verb|hal.inria.fr/inria-00072643/PDF/RR-4401.pdf| +%\verb|axiom-developer.org/axiom-website/papers/FDN00b.pdf| keywords = "axiomref", abstract = " - The Axiom language is based on the notions of categories'', - domains'', and packages''. These concepts are used to build an - interface between symbolic and numeric calculations. In particular, an - interface to the NAG Fortran Library and Axiom's algebra and graphics - facilities is presented. Some examples of numerical calculations in a - symbolic computational environment are also included using the finite - element method. While the examples are elementary, we believe that - they point to very powerful methods for combining numeric and symbolic - computational techniques." - -\end{chunk} + One of the main strengths of computer algebra is being able to solve a + family of problems with one computation. In order to express not only + one problem but a family of problems, one introduces some symbols + which are in fact the parameters common to all the problems of the + family. -\begin{chunk}{ignore} -\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri -OpenMath Library for Computing on Riemann Surfaces'' -PhD thesis, Nov 2008 Florida State University -\verb|www.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html| - keywords = "axiomref", + The user must be able to understand in which way these parameters + affect the result when he looks at the answer. Otherwise it may lead + to completely wrong calculations, which when used for numerical + applications bring nonsensical answers. This is the case in most + current Computer Algebra Systems we know because the form of the + answer is never explicitly conditioned by the values of the + parameters. The user is not even informed that the given answer may be + wrong in some cases then computer algebra systems can not be entirely + trustworthy. We have introduced multi-valued expressions called {\sl + conditional} expressions, in which each potential value is associated + with a condition on some parameters. This is used, in particular, to + capture the situation in integration, where the form of the answer can + depend on whether certain quantities are positive, negative or + zero. We show that it is also necessary when solving modular linear + equations or deducing congruence conditions from complex expressions." \end{chunk} \begin{chunk}{ignore} -\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E. -The use of MathCAD and Theorist in the ChE classroom'' -In Anonymous [Ano91], pp287-299 (vol. 1) 2 vols. +\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed) +EUROSAM '84: International Symposium on Symbolic and +Algebraic Computation, Cambridge, England, July 9-11, 1984, volume 174 of +Lecture Notes in Computer Science. Springer-Verlag, Berlin, Germany / +Heildelberg, Germany / London, UK / etc., 1984 ISBN 0-387-13350-X +LCCN QA155.7.E4 I57 1984 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire -Dynamic Evaluation and Real Closure Implementation in Axiom'' -June 29, 1996 -\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps| -%\verb|axiom-developer.org/axiom-website/papers/Le96.ps| +\bibitem[Fitch 93]{Fit93} Fitch, J. (ed) +Design and Implementation of Symbolic Computation Systems +International Symposium DISCO '92 Proceedings. Springer-Verlag, Berlin, +Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0-387-57272-4 +(New York), 3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire -The Dynamic Real Closure implemented in Axiom'' -\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps| +\bibitem[Fogus 11]{Fog11} Fogus, Michael +UnConj'' +\verb|clojure.com/blog/2011/11/22/unconj.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed) -ISSAC '95: Proceedings of the 1995 International -Symposium on Symbolic and Algebraic Computation: July 10-12, 1995, Montreal, -Canada ISSAC-PROCEEDINGS-1995. ACM Press, New York, NY 10036, USA, 1995 -ISBN 0-89791-699-9 LCCN QA76.95 I59 1995 ACM order number 505950 +\bibitem[Fortenbacher 90]{For90} Fortenbacher, A. +Efficient type inference and coercion in computer algebra'' +In Miola [Mio90], pp56-60. ISBN 0-387-52531-9 (New York), 3-540-52531-9 +(Berlin). LCCN QA76.9.S88I576 1990 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno -Efficient Implementation of Polynomial Arithmetic in a Multiple-Level -Programming Environment'' -Lecture Notes in -Computer Science Springer Vol 4151/2006 ISBN 978-3-540-38084-9 pp12-23 -Proceedings of International Congress of Mathematical Software ICMS 2006 -\verb|www.csd.uwo.ca/~moreno//Publications/Li-MorenoMaza-ICMS-06.pdf| +\bibitem[Fouche 90]{Fou90} Fouche, Francois +Une implantation de l'algorithme de Kovacic en Scratchpad'' +Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e'' +Strasbourg, France, 1990 31pp keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel -A Quantitative Study of Reductions in Algebraic Libraries'' -PASCO 2010 -\verb|www.axiomatics.org/~gdr/concurrency/quant-pasco10.pdf| +\bibitem[FSF 14]{FSF14} FSF +Free Software Directory'' +\verb|directory.fsf.org/wiki/Axiom| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel -An Automatic Parallelization Framework for Algebraic Computation -Systems'' -ISSAC 2011 -\verb|www.axiomatics.org/~gdr/concurrency/oa-conc-issac11.pdf| -%\verb|axiom-developer.org/axiom-website/papers/YL11.pdf| +\bibitem[Frisco ]{Fris} Frisco +Objectives and Results'' +\verb|www.nag.co.uk/projects/frisco/frisco/node3.htm| keywords = "axiomref", - abstract = " - This paper proposes a non-intrusive automatic parallelization - framework for typeful and property-aware computer algebra systems." \end{chunk} +\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud; -Roy, Marie Francoise -Generic computation of the real closure of an ordered field'' -Math. and Computers in Simulation 42 pp 541-549 (1996) -%\verb|axiom-developer.org/axiom-website/papers/Liga96.pdf| +\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael +Buchberger's algorithm and staggered linear bases'' +In Bruce W. Char, editor. Proceedings of the 1986 +Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, 1986 +Waterloo, Ontario, pp218-221 ACM Press, New York, NY 10036, USA, 1986. +ISBN 0-89791-199-7 LCCN QA155.7.E4 A281 1986 ACM order number 505860 keywords = "axiomref", - abstract = " - This paper describes a generalization of the real closure computation - of an ordered field (Rioboo, 1991) enabling to use different technques - to code a single real algebraic number." \end{chunk} \begin{chunk}{ignore} -\bibitem[Linton 93]{Lin93} Linton, Steve -Vector Enumeration Programs, version 3.04'' -\verb|www.cs.st-andrews.ac.uk/~sal/nme/nme_toc.html#SEC1| +\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M. +On an installation of Buchberger's algorithm'' +Journal of Symbolic Computation, 6(2-3) pp275-286 1988 +CODEN JSYCEH ISSN 0747-7171 +\verb|www.sciencedirect.com/science/article/pii/S0747717188800488/pdf| +\verb|?md5=f6ccf63002ef3bc58aaa92e12ef18980&| +\verb|pid=1-s2.0-S0747717188800488-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/GM88.pdf| keywords = "axiomref", + abstract = " + Buchberger's algorithm calculates Groebner bases of polynomial + ideals. Its efficiency depends strongly on practical criteria for + detecting superfluous reductions. Buchberger recommends two + criteria. The more important one is interpreted in this paper as a + criterion for detecting redundant elements in a basis of a module of + syzygies. We present a method for obtaining a reduced, nearly minimal + basis of that module. The simple procedure for detecting (redundant + syzygies and )superfluous reductions is incorporated now in our + installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE + 3.3. The paper concludes with statistics stressing the good + computational properties of these installations." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri; -Sinor, Milan; Wester, Michael; Winkler, Franz -Computer Algebra - algorithms, systems and applications'' -June 2, 1997 -\verb|kfe.fjfi.cvut.cz/~liska/ca/all.html| - keywords = "axiomref", +\begin{chunk}{axiom.bib} +@book{Gedd92, + author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George", + title = "Algorithms For Computer Algebra", + publisher = "Kluwer Academic Publishers", + isbn = "0-7923-9259-0", + month = "September", + year = "1992", + keywords = "axiomref" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Lucks 86]{Luc86} Lucks, Michael -A fast implementation of polynomial factorization'' -In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic -and Algebraic Computation: SYMSAC '86, July 21-23, 1986, Waterloo, Ontario, -pp228-232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 -LCCN QA155.7.E4 A281 1986 ACM order number 505860 +\bibitem[Gianni 87]{Gia87} Gianni, Patrizia +Primary Decomposition of Ideals'' +in [Wit87], pp12-13 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lueken 77]{Lue77} Lueken, E. -Ueberlegungen zur Implementierung eines Formelmanipulationssystems'' -Master's thesis, Technischen Universit{\"{a}}t Carolo-Wilhelmina zu -Braunschweig. Braunschweig, Germany, 1977 +\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.; +Zacharias, Gail. +Gr\"obner Bases and Primary Decomposition of Polynomial Ideals'' +J. Symbolic Computation 6, 149-167 (1988) +\verb|www.sciencedirect.com/science/article/pii/S0747717188800403/pdf| +\verb|?md5=40c29b67947035884904fd4597ddf710&| +\verb|pid=1-s2.0-S0747717188800403-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Gia88.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A. -New quantum mechanical perturbation technique -using an 'electronic scratchpad' on an inexpensive computer'' -American Journal of Pyhsics, 59(3) pp270-273, March 1991. -CODEN AJPIAS ISSN 0002-9505 +\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed) +Symbolic and Algebraic Computation. +International Symposium ISSAC '88, Rome, Italy, July 4-8, 1988. Proceedings, +volume 358 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, +Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3-540-51084-2 +LCCN QA76.95.I57 1988 Conference held jointly with AAECC-6 keywords = "axiomref", \end{chunk} -\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia -Programming and certifying the CAD algorithm inside the coq system'' -Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl -Seminar Proceedings, Schloss Dagstuhl (2005) +\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T. +Algebraic solution of systems of polynomial equations using +Gr{\"o}bner bases.'' +In Huguet and Poli [HP89], pp247-257 ISBN 3-540-51082-6 LCCN QA268.A35 1987 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Mathews 89]{Mat89} Mathews, J. -Symbolic computational algebra applied to Picard iteration'' -Mathematics and computer education, 23(2) pp117-122 Spring 1989 CODEN MCEDDA, -ISSN 0730-8639 +\bibitem[Gil 92]{Gil92} Gil, I. +Computation of the Jordan canonical form of a square matrix (using +the Axiom programming language)'' +In Wang [Wan92], pp138-145. +ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover) +LCCN QA76.95.I59 1992 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[McJones 11]{McJ11} McJones, Paul -Software Presentation Group -- Common Lisp family'' -\verb|www.softwarepreservation.org/projects/LISP/common_lisp_family| +\bibitem[Gomez-Diaz 92]{Gom92} G\'omez-D'iaz, Teresa +Quelques applications de l\'evaluation dynamique'' +Ph.D. Thesis L'Universite De Limoges March 1992 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L. -Teaching advantages of transparent computer software -- MathCAD'' -CoED, 10(1) pp71-76, January-March 1990 CODEN CWLJDP ISSN 0736-8607 +\bibitem[Gomez-Diaz 93]{Gom93} G\'omez-D\'iaz, Teresa +Examples of using Dynamic Constructible Closure'' +IMACS Symposium SC-1993 +%\verb|axiom-developer.org/axiom-website/papers/Gom93.pdf| keywords = "axiomref", + abstract = " + We present here some examples of using the Dynamic Constructible + Closure'' program, which performs automatic case distinction in + computations involving parameters over a base field $K$. This program + is an application of the Dynamic Evaluation'' principle, which + generalizes traditional evaluation and was first used to deal with + algebraic numbers." \end{chunk} \begin{chunk}{ignore} -\bibitem[Miola 90]{Mio90} Miola, A. (ed) -Design and Implementation of Symbolic Computation Systems'' -International Symposium DISCO '90, Capri, Italy, April 10-12, 1990, Proceedings -volume 429 of Lecture Notes in Cmputer Science, -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -1990 ISBN 0-387-52531-9 (New York), 3-540-52531-9 (Berlin) LCCN QA76.9.S88I576 -1990 +\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A. +Using MathCAD in teaching material and energy balance concepts'' +In Anonymous [Ano91], pp345-349 (vol. 1) 2 vols. keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Miola 93]{Mio93} Miola, A. (ed) -Design and Implementation of Symbolic Computation Systems'' -International Symposium DISCO '93 Gmunden, Austria, September 15-17, 1993: -Proceedings. -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -1993 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993 +\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds) +Proceedings of the 1984 MACSYMA Users' Conference: +Schenectady, New York, July 23-25, 1984, General Electric, +Schenectady, NY, USA, 1984 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas -Using Commutativity Properties for Controlling Coercions'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/| -\verb|WeberA/MissuraWeber94a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Miss94.pdf| +\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H. +Official verion 1.0 of the Meta Content Dictionary'' +\verb|www.inf.ethz.ch/personal/gonnet/ContDict/Meta| keywords = "axiomref", - abstract = " - This paper investigates some soundness conditions which have to be - fulfilled in systems with coercions and generic operators. A result of - Reynolds on unrestricted generic operators is extended to generic - operators which obey certain constraints. We get natural conditions - for such operators, which are expressed within the theoretic framework - of category theory. However, in the context of computer algebra, there - arise examples of coercions and generic operators which do not fulfil - these conditions. We describe a framework -- relaxing the above - conditions -- that allows distinguishing between cases of ambiguities - which can be resolved in a quite natural sense and those which - cannot. An algorithm is presented that detects such unresolvable - ambiguities in expressions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Monagan 87]{Mon87} Monagan, Michael B. -Support for Data Structures in Scratchpad II'' -in [Wit87], pp17-18 +\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P. +An abstract data type development of graded rings'' +In Fitch [Fit93], pp193-202. ISBN 0-387-57272-4 (New York), +3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Monagan 93]{Mon93} Monagan, M. B. -Gauss: a parameterized domain of computation system with -support for signature functions'' -In Miola [Mio93], pp81-94 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993 +\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom; +Martin, Ursula +Hidden verification for computational mathematics'' +Journal of Symbolic Computation, Vol39, Num 5, pp539-567 (2005) +\verb|www.sciencedirect.com/science/article/pii/S0747717105000295| +%\verb|axiom-developer.org/axiom-website/papers/GKM05.pdf| keywords = "axiomref", + abstract = " + We present hidden verification as a means to make the power of + computational logic available to users of computer algebra systems + while shielding them from its complexity. We have implemented in PVS a + library of facts about elementary and transcendental function, and + automatic procedures to attempt proofs of continuity, convergence and + differentiability for functions in this class. These are called + directly from Maple by a simple pipe-lined interface. Hence we are + able to support the analysis of differential equations in Maple by + direct calls to PVS for: result refinement and verification, discharge + of verification conditions, harnesses to ensure more reliable + differential equation solvers, and verifiable look-up tables." \end{chunk} \begin{chunk}{ignore} -\bibitem[Mora 89]{Mor89} Mora, T. (ed) -Applied Algebra, Algebraic Algorithms and Error-Correcting -Codes, 6th International Conference, AAECC-6, Rome, Italy, July 4-8, 1998, -Proceedings, volume 357 of Lecture Notes in Computer Science -Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., -1989 ISBN 3-540-51083-4, LCCN QA268.A35 1988 Conference held jointly with -ISSAC '88 +\bibitem[Grabe 98]{Gra98} Gr\"abe, Hans-Gert +About the Polynomial System Solve Facility of Axiom, Macyma, Maple +Mathematica, MuPAD, and Reduce'' +%\verb|axiom-developer.org/axiom-website/papers/Gra98.pdf| keywords = "axiomref", + abstract = " + We report on some experiences with the general purpose Computer + Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and + Reduce solving systems of polynomial equations and the way they + present their solutions. This snapshot (taken in the spring of 1996) + of the current power of the different systems in a special area + concentrates on both CPU-times and the quality of the output." \end{chunk} \begin{chunk}{ignore} -\bibitem[Moses 71]{Mos71} Moses, Joel -Algebraic Simplification: A Guide for the Perplexed'' -CACM August 1971 Vol 14 No. 8 pp527-537 +\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U. +Das ComputeralgebraSystem AXIOM bei kryptologischen und +verkehrstheoretischen Untersuchungen des +Forschunginstituts der Deutschen Bundespost TELEKOM'' +Technischer Report TR 75.91.20, IBM Wissenschaftliches +Zentrum, Heidelberg, Germany, 1991 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Moses 08]{Mos08} Moses, Joel -Macsyma: A Personal History'' -Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago -\verb|esd.mit.edu/Faculty_Pages/moses/Macsyma.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Mos08.pdf| +\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A. +Finite fields in Axiom'' +AXIOM Technical Report TR7/92 (ATR/5)(NP2522), +Numerical Algorithms Group, Inc., Downer's +Grove, IL, USA and Oxford, UK, 1992 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +and Technical Report, IBM Heidelberg Scientific Center, 1992 keywords = "axiomref", - abstract = " - The Macsyma system arose out of research on mathematical software in - the AI group at MIT in the 1960's. Algorithm development in symbolic - integration and simplification arose out of the interest of people, - such as the author, who were also mathematics students. The later - development of algorithms for the GCD of sparse polynomials, for - example, arose out of the needs of our user community. During various - times in the 1970's the computer on which Macsyma ran was one of the - most popular notes on the ARPANET. We discuss the attempts in the late - 70's and the 80's to develop Macsyma systems that ran on popular - computer architectures. Finally, we discuss the impact of the - fundamental ideas in Macsyma on current research on large scale - engineering systems." \end{chunk} -\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian -From Untyped to Polymorphically Typed Objects in Mathematical Web -Services'' -%\verb|axiom-developer.org/axiom-website/papers/NPxx.pdf| +\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich; +Weispfenning, Volker (eds) +Computer algebra handbook: foundations, applications, systems. +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +2003. ISBN 3-540-65466-6 637pp Includes CDROM +\verb|www.springer.com/sgw/cda/frontpage/| +\verb|0,11855,1-102-22-1477871-0,00.html| keywords = "axiomref", - abstract = " - OpenMath is a widely recognized approach to the semantic markup of - mathematics that is often used for communication between OpenMath - compliant systems. The Aldor language has a sophisticated - category-based type system that was specifically developed for the - purpose of modelling mathematical structures, while the system itself - supports the creation of small-footprint applications suitable for - deployment as web services. In this paper we present our first results - of how one may perform translations from generic OpenMath objects into - values in specific Aldor domains, describing how the Aldor interfae - domain ExpresstionTree is used to achieve this. We outline our Aldor - implementation of an OpenMath translator, and describe an efficient - extention of this to the Parser category. In addition, the Aldor - service creation and invocation mechanism are explained. Thus we are - in a position to develop and deploy mathematical web services whose - descriptions may be directly derived from Aldor's rich type language." \end{chunk} \begin{chunk}{ignore} -\bibitem[Naylor 95]{N95} Naylor, Bill -Symbolic Interface for an advanced hyperbolic PDE solver'' -\verb|www.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps| -%\verb|axiom-developer.org/axiom-website/papers/N95.pdf| +\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D. +SCRATCHPAD/1 -- an interactive facility for symbolic mathematics'' +In Petrick [Pet71], pp42-58. LCCN QA76.5.S94 1971 +\verb|delivery.acm.org/10.1145/810000/806266/p42-griesmer.pdf| +SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic +Manipulation pp45-48 +%\verb|axiom-developer.org/axiom-website/papers/GJ71.pdf| REF:00027 keywords = "axiomref", abstract = " - An Axiom front end is described, which is used to generate - mathematical objects needed by one of the latest NAG routines, to be - included in the Mark 17 version of the NAG Numerical library. This - routine uses powerful techniques to find the solution to Hyperbolic - Partial Differential Equations in conservation form and in one spatial - dimension. These mathematical objects are non-trivial, requiring much - mathematical knowledge on the part of the user, which is otherwise - irrelvant to the physical problem which is to be solved. We discuss - the individual mathematical objects, considering the mathematical - theory which is relevant, and some of the problems which have been - encountered and solved during the FORTRAN generation necessary to - realise the object. Finally we display some of our results." + The SCRATCHPAD/1 system is designed to provide an interactive symbolic + computational facility for the mathematician user. The system features + a user language designed to capture the style and succinctness of + mathematical notation, together with a facility for conveniently + introducing new notations into the language. A comprehensive system + library incorporates symbolic capabilities provided by such systems as + SIN, MATHLAB, and REDUCE." \end{chunk} \begin{chunk}{ignore} -\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H. -A Monte-Carlo Extension to a Category-Based Type System'' -\verb|www.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps| -%\verb|axiom-developer.org/axiom-website/papers/ND00.pdf| +\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R. +Experience with an online symbolic math system SCRATCHPAD'' +in Online'72 [Onl72] ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes keywords = "axiomref", - abstract = " - The normal claim for mathematics is that all calculations are 100\% - accurate and therefore one calculation can rely completely on the - results of sub-calculations, hoever there exist {\sl Monte-Carlo} - algorithms which are often much faster than the equivalent - deterministic ones where the results will have a prescribed - probability (presumably small) of being incorrect. However there has - been little discussion of how such algorithms can be used as building - blocks in Computer Algebra. In this paper we describe how the - computational category theory which is the basis of the type structure - used in the Axiom computer algebra system may be extended to cover - probabilistic algorithms, which use Monte-Carlo techniques. We follow - this with a specific example which uses Straight Line Program - representation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Norman 75]{Nor75} Norman, A. C. -Computing with formal power series'' -ACM Transactions on Mathematical Software, 1(4) pp346-356 -Dec. 1975 CODEN ACMSCU ISSN 0098-3500 +\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D. +SCRATCHPAD: A capsule view'' +ACM SIGPLAN Notices, 7(10) pp93-102, 1972. Proceedings of the symposium +on Two-dimensional man-machine communications. Mark B. Wells and +James B. Morris (eds.). keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Norman 75a]{Nor75a} Norman, A.C. -The SCRATCHPAD Power Series Package'' -IBM T.J. Watson Research RC4998 +\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y +SCRATCHPAD User's Manual'' +IBM Research Publication RA70 June 1975 keywords = "axiomref", \end{chunk} -\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Ollivier 89]{Oll89} Ollivier, F. -Inversibility of rational mappings and structural -identifiablility in automatics'' -In ACM [ACM89], pp43-54 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989 +\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y +A Set of SCRATCHPAD Examples'' +April 1976 (private copy) keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Online 72]{Onl72}. -Online 72: conference proceedings ... international conference on online -interactive computing, Brunel University, Uxbridge, England, 4-7 September -1972 ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes. +\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M. +Introduction to Gauss'' +SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic +Manipulation), 28(3) pp3-19 August 1994 CODEN SIGSBZ ISSN 0163-5824 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[OpenMath]{OpenMa}. -OpenMath Technical Overview'' -\verb|www.openmath.org/overview/technical.html| +\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik +On Computing Limits in a Symbolic Manipulation System'' +Thesis, Swiss Federal Institute of Technology Z\"urich 1996 +Diss. ETH No. 11432 +\verb|www.cybertester.com/data/gruntz.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Gru96.pdf| keywords = "axiomref", + abstract = " + This thesis presents an algorithm for computing (one-sided) limits + within a symbolic manipulation system. Computing limtis is an + important facility, as limits are used both by other functions such as + the definite integrator and to get directly some qualitative + information about a given function. + + The algorithm we present is very compact, easy to understand and easy + to implement. It overcomes the cancellation problem other algorithms + suffer from. These goals were achieved using a uniform method, namely + by expanding the whole function into a series in terms of its most + rapidly varying subexpression instead of a recursive bottom up + expansion of the function. In the latter approach exact error terms + have to be kept with each approximation in order to resolve the + cancellation problem, and this may lead to an intermediate expression + swell. Our algorithm avoids this problem and is thus suited to be + implemented in a symbolic manipulation system." \end{chunk} -\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Page 07]{Pa07} Page, William S. -Axiom - Open Source Computer Algebra System'' -Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114 +\bibitem[Boyle 88]{Boyl88} Boyle, Ann +Future Directions for Research in Symbolic Computation'' +Soc. for Industrial and Applied Mathematics, Philadelphia (1990) +\verb|www.eecis.udel.edu/~caviness/wsreport.pdf| +%\verb|axiom-developer.org/axiom-website/Boyl88.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Petitot 90]{Pet90} Petitot, Michel -Types r\'ecursifs en scratchpad, application aux polyn\^omes non -commutatifs'' -LIFL, 1990 +\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.; +Watt, Stephen M. +Construction of Algebraic Error Control Codes (ECC) on the Elliptic +Riemann Surface'' +in [Wit87], pp5-8 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Petitot 93]{Pet93} Petitot, M. -Experience with Axiom'' -In Jacob et al. [JOS93], page 240 +\bibitem[Heck 01]{Hec01} Heck, A. +Variables in computer algebra, mathematics and science'' +The International Journal of Computer Algebra in Mathematics Education +Vol. 8 No. 3 pp195-210 (2001) keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Petric 71]{Pet71} Petric, S. R. (ed) -Proceedings of the second symposium on Symbolic and -Algebraic Manipulation, March 23-25, 1971, Los Angeles, California, ACM Press, -New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971 +\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds). +Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. +5th International Conference AAECC-5 Proceedings. +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +1989. ISBN 3-540-51082-6. LCCN QA268.A35 1987 keywords = "axiomref", \end{chunk} +\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Pinch 93]{Pin93} Pinch, R.G.E. -Some Primality Testing Algorithms'' -Devlin, Keith (ed.) -Computers and Mathematics November 1993, Vol 40, Number 9 pp1203-1210 +\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds) +Proceedings SC 93 +International IMACS Symposium on Symbolic Computation. New Trends and +Developments. LIFL Univ. Lille, Lille France, 1993 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Poll (b)]{Polxx} Poll, Erik -The type system of Axiom'' -%\verb|axiom-developer.org/axiom-website/papers/Polxx.pdf| +\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed) +Trends in Computer Algebra, International Symposium +Bad Neuenahr, May 19-21, 1987, Proceedings, volume 296 of Lecture Notes in +Computer Science. +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +1988 ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Purtilo 86]{Pur86} Purtilo, J. -Applications of a software interconnection system in mathematical -problem solving environments'' In Bruce W. Char, editor. Proceedings of the -1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, -ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 LCCN QA155.7.E4 -A281 1986 ACM order number 505860 +\bibitem[Jenks 69]{Jen69} Jenks, R. D. +META/LISP: An interactive translator writing system'' +Research Report International Business Machines, Inc., Thomas J. +Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970 keywords = "axiomref", \end{chunk} -\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Rainer 14]{Rain14} Joswig, Rainer -2014: 30+ Years Common Lisp the Language'' -\verb|lispm.de/30ycltl| +\bibitem[Jenks 71]{Jen71} Jenks, R. D. +META/PLUS: The syntax extension facility for SCRATCHPAD'' +Research Report RC 3259, International Business Machines, Inc., Thomas J. +Watson Research Center, Yorktown Heights, NY, USA, 1971 +% REF:00040 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud -Quelques aspects du calcul exact avec des nombres r\'eels'' -Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg -%\verb|axiom-developer.org/axiom-website/papers/Riob03a.ps| +\bibitem[Jenks 74]{Jen74} Jenks, R. D. +The SCRATCHPAD language'' +ACM SIGPLAN Notices, 9(4) pp101-111 1974 CODEN SINODQ. ISSN 0362-1340 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud -Towards Faster Real Algebraic Numbers'' -J. of Symbolic Computation 36 pp 513-533 (2003) -%\verb|axiom-developer.org/axiom-website/papers/Riob03.pdf| +\bibitem[Jen76]{Jen76} Jenks, Richard D. +A pattern compiler'' +In Richard D. Jenks, editor, +SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic +Computation, August 10-12, 1976, Yorktown Heights, New York, pp60-65, +ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976 +QA9.58.A11 1976 keywords = "axiomref", - abstract = " - This paper presents a new encoding scheme for real algebraic number - manipulations which enhances current Axiom's real closure. Algebraic - manipulations are performed using different instantiations of - sub-resultant-like algorithms instead of Euclidean-like algorithms. - We use these algorithms to compute polynomial gcds and Bezout - relations, to compute the roots and the signs of algebraic - numbers. This allows us to work in the ring of real algebraic integers - instead of the field of read algebraic numbers avoiding many - denominators." \end{chunk} \begin{chunk}{ignore} -\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas -Does Axiom Solve Systems of O.D.E's Like Mathematica?'' -July 1993 -%\verb|axiom-developer.org/axiom-website/papers/Rob93.pdf| +\bibitem[Jenks 79]{Jen79} Jenks, R. D. +MODLISP: An Introduction'' +Proc EUROSAM 79, pp466-480, 1979 and IBMRC8073 Jan 1980 keywords = "axiomref", - abstract = " - If I were demonstrating Axiom and were asked this question, my reply - would be No, but I am not sure that this is a bad thing''. And I - would illustrate this with the following example. - - Consider the following system of O.D.E.'s - $- \begin{array}{rcl} - \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\ - \frac{dx_2}{dt} & = & x_1 - x_2 - \end{array} -$ - This is a very simple system: $x_1$ is actually uncoupled from $x_2$" \end{chunk} \begin{chunk}{ignore} -\bibitem[Rioboo 92]{Rio92} Rioboo, R. -Real algebraic closure of an ordered field, implementation in Axiom'' -In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover) -0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992 -%\verb|axiom-developer.org/axiom-website/papers/Rio92.pdf| +\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M. +A Language for Computational Algebra'' +Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation, +Snowbird, Utah August, 1981 keywords = "axiomref", - abstract = " - Real algebraic numbers appear in many Computer Algebra problems. For - instance the determination of a cylindrical algebraic decomposition - for an euclidean space requires computing with real algebraic numbers. - This paper describes an implementation for computations with the real - roots of a polynomial. This process is designed to be recursively - used, so the resulting domain of computation is the set of all real - algebraic numbers. An implementation for the real algebraic closure - has been done in Axiom (previously called Scratchpad)." \end{chunk} \begin{chunk}{ignore} -\bibitem[Roesner 95]{Roe95} Roesner, K. G. -Verified solutions for parameters of an exact solution for -non-Newtonian liquids using computer algebra'' Zeitschrift fur Angewandte -Mathematik und Physik, 75 (suppl. 2):S435-S438, 1995 ISSN 0044-2267 +\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M. +A Language for Computational Algebra'' +SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981 keywords = "axiomref", \end{chunk} -\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Sage 14]{Sage14} Stein, William -Sage'' -\verb|www.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html| +\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M. +A Language for Computational Algebra'' +IBM Research Report RC8930 IBM Yorktown Heights, NY keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Salvy 89]{Sal89} Salvy, B. -Examples of automatic asymptotic expansions'' -Technical Report 114, -Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp +\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D. +The new SCRATCHPAD language and system for computer algebra'' +In Golden and Hussain [GH84], pp409-?? keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Salvy 91]{Sal91} Salvy, B. -Examples of automatic asymptotic expansions'' -SIGSAM Bulletin (ACM Special Interest Group on Symbolic and -Algebraic Manipulation), 25(2) pp4-17 -April 1991 CODEN SIGSBZ ISSN 0163-5824 +\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D. +A primer: 11 keys to New Scratchpad'' +In Fitch [Fit84], pp123-147. ISBN 0-387-13350-X LCCN QA155.7.E4 I57 1984 keywords = "axiomref", \end{chunk} -\begin{chunk}{axiom.bib} -@article{Saun80, - author = "Saunders, B. David", - title = "A Survey of Available Systems", - journal = "SIGSAM Bull.", - issue_date = "November 1980", - volume = "14", - number = "4", - month = "November", - year = "1980", - issn = "0163-5824", - pages = "12--28", - numpages = "17", - url = "http://doi.acm.org/10.1145/1089235.1089237", - doi = "10.1145/1089235.1089237", - acmid = "1089237", - publisher = "ACM", - address = "New York, NY, USA", - keywords = "axiomref,survey", - paper = "Saun80.pdf" -} +\begin{chunk}{ignore} +\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.; +Watt, Stephen M. +Scratchpad II: An Abstract Datatype System for Mathematical Computation'' +Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc., +Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp +\verb|www.csd.uwo.ca/~watt/pub/reprints/1987-ima-spadadt.pdf| +%\verb|axiom-developer.org/axiom-website/papers/JWS86.pdf| + keywords = "axiomref", + abstract = " + Scratchpad II is an abstract datatype language and system that is + under development in the Computer Algebra Group, Mathematical Sciences + Department, at the IBM Thomas J. Watson Research Center. Some features + of APL that made computation particularly elegant have been borrowed. + Many different kinds of computational objects and data structures are + provided. Facilities for computation include symbolic integration, + differentiation, factorization, solution of equations and linear + algebra. Code economy and modularity is achieved by having + polymorphic packages of functions that may create datatypes. The use + of categories makes these facilities as general as possible." \end{chunk} \begin{chunk}{ignore} -\bibitem[Schu 92]{Sch92} Sch\"u, J. -Implementing des Cartan-Kuranishi-Theorems in AXIOM'' -Master's diploma thesis (in german), Institut f\"ur Algorithmen und -Kognitive Systeme, Universit\"t Karlsruhe 1992 +\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.; +Watt, Stephen M. +Scratchpad II: an Abstract Datatype System for Mathematical Computation'' +Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296, +Springer Verlag, (1987) keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Schwarz 88]{Sch88} Schwarz, F. -Programming with abstract data types: the symmetry package SPDE -in Scratchpad'' -In Jan{\ss}en [Jan88], pp167-176, ISBN 3-540-18928-9, -0-387-18928-9 LCCN QA155.7.E4T74 1988 +\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M. +Scratchpad II: An abstract datatype system for mathematical computation'' +In Jan{\ss}en [Jan88], +pp12-?? ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Schwarz 89]{Sch89} Schwarz, F. -A factorization algorithm for linear ordinary differential equations'' -In ACM [ACM89], pp17-25 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989 +\bibitem[Jenks 88a]{Jen88a} Jenks, R. D. +A Guide to Programming in BOOT'' +Computer Algebra Group, Mathematical Sciences Department, IBM Research +Draft September 5, 1988 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Schwarz 91]{Sch91} Schwarz, F. -Monomial orderings and Gr{\"o}bner bases'' -SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic -Manipulation) 2591) pp10-23 Jan. 1991 CODEN SIGSBZ ISSN 0163-5824 +\bibitem[Jenks 88b]{Jen88b} Jenks, Richard +The Scratchpad II Computer Algebra System Interactive Environment Users +Guide'' + Spring 1988 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus -Analysis and Application of the Formal Theory of Partial Differential -Equations'' -PhD thesis, School of Physics and Materials, Lancaster University (1994) -\verb|www.mathematik.uni-kassel.de/~seiler/Papers/Diss/diss.ps.gz| -%\verb|axiom-developer.org/axiom-website/papers/Sei94.pdf| +\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M. +Scratchpad II: an abstract datatype system for mathematical computation'' +In Jan{\ss}en +[Jan88], pp12-37. ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988 keywords = "axiomref", - abstract = " - An introduction to the formal theory of partial differential equations - is given emphasizing the properties of involutive symbols and - equations. An algorithm to complete any differential equation to an - involutive one is presented. For an involutive equation possible - values for the number of arbitrary functions in its general solution - are determined. The existence and uniqueness of solutions for analytic - equations is proven. Applications of these results include an - analysis of symmetry and reduction methods and a study of gauge - systems. It is show that the Dirac algorithm for systems with - constraints is closely related to the completion of the equation of - motion to an involutive equation. Specific examples treated comprise - the Yang-Mills Equations, Einstein Equations, complete and Jacobian - systems, and some special models in two and three dimensions. To - facilitate the involved tedious computations an environment for - geometric approaches to differential equations has been developed in - the computer algebra system Axiom. The appendices contain among others - brief introductions into Carten-K{\"a}hler Theory and Janet-Riquier - Theory." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@book{Jenk92, + author = "Jenks, Richard D. and Sutor, Robert S.", + title = "AXIOM: The Scientific Computation System", + publisher = "Springer-Verlag, Berlin, Germany", + year = "1992", + isbn = "0-387-97855-0", + keywords = "axiomref" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler 94a]{Sei94a} Seiler, W.M. -Completion to involution in AXIOM'' -in Calmet [Cal94] pp103-104 +\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M. +How to make AXIOM into a Scratchpad'' +In ACM [ACM94], pp32-40 ISBN 0-89791-638-7 LCCN QA76.95.I59 1994 +%\verb|axiom-developer.org/axiom-website/papers/JT94.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Sieler 94b]{Sei94b} Seiler, W.M. -Pseudo differential operators and integrable systems in AXIOM'' -Computer Physics Communications, 79(2) pp329-340 April 1994 CODEN CPHCBZ -ISSN 0010-4655 -%\verb|axiom-developer.org/axiom-website/papers/Sei94b.pdf| +\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki +Algebra, geometry, and software systems'' +Springer-Verlag ISBN 3-540-00256-1 p291 keywords = "axiomref", - abstract = " - An implementation of the algebra of pseudo differential operators in - the computer algebra system Axiom is described. In several exmaples - the application of the package to typical computations in the theory - of integrable systems is demonstrated." \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler 95]{Sei95} Seiler, W.M. -Applying AXIOM to partial differential equations'' -Internal Report 95-17, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik -1995 -%\verb|axiom-developer.org/axiom-website/papers/Sei95.pdf| +\bibitem[Joyner 06]{J006} Joyner, David +OSCAS - Maxima'' +SIGSAM Communications in Computer Algebra, 157 2006 +\verb|sage.math.washington.edu/home/wdj/sigsam/oscas-cca1.pdf| keywords = "axiomref", - abstract = " - We present an Axiom environment called JET for geometric computations - with partial differential equations within the framework of the jet - bundle formalism. This comprises expecially the completion of a given - differential equation to an involutive one according to the - Cartan-Kuranishi Theorem and the setting up of the determining system - for the generators of classical and non-classical Lie - symmetries. Details of the implementations are described and - applications are given. An appendix contains tables of all exported - functions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J. -JET -- An Axiom Environment for Geometric Computations with Differential -Equations'' -%\verb|axiom-developer.org/axiom-website/papers/SC95.pdf| +\bibitem[Joyner 14]{JO14} Joyner, David +Links to some open source mathematical programs'' +\verb|www.opensourcemath.org/opensource_math.html| keywords = "axiomref", - abstract = " - JET is an environment within the computer algebra system Axiom to - perform such computations. The current implementation emphasises the - two key concepts involution and symmetry. It provides some packages - for the completion of a given system of differential equations to an - equivalent involutive one based on the Cartan-Kuranishi theorem and - for setting up the determining equations for classical and - non-classical point symmetries." \end{chunk} +\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Seiler 97]{Sei97} Seiler, Werner M. -Computer Algebra and Differential Equations: An Overview'' -\verb|www.mathematik.uni-kassel.di/~seiler/Papers/Postscript/CADERep.ps.gz| +\bibitem[Kauers 08]{Kau08} Kauers, Manuel +Integration of Algebraic Functions: A Simple Heuristic for Finding +the Logarithmic Part'' +ISSAC July 2008 ACM 978-1-59593-904 pp133-140 +\verb|www.risc.jku.at/publications/download/risc_3427/Ka01.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Kau08.pdf| keywords = "axiomref", abstract = " - We present an informal overview of a number of approaches to - differential equations which are popular in computer algebra. This - includes symmetry and completion theory, local analysis, differential - ideal and Galois theory, dynamical systems and numerical analysis. A - large bibliography is provided." + A new method is proposed for finding the logarithmic part of an + integral over an algebraic function. The method uses Gr{\"o}bner bases + and is easy to implement. It does not have the feature of finding a + closed form of an integral whenever there is one. But it very often + does, as we will show by a comparison with the built-in integrators of + some computer algebra systems." \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler (a)]{Seixx} Seiler, W.M. -DETools: A Library for Differential Equations'' -\verb|iaks-www.ira.uka.de/iaks-calmet/werner/werner.html| +\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G. +Production of Argument SubPrograms in the AXIOM -- NAG +link: examples involving nonleanr systems'' +Technical Report TR1/94 +ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and +Oxford, UK, 1994 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M. -Using Gr{\"o}bner bases to determine algebra -membership, split surjective algebra homomorphisms determine birational -equivalence'' -Journal of Symbolic Computation 6(2-3) pp267-273 -Oct.-Dec. 1988 CODEN JSYCEH ISSN 0747-7171 +\bibitem[Kelsey 99]{Kel99} Kelsey, Tom +Formal Methods and Computer Algebra: A Larch Specification of AXIOM +Categories and Functors'' +Ph.D. Thesis, University of St Andrews, 1999 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Sit 89]{Sit89} Sit, W.Y. -On Goldman's algorithm for solving first-order multinomial -autonomous systems'' In Mora [Mor89], pp386-395 ISBN 3-540-51083-4 -LCCN QA268.A35 1998 Conference held jointly with ISSAC '88 +\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom +Formal specification of computer algebra'' +University of St Andrews, 6th April 2000 +\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbs.ps| +%\verb|axiom-developer.org/axiom-website/papers/Kel00a.pdf| keywords = "axiomref", + abstract = " + We investigate the use of formal methods languages and tools in the + design and development of computer algebra systems (henceforth CAS). + We demonstrate that errors in CAS design can be identified and + corrected by the use of (i) abstract specifications of types and + procedures, (ii) automated proofs of properties of the specifications, + and (iii) interface specifications which assist the verification of + pre- and post conditions of implemented code." \end{chunk} \begin{chunk}{ignore} -\bibitem[Sit 92]{Sit92} Sit, W.Y. -An algorithm for solving parametric linear systems'' -Journal of Symbolic Computations, 13(4) pp353-394, April 1992 CODEN JSYCEH -ISSN 0747-7171 -\verb|www.sciencedirect.com/science/article/pii/S0747717108801046/pdf| -\verb|?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&| -\verb|pid=1-s2.0-S0747717108801046-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Sit92.pdf| +\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom +Formal specification of computer algebra'' +(slides) University of St Andrews, Sept 21, 2000 +\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbstalk.ps| keywords = "axiomref", - abstract = " - We present a theoretical foundation for studying parametric systesm of - linear equations and prove an efficient algorithm for identifying all - parametric values (including degnerate cases) for which the system is - consistent. The algorithm gives a small set of regimes where for each - regime, the solutions of the specialized systems may be given - uniformly. For homogeneous linear systems, or for systems were the - right hand side is arbitrary, this small set is irredunant. We discuss - in detail practical issues concerning implementations, with particular - emphasis on simplification of results. Examples are given based on a - close implementation of the algorithm in SCRATCHPAD II. We also give a - complexity analysis of the Gaussian elimination method and compare - that with our algorithm." \end{chunk} \begin{chunk}{ignore} -\bibitem[Sit 06]{Sit06} Sit, Emil -Tools for Repeatable Research'' -\verb|www.emilsit.net/blog/archives/tools-for-repeatable-research| +\bibitem[Kendall 99a]{Ken99a} Kendall, W.S. +Itovsn3 in AXIOM: modules, algebras and stochastic differentials'' +\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/| +\verb|kendall/personal/ppt/328.ps.gz| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Smedley 92]{Sme92} Smedley, Trevor J. -Using pictorial and object oriented programming for computer algebra'' -In Hal Berghel et al., editors. Applied computing -- -technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP -Symposium on Applied Computing, Kansas City Convention Center, March 1-3, 1992 -pp1243-1247. ACM Press, New York, NY 10036, USA, 1992. ISBN 0-89791-502-X -LCCN QA76.76.A65 S95 1992 +\bibitem[Kendall 99b]{Ken99b} Kendall, W.S. +Symbolic It\^o calculus in AXIOM: an ongoing story +\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/| +\verb|kendall/personal/ppt/327.ps.gz| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko -Algorithmic differentiation in Axiom'' -ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347-354 -ISBN 978-1-59593-743-8 -%\verb|axiom-developer.org/axiom-website/papers/SDJ07.pdf| +\bibitem[Kosleff 91]{Kos91} P.-V. Koseleff +Word games in free Lie algebras: several bases and formulas'' +Theoretical Computer Science 79(1) pp241-256 Feb. 1991 CODEN TCSCDI +ISSN 0304-3975 + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H. +Implementation of a geometry theorem proving package in SCRATCHPAD II'' +In Davenport [Dav89] pp246-257 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987 keywords = "axiomref", - abstract = " - This paper describes the design and implementation of an algorithmic - differentiation framework in the Axiom computer algebra system. Our - implementation works by transformations on Spad programs at the level - of the typed abstract syntax tree." \end{chunk} +\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[SSC92]{SSC92}. -Algorithmic Methods For Lie Pseudogroups'' -In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group -Analysis: Advanced Analytical and Computational Methods in Mathematical -Physics, pp337-344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993 -\verb|iaks-www.ira.uka.de/iaks-calmet/werner/Papers/Acireale92.ps.gz| +\bibitem[Lahey 08]{Lah08} Lahey, Tim +Sage Integration Testing'' +\verb|github.com/tjl/sage_int_testing| Dec. 2008 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G. -Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel'' -Technical Report 640-M, TIM 3 (IMAG), Grenoble, France, Feb 1987 +\bibitem[Lambe 89]{Lam89} Lambe, L. A. +Scratchpad II as a tool for mathematical research'' +Notices of the AMS, February 1928 pp143-147 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P. -The Evolution of Lisp'' -\verb|www.dreamsongs.com/Files/HOPL2-Uncut.pdf| +\bibitem[Lambe 91]{Lam91} Lambe, L. A. +Resolutions via homological perturbation'' +Journal of Symbolic Computation 12(1) pp71-87 July 1991 +CODEN JSYCEH ISSN 0747-7171 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Sutor 85]{Sut85} Sutor, R.S. -The Scratchpad II computer algebra language and system'' -In Buchberger and Caviness [BC85], pp32-33 ISBN 0-387-15983-5 (vol. 1), -0-387-15984-3 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes. +\bibitem[Lambe 92]{Lam92} Lambe, Larry +Next Generation Computer Algebra Systems AXIOM and the Scratchpad +Concept: Applications to Research in Algebra'' +$21^{st}$ Nordic Congress of Mathematicians 1992 +%\verb|axiom-developer.org/axiom-website/papers/Lam92.pdf| keywords = "axiomref", + abstract = " + One way in which mathematicians deal with infinite amounts of data is + symbolic representation. A simple example is the quadratic equation + $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ + a formula which uses symbolic representation to describe the solutions + to an infinite class of equations. Most computer algebra systems can + deal with polynomials with symbolic coefficients, but what if symbolic + exponents are called for (e.g. $1+t^i$)? What if symbolic limits on + summations are also called for, for example + $1+t+\ldots+t^i=\sum_j{t^j}$ + + The Scratchpad Concept'' is a theoretical ideal which allows the + implementation of objects at this level of abstraction and beyond in a + mathematically consistent way. The Axiom computer algebra system is an + implementation of a major part of the Scratchpad Concept. Axiom + (formerly called Scratchpad) is a language with extensible + parameterized types and generic operators which is based on the + notions of domains and categories. By examining some aspects of the + Axiom system, the Scratchpad Concept will be illustrated. It will be + shown how some complex problems in homologicial algebra were solved + through the use of this system." \end{chunk} \begin{chunk}{ignore} -\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D. -The type inference and coercion facilities in -the Scratchpad II interpreter'' In Wexelblat [Wex87], pp56-63 -ISBN 0-89791-235-7 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987) -%\verb|axiom-developer.org/axiom-website/papers/SJ87a.pdf| +\bibitem[Lambe 93]{Lam93} Lambe, Larry +On Using Axiom to Generate Code'' +(preprint) 1993 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Sutor 87b]{Su87} Sutor, Robert S. -The Scratchpad II Computer Algebra System. Using and -Programming the Interpreter'' -IBM Course presentation slide deck Spring 1987 +\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard +Object-Oriented Mathematical Programming and Symbolic/Numeric Interface'' +$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993 +%\verb|axiom-developer.org/axiom-website/papers/LL93.pdf| keywords = "axiomref", + abstract = " + The Axiom language is based on the notions of categories'', + domains'', and packages''. These concepts are used to build an + interface between symbolic and numeric calculations. In particular, an + interface to the NAG Fortran Library and Axiom's algebra and graphics + facilities is presented. Some examples of numerical calculations in a + symbolic computational environment are also included using the finite + element method. While the examples are elementary, we believe that + they point to very powerful methods for combining numeric and symbolic + computational techniques." \end{chunk} \begin{chunk}{ignore} -\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard -The type inference and coercion facilities -in the Scratchpad II interpreter'' -Research report RC 12595 (\#56575), -IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp -%\verb|axiom-developer.org/axiom-website/papers/SJ87c.pdf| +\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri +OpenMath Library for Computing on Riemann Surfaces'' +PhD thesis, Nov 2008 Florida State University +\verb|www.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html| keywords = "axiomref", - abstract = " - The Scratchpad II system is an abstract datatype programming language, - a compiler for the language, a library of packages of polymorphic - functions and parameterized abstract datatypes, and an interpreter - that provides sophisticated type inference and coercion facilities. - Although originally designed for the implementation of symbolic - mathematical algorithms, Scratchpad II is a general purpose - programming language. This paper discusses aspects of the - implementation of the intepreter and how it attempts to provide a user - friendly and relatively weakly typed front end for the strongly typed - programming language." \end{chunk} \begin{chunk}{ignore} -\bibitem[Sutor 88]{Su88} Sutor, Robert S. -A guide to programming in the scratchpad 2 interpreter'' -IBM Manual, March 1988 +\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E. +The use of MathCAD and Theorist in the ChE classroom'' +In Anonymous [Ano91], pp287-299 (vol. 1) 2 vols. keywords = "axiomref", \end{chunk} -\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Thompson 00]{Tho00} Thompson, Simon -Logic and dependent types in the Aldor Computer Algebra System'' -%\verb|axiom-developer.org/axiom-website/papers/Tho00.pdf| +\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire +Dynamic Evaluation and Real Closure Implementation in Axiom'' +June 29, 1996 +\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps| +%\verb|axiom-developer.org/axiom-website/papers/Le96.ps| keywords = "axiomref", - abstract = " - We show how the Aldor type system can represent propositions of - first-order logic, by means of the 'propositions as types' - correspondence. The representation relies on type casts (using - pretend) but can be viewed as a prototype implementation of a modified - type system with {\sl type evaluation} reported elsewhere. The logic - is used to provide an axiomatisation of a number of familiar Aldor - categories as well as a type of vectors." \end{chunk} \begin{chunk}{ignore} -\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid -The Aldor\-\- language'' -%\verb|axiom-developer.org/axiom-website/papers/TTxx.pdf| +\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire +The Dynamic Real Closure implemented in Axiom'' +\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps| keywords = "axiomref", - abstract = " - This paper introduces the \verb|Aldor--| language, which is a - functional programming language with dependent types and a powerful, - type-based, overloading mechanism. The language is built on a subset - of Aldor, the 'library compiler' language for the Axiom computer - algebra system. \verb|Aldor--| is designed with the intention of - incorporating logical reasoning into computer algebra computations. - - The paper contains a formal account of the semantics and type system - of \verb|Aldor--|; a general discussion of overloading and how the - overloading in \verb|Aldor--| fits into the general scheme; examples - of logic within \verb|Aldor--| and notes on the implementation of the - system." \end{chunk} \begin{chunk}{ignore} -\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel -Etude du typage dans le syst\eme de calcul scientifique Aldor'' -Universit\'e de Limoges 1998 -%\verb|axiom-developer.org/axiom-website/papers/Tou98.pdf| +\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed) +ISSAC '95: Proceedings of the 1995 International +Symposium on Symbolic and Algebraic Computation: July 10-12, 1995, Montreal, +Canada ISSAC-PROCEEDINGS-1995. ACM Press, New York, NY 10036, USA, 1995 +ISBN 0-89791-699-9 LCCN QA76.95 I59 1995 ACM order number 505950 keywords = "axiomref", \end{chunk} -\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris -Computer algebra systems and TeXmacs'' -\verb|www.texmacs.org/tmweb/plugins/cas.en.html| +\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno +Efficient Implementation of Polynomial Arithmetic in a Multiple-Level +Programming Environment'' +Lecture Notes in +Computer Science Springer Vol 4151/2006 ISBN 978-3-540-38084-9 pp12-23 +Proceedings of International Congress of Mathematical Software ICMS 2006 +\verb|www.csd.uwo.ca/~moreno//Publications/Li-MorenoMaza-ICMS-06.pdf| keywords = "axiomref", \end{chunk} -\begin{chunk}{axiom.bib} -@article{Hoei94, - author = "{van Hoeij}, M.", - title = "An algorithm for computing an integral basis in an algebraic - function field", - journal = "Journal of Symbolic Computation", - volume = "18", - number = "4", - year = "1994", - pages = "353-363", - issn = "0747-7171", +\begin{chunk}{ignore} +\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel +A Quantitative Study of Reductions in Algebraic Libraries'' +PASCO 2010 +\verb|www.axiomatics.org/~gdr/concurrency/quant-pasco10.pdf| keywords = "axiomref", - paper = "Hoei94.pdf", - abstract = " - Algorithms for computing integral bases of an algebraic function field - are implemented in some computer algebra systems. They are used e.g. - for the integration of algebraic functions. The method used by Maple - 5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm - of Ford and Zassenhaus [Ford, 1978], that computes the ring of - integers in an algebraic number field, to the case of a function field. - It turns out that using algebraic geometry one can write a faster - algorithm. The method we will give is based on Puiseux expansions. - One cas see this as a variant on the Coates' algorithm as it is - described in [Davenport, 1981]. Some difficulties in computing with - Puiseux expansions can be avoided using a sharp bound for the number - of terms required which will be given in Section 3. In Section 5 we - derive which denominator is needed in the integral basis. Using this - result 'intermediate expression swell' can be avoided. +\end{chunk} - The Puiseux expansions generally introduce algebraic extensions. These - extensions will not appear in the resulting integral basis." -} +\begin{chunk}{ignore} +\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel +An Automatic Parallelization Framework for Algebraic Computation +Systems'' +ISSAC 2011 +\verb|www.axiomatics.org/~gdr/concurrency/oa-conc-issac11.pdf| +%\verb|axiom-developer.org/axiom-website/papers/YL11.pdf| + keywords = "axiomref", + abstract = " + This paper proposes a non-intrusive automatic parallelization + framework for typeful and property-aware computer algebra systems." \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Hoei08, - author = "{van Hoeij}, Mark and Novocin, Andrew", - title = "A Reduction Algorithm for Algebraic Function Fields", - year = "2008", - month = "April", - url = "http://andy.novocin.com/pro/algext.pdf", - paper = "Hoei08.pdf", +\begin{chunk}{ignore} +\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud; +Roy, Marie Francoise +Generic computation of the real closure of an ordered field'' +Math. and Computers in Simulation 42 pp 541-549 (1996) +%\verb|axiom-developer.org/axiom-website/papers/Liga96.pdf| + keywords = "axiomref", abstract = " - Computer algebra systesm often produce large expressions involving - complicated algebraic numbers. In this paper we study variations of - the {\tt polred} algorithm that can often be used to find better - representations for algebraic numbers. The main new algorithm - presented here is an algorithm that treats the same problem for the - function field case." -} + This paper describes a generalization of the real closure computation + of an ordered field (Rioboo, 1991) enabling to use different technques + to code a single real algebraic number." \end{chunk} \begin{chunk}{ignore} -\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer -Computational Methods in Commutative Algebra and Algebraic Geometry'' -Springer, Algorithms and Computation in Mathematics, Vol 2 1999 -ISBN 3-540-21311-2 +\bibitem[Linton 93]{Lin93} Linton, Steve +Vector Enumeration Programs, version 3.04'' +\verb|www.cs.st-andrews.ac.uk/~sal/nme/nme_toc.html#SEC1| keywords = "axiomref", \end{chunk} -\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Wang 89]{Wan89} Wang, D. -A program for computing the Liapunov functions and Liapunov -constants in Scratchpad II'' -SIGSAM Bulletin (ACM Special Interest Group -on Symbolic and Algebraic Manipulation), 23(4) pp25-31, Oct. 1989, -CODEN SIGSBZ ISSN 0163-5824 +\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri; +Sinor, Milan; Wester, Michael; Winkler, Franz +Computer Algebra - algorithms, systems and applications'' +June 2, 1997 +\verb|kfe.fjfi.cvut.cz/~liska/ca/all.html| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Wang 91]{Wan91} Wang, Dongming -Mechanical manipulation for a class of differential systems'' -Journal of Symbolic Computation, 12(2) pp233-254 Aug. 1991 -CODEN JSYCEH ISSN 0747-7171 +\bibitem[Lucks 86]{Luc86} Lucks, Michael +A fast implementation of polynomial factorization'' +In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic +and Algebraic Computation: SYMSAC '86, July 21-23, 1986, Waterloo, Ontario, +pp228-232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 +LCCN QA155.7.E4 A281 1986 ACM order number 505860 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed) -International System Symposium on Symbolic and -Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992 -ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover), -LCCN QA76.95.I59 1992 +\bibitem[Lueken 77]{Lue77} Lueken, E. +Ueberlegungen zur Implementierung eines Formelmanipulationssystems'' +Master's thesis, Technischen Universit{\"{a}}t Carolo-Wilhelmina zu +Braunschweig. Braunschweig, Germany, 1977 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed) -ISSAC '90 Proceedings of the -International Symposium on Symbolic and Algebraic Computation ACM Press, -New York, NY, 10036, USA. 1990 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990 +\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A. +New quantum mechanical perturbation technique +using an 'electronic scratchpad' on an inexpensive computer'' +American Journal of Pyhsics, 59(3) pp270-273, March 1991. +CODEN AJPIAS ISSN 0002-9505 keywords = "axiomref", \end{chunk} +\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Watt 85]{Wat85} Watt, Stephen -Bounded Parallelism in Computer Algebra'' -PhD Thesis, University of Waterloo -\verb|www.csd.uwo.ca/~watt/pub/reprints/1985-smw-phd.pdf| +\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia +Programming and certifying the CAD algorithm inside the coq system'' +Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl +Seminar Proceedings, Schloss Dagstuhl (2005) keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J. -Algebra Snapshot: Linear Ordinary Differential Operators'' -Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986) -\verb|www.csd.uwo.ca/~watt/pub/reprints/1986-snews-lodo.pdf| +\bibitem[Mathews 89]{Mat89} Mathews, J. +Symbolic computational algebra applied to Picard iteration'' +Mathematics and computer education, 23(2) pp117-122 Spring 1989 CODEN MCEDDA, +ISSN 0730-8639 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 87]{Wat87} Watt, Stephen -Domains and Subdomains in Scratchpad II'' -in [Wit87], pp3-5 +\bibitem[McJones 11]{McJ11} McJones, Paul +Software Presentation Group -- Common Lisp family'' +\verb|www.softwarepreservation.org/projects/LISP/common_lisp_family| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H. -Mapping as First Class Objects'' -in [Wit87], pp13-17 +\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L. +Teaching advantages of transparent computer software -- MathCAD'' +CoED, 10(1) pp71-76, January-March 1990 CODEN CWLJDP ISSN 0736-8607 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 89]{Wat89} Watt, S. M. -A fixed point method for power series computation'' -In Gianni [Gia89], pp206-217 ISBN 3-540-51084-2 LCCN QA76.95.I57 -1988 Conference held jointly with AAECC-6 +\bibitem[Miola 90]{Mio90} Miola, A. (ed) +Design and Implementation of Symbolic Computation Systems'' +International Symposium DISCO '90, Capri, Italy, April 10-12, 1990, Proceedings +volume 429 of Lecture Notes in Cmputer Science, +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +1990 ISBN 0-387-52531-9 (New York), 3-540-52531-9 (Berlin) LCCN QA76.9.S88I576 +1990 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M. -The Scratchpad II type system: Domains and subdomains'' -in A.M. Miola, editor Computing Tools -for Scientific Problem Solving, Academic Press, New York, 1990 +\bibitem[Miola 93]{Mio93} Miola, A. (ed) +Design and Implementation of Symbolic Computation Systems'' +International Symposium DISCO '93 Gmunden, Austria, September 15-17, 1993: +Proceedings. +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +1993 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed) -Proceedings of the 1991 International Symposium on -Symbolic and Algebraic Computation, ISSAC'91, July 15-17, 1991, Bonn, Germany, -ACM Press, New York, NY 10036, USA, 1991 ISBN 0-89791-437-6 -LCCN QA76.95.I59 1991 +\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas +Using Commutativity Properties for Controlling Coercions'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/| +\verb|WeberA/MissuraWeber94a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Miss94.pdf| keywords = "axiomref", + abstract = " + This paper investigates some soundness conditions which have to be + fulfilled in systems with coercions and generic operators. A result of + Reynolds on unrestricted generic operators is extended to generic + operators which obey certain constraints. We get natural conditions + for such operators, which are expressed within the theoretic framework + of category theory. However, in the context of computer algebra, there + arise examples of coercions and generic operators which do not fulfil + these conditions. We describe a framework -- relaxing the above + conditions -- that allows distinguishing between cases of ambiguities + which can be resolved in a quite natural sense and those which + cannot. An algorithm is presented that detects such unresolvable + ambiguities in expressions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.; -Steinback, J.M.; Sutor, R.S. -A\# User's Guide'' -Version 1.0.0 O($\epsilon{}^1$) June 8, 1994 +\bibitem[Monagan 87]{Mon87} Monagan, Michael B. +Support for Data Structures in Scratchpad II'' +in [Wit87], pp17-18 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.; -Dooley, Samuel S.; Iglio, Pietro -A First Report on the A\# Compiler (including benchmarks)'' -IBM Research Report RC19529 (85075) May 12, 1994 -%\verb|axiom-developer.org/axiom-website/papers/Wat94.pdf| +\bibitem[Monagan 93]{Mon93} Monagan, M. B. +Gauss: a parameterized domain of computation system with +support for signature functions'' +In Miola [Mio93], pp81-94 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 94c]{Wat94c} Watt, Stephen M. -A\# Language Reference Version 0.35'' -IBM Research Division Technical Report RC19530 May 1994 +\bibitem[Mora 89]{Mor89} Mora, T. (ed) +Applied Algebra, Algebraic Algorithms and Error-Correcting +Codes, 6th International Conference, AAECC-6, Rome, Italy, July 4-8, 1998, +Proceedings, volume 357 of Lecture Notes in Computer Science +Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc., +1989 ISBN 3-540-51083-4, LCCN QA268.A35 1988 Conference held jointly with +ISSAC '88 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P. -Steinbach, J.M.; Morrison, S.C.; Sutor, R.S. -AXIOM Library Compiler Users Guide'' -The Numerical Algorithms Group (NAG) Ltd, 1994 +\bibitem[Moses 71]{Mos71} Moses, Joel +Algebraic Simplification: A Guide for the Perplexed'' +CACM August 1971 Vol 14 No. 8 pp527-537 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro; -Morrison, Scott C.; Steinbach, Jonathan M. -FOAM: A First Order Abstract Machine Version 0.35'' -IBM T. J. Watson Research Center (2001) -%\verb|axiom-developer.org/axiom-website/papers/Wat01.pdf| +\bibitem[Moses 08]{Mos08} Moses, Joel +Macsyma: A Personal History'' +Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago +\verb|esd.mit.edu/Faculty_Pages/moses/Macsyma.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Mos08.pdf| keywords = "axiomref", + abstract = " + The Macsyma system arose out of research on mathematical software in + the AI group at MIT in the 1960's. Algorithm development in symbolic + integration and simplification arose out of the interest of people, + such as the author, who were also mathematics students. The later + development of algorithms for the GCD of sparse polynomials, for + example, arose out of the needs of our user community. During various + times in the 1970's the computer on which Macsyma ran was one of the + most popular notes on the ARPANET. We discuss the attempts in the late + 70's and the 80's to develop Macsyma systems that ran on popular + computer architectures. Finally, we discuss the impact of the + fundamental ideas in Macsyma on current research on large scale + engineering systems." \end{chunk} +\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Weber 92]{Webe92} Weber, Andreas -Type Systems for Computer Algebra'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe92.pdf| +\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian +From Untyped to Polymorphically Typed Objects in Mathematical Web +Services'' +%\verb|axiom-developer.org/axiom-website/papers/NPxx.pdf| keywords = "axiomref", abstract = " - An important feature of modern computer algebra systems is the support - of a rich type system with the possibility of type inference. Basic - features of such a type system are polymorphism and coercion between - types. Recently the use of order-sorted rewrite systems was proposed - as a general framework. We will give a quite simple example of a - family of types arising in computer algebra whose coercion relations - cannot be captured by a finite set of first-order rewrite rules." + OpenMath is a widely recognized approach to the semantic markup of + mathematics that is often used for communication between OpenMath + compliant systems. The Aldor language has a sophisticated + category-based type system that was specifically developed for the + purpose of modelling mathematical structures, while the system itself + supports the creation of small-footprint applications suitable for + deployment as web services. In this paper we present our first results + of how one may perform translations from generic OpenMath objects into + values in specific Aldor domains, describing how the Aldor interfae + domain ExpresstionTree is used to achieve this. We outline our Aldor + implementation of an OpenMath translator, and describe an efficient + extention of this to the Parser category. In addition, the Aldor + service creation and invocation mechanism are explained. Thus we are + in a position to develop and deploy mathematical web services whose + descriptions may be directly derived from Aldor's rich type language." \end{chunk} \begin{chunk}{ignore} -\bibitem[Weber 92b]{Webe92b} Weber, Andreas -Structuring the Type System of a Computer Algebra System'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe92b.pdf| +\bibitem[Naylor 95]{N95} Naylor, Bill +Symbolic Interface for an advanced hyperbolic PDE solver'' +\verb|www.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps| +%\verb|axiom-developer.org/axiom-website/papers/N95.pdf| keywords = "axiomref", abstract = " - Most existing computer algebra systems are pure symbol manipulating - systems without language support for the occuring types. This is - mainly due to the fact taht the occurring types are much more - complicated than in traditional programming languages. In the last - decade the study of type systems has become an active area of - research. We will give a proposal for a type system showing that - several problems for a type system of a symbolic computation system - can be solved by using results of this research. We will also provide - a variety of examples which will show some of the problems that remain - and that will require further research." + An Axiom front end is described, which is used to generate + mathematical objects needed by one of the latest NAG routines, to be + included in the Mark 17 version of the NAG Numerical library. This + routine uses powerful techniques to find the solution to Hyperbolic + Partial Differential Equations in conservation form and in one spatial + dimension. These mathematical objects are non-trivial, requiring much + mathematical knowledge on the part of the user, which is otherwise + irrelvant to the physical problem which is to be solved. We discuss + the individual mathematical objects, considering the mathematical + theory which is relevant, and some of the problems which have been + encountered and solved during the FORTRAN generation necessary to + realise the object. Finally we display some of our results." \end{chunk} \begin{chunk}{ignore} -\bibitem[Weber 93b]{Webe93b} Weber, Andreas -Type Systems for Computer Algebra'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber93b.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe93b.pdf| +\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H. +A Monte-Carlo Extension to a Category-Based Type System'' +\verb|www.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps| +%\verb|axiom-developer.org/axiom-website/papers/ND00.pdf| keywords = "axiomref", abstract = " - We study type systems for computer algebra systems, which frequently - correspond to the pragmatically developed'' typing constructs used - in AXIOM. A central concept is that of {\sl type classes} which - correspond to AXIOM categories. We will show that types can be - syntactically described as terms of a regular order-sorted signature - if no type parameters are allowed. Using results obtained for the - functional programming language Haskell we will show that the problem - of {\sl type inference} is decidable. This result still holds if - higher-order functions are present and {\sl parametric polymorphism} - is used. These additional typing constructs are useful for further - extensions of existing computer algebra systems: These typing concepts - can be used to implement category theoretic constructs and there are - many well known constructive interactions between category theory and - algebra." + The normal claim for mathematics is that all calculations are 100\% + accurate and therefore one calculation can rely completely on the + results of sub-calculations, hoever there exist {\sl Monte-Carlo} + algorithms which are often much faster than the equivalent + deterministic ones where the results will have a prescribed + probability (presumably small) of being incorrect. However there has + been little discussion of how such algorithms can be used as building + blocks in Computer Algebra. In this paper we describe how the + computational category theory which is the basis of the type structure + used in the Axiom computer algebra system may be extended to cover + probabilistic algorithms, which use Monte-Carlo techniques. We follow + this with a specific example which uses Straight Line Program + representation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Weber 94]{Web94} Weber, Andreas -Algorithms for Type Inference with Coercions'' -ISSAC 94 ACM 0-89791-638-7/94/0007 -%\verb|axiom-developer.org/axiom-website/papers/Web94.pdf| +\bibitem[Norman 75]{Nor75} Norman, A. C. +Computing with formal power series'' +ACM Transactions on Mathematical Software, 1(4) pp346-356 +Dec. 1975 CODEN ACMSCU ISSN 0098-3500 keywords = "axiomref", - abstract = " - This paper presents algorithms that perform a type inference for a - type system occurring in the context of computer algebra. The type - system permits various classes of coercions between types and the - algorithms are complete for the precisely defined system, which can be - seen as a formal description of an important subset of the type system - supported by the computer algebra program Axiom. - - Previously only algorithms for much more restricted cases of coercions - have been described or the frameworks used have been so general that - the corresponding type inference problems were known to be - undecidable." \end{chunk} \begin{chunk}{ignore} -\bibitem[Weber 95]{Webe95} Weber, A. -On coherence in computer algebra'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber94e.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe95.pdf| +\bibitem[Norman 75a]{Nor75a} Norman, A.C. +The SCRATCHPAD Power Series Package'' +IBM T.J. Watson Research RC4998 keywords = "axiomref", - abstract = " - Modern computer algebra systems (e.g. AXIOM) support a rich type - system including parameterized data types and the possibility of - implicit coercions between types. In such a type system it will be - frequently the case that there are different ways of building - coercions between types. An important requirement is that all - coercions between two types coincide, a property which is called {\sl - coherence}. We will prove a coherence theorem for a formal type system - having several possibilities of coercions covering many important - examples. Moreover, we will give some informal reasoning why the - formally defined restrictions can be satisfied by an actual system." \end{chunk} +\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Weber 96]{Webe96} Weber, Andreas -Computing Radical Expressions for Roots of Unity'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber96a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe96.pdf| +\bibitem[Ollivier 89]{Oll89} Ollivier, F. +Inversibility of rational mappings and structural +identifiablility in automatics'' +In ACM [ACM89], pp43-54 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989 keywords = "axiomref", - abstract = " - We present an improvement of an algorithm given by Gauss to compute a - radical expression for a $p$-th root of unity. The time complexity of - the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime - factor of $p-1$." \end{chunk} - \begin{chunk}{ignore} -\bibitem[Weber 99]{Webe99} Weber, Andreas -Solving Cyclotomic Polynomials by Radical Expressions'' -\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/| -\verb|WeberA/WeberKeckeisen99a.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Webe99.pdf| +\bibitem[Online 72]{Onl72}. +Online 72: conference proceedings ... international conference on online +interactive computing, Brunel University, Uxbridge, England, 4-7 September +1972 ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes. keywords = "axiomref", - abstract = " - We describe a Maple package that allows the solution of cyclotomic - polynomials by radical expressions. We provide a function that is an - extension of the Maple {\sl solve} command. The major algorithmic - ingredient of the package is an improvement of a method due to Gauss - which gives radical expressions for roots of unity. We will give a - summary for computations up to degree 100, which could be done within - a few hours of cpu time on a standard workstation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Wei-Jiang 12]{WJ12} Wei-Jiang -Top free algebra System'' -\verb|wei-jiang.com/it/software/top-free-algebra-system-bye-mathematica-bye-maple| +\bibitem[OpenMath]{OpenMa}. +OpenMath Technical Overview'' +\verb|www.openmath.org/overview/technical.html| keywords = "axiomref", \end{chunk} +\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Wester 99]{Wes99} Wester, Michael J. -Computer Algebra Systems'' -John Wiley and Sons 1999 ISBN 0-471-98353-5 +\bibitem[Page 07]{Pa07} Page, William S. +Axiom - Open Source Computer Algebra System'' +Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed) -Proceedings of the SIGPLAN '87 Symposium on -Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 24-26, 1987 -ACM Press, New York, NY 10036, USA, 1987 ISBN 0-89791-235-7 -LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987) +\bibitem[Petitot 90]{Pet90} Petitot, Michel +Types r\'ecursifs en scratchpad, application aux polyn\^omes non +commutatifs'' +LIFL, 1990 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Wityak 87]{Wit87} Wityak, Sandra -Scratchpad II Newsletter'' -Volume 2, Number 1, Nov 1987 +\bibitem[Petitot 93]{Pet93} Petitot, M. +Experience with Axiom'' +In Jacob et al. [JOS93], page 240 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[WWW1]{WWW1}. -Software Preservation Group -\verb|www.softwarepresentation.org/projects/LISP/common_lisp_family| +\bibitem[Petric 71]{Pet71} Petric, S. R. (ed) +Proceedings of the second symposium on Symbolic and +Algebraic Manipulation, March 23-25, 1971, Los Angeles, California, ACM Press, +New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971 keywords = "axiomref", \end{chunk} -\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Yap 00]{Yap00} Yap, Chee Keng -Fundamental Problems of Algorithmic Algebra'' -Oxford University Press (2000) ISBN0-19-512516-9 +\bibitem[Pinch 93]{Pin93} Pinch, R.G.E. +Some Primality Testing Algorithms'' +Devlin, Keith (ed.) +Computers and Mathematics November 1993, Vol 40, Number 9 pp1203-1210 keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai -Literate Programming Tools Implemented in ANSI Common Lisp'' -\verb|brlcad.org/~starseeker/cl-web-v0.8.lisp.pamphlet| +\bibitem[Poll (b)]{Polxx} Poll, Erik +The type system of Axiom'' +%\verb|axiom-developer.org/axiom-website/papers/Polxx.pdf| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Yun 83]{Yun83} Yun, David Y.Y. -Computer Algebra and Complex Analysis'' -Computational Aspects of Complex Analysis pp379-393 -D. Reidel Publishing Company H. Werner et. al. (eds.) +\bibitem[Purtilo 86]{Pur86} Purtilo, J. +Applications of a software interconnection system in mathematical +problem solving environments'' In Bruce W. Char, editor. Proceedings of the +1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, +ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 LCCN QA155.7.E4 +A281 1986 ACM order number 505860 keywords = "axiomref", \end{chunk} -\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Zen92]{Zen92} Zenger, Ch. -Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre -Implementierung in AXIOM'' -Diplomarbeit, Universit{\"a}t Karlsruhe, -Karlsruhe, Germany, 1992 +\bibitem[Rainer 14]{Rain14} Joswig, Rainer +2014: 30+ Years Common Lisp the Language'' +\verb|lispm.de/30ycltl| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Zip92]{Zip92} Zippel, Richard -Algebraic Computation'' -(unpublished) Cornell University Ithaca, NY Sept 1992 +\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud +Quelques aspects du calcul exact avec des nombres r\'eels'' +Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg +%\verb|axiom-developer.org/axiom-website/papers/Riob03a.ps| keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel -Handbook of Integration'' -Jones and Bartlett, 1992, ISBN 0-86720-293-9 +\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud +Towards Faster Real Algebraic Numbers'' +J. of Symbolic Computation 36 pp 513-533 (2003) +%\verb|axiom-developer.org/axiom-website/papers/Riob03.pdf| keywords = "axiomref", + abstract = " + This paper presents a new encoding scheme for real algebraic number + manipulations which enhances current Axiom's real closure. Algebraic + manipulations are performed using different instantiations of + sub-resultant-like algorithms instead of Euclidean-like algorithms. + We use these algorithms to compute polynomial gcds and Bezout + relations, to compute the roots and the signs of algebraic + numbers. This allows us to work in the ring of real algebraic integers + instead of the field of read algebraic numbers avoiding many + denominators." \end{chunk} -\section{Axiom Citations of External Sources} -\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{ignore} +\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas +Does Axiom Solve Systems of O.D.E's Like Mathematica?'' +July 1993 +%\verb|axiom-developer.org/axiom-website/papers/Rob93.pdf| + keywords = "axiomref", + abstract = " + If I were demonstrating Axiom and were asked this question, my reply + would be No, but I am not sure that this is a bad thing''. And I + would illustrate this with the following example. -\begin{chunk}{axiom.bib} -@article{Abla98, - author = "Ablamowicz, Rafal", - title = "Spinor Representations of Clifford Algebras: A Symbolic Approach", - journal = "Computer Physics Communications", - volume = "115", - number = "2-3", - month = "December", - year = "1998", - pages = "510-535" -} + Consider the following system of O.D.E.'s + $+ \begin{array}{rcl} + \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\ + \frac{dx_2}{dt} & = & x_1 - x_2 + \end{array} +$ + This is a very simple system: $x_1$ is actually uncoupled from $x_2$" \end{chunk} -\begin{chunk}{axiom.bib} -@article{Abra06, - author = "Abramov, Sergey A.", - title = "In Memory of Manuel Bronstein", - journal = "Programming and Computer Software", - volume = "32", - number = "1", - pages = "56-58", - publisher = "Pleiades Publishing Inc", - year = "2006", - paper = "Abra06.pdf" -} +\begin{chunk}{ignore} +\bibitem[Rioboo 92]{Rio92} Rioboo, R. +Real algebraic closure of an ordered field, implementation in Axiom'' +In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover) +0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992 +%\verb|axiom-developer.org/axiom-website/papers/Rio92.pdf| + keywords = "axiomref", + abstract = " + Real algebraic numbers appear in many Computer Algebra problems. For + instance the determination of a cylindrical algebraic decomposition + for an euclidean space requires computing with real algebraic numbers. + This paper describes an implementation for computations with the real + roots of a polynomial. This process is designed to be recursively + used, so the resulting domain of computation is the set of all real + algebraic numbers. An implementation for the real algebraic closure + has been done in Axiom (previously called Scratchpad)." \end{chunk} \begin{chunk}{ignore} -\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A. -Handbook of Mathematical Functions'' -(1964) Dover Publications, NY ISBN 0-486-61272-4 +\bibitem[Roesner 95]{Roe95} Roesner, K. G. +Verified solutions for parameters of an exact solution for +non-Newtonian liquids using computer algebra'' Zeitschrift fur Angewandte +Mathematik und Physik, 75 (suppl. 2):S435-S438, 1995 ISSN 0044-2267 + keywords = "axiomref", \end{chunk} +\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A -Handbook of Mathematical Functions'' -Dover Publications. (1968) +\bibitem[Sage 14]{Sage14} Stein, William +Sage'' +\verb|www.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html| + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Salvy 89]{Sal89} Salvy, B. +Examples of automatic asymptotic expansions'' +Technical Report 114, +Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Salvy 91]{Sal91} Salvy, B. +Examples of automatic asymptotic expansions'' +SIGSAM Bulletin (ACM Special Interest Group on Symbolic and +Algebraic Manipulation), 25(2) pp4-17 +April 1991 CODEN SIGSBZ ISSN 0163-5824 + keywords = "axiomref", \end{chunk} \begin{chunk}{axiom.bib} -@book{Altm05, - author = "Altmann, Simon L.", - title = "Rotations, Quaternions, and Double Groups", - publisher = "Dover Publications, Inc.", - year = "2005", - isbn = "0-486-44518-6" +@article{Saun80, + author = "Saunders, B. David", + title = "A Survey of Available Systems", + journal = "SIGSAM Bull.", + issue_date = "November 1980", + volume = "14", + number = "4", + month = "November", + year = "1980", + issn = "0163-5824", + pages = "12--28", + numpages = "17", + url = "http://doi.acm.org/10.1145/1089235.1089237", + doi = "10.1145/1089235.1089237", + acmid = "1089237", + publisher = "ACM", + address = "New York, NY, USA", + keywords = "axiomref,survey", + paper = "Saun80.pdf" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Ames 77]{Ames77} Ames W F -Nonlinear Partial Differential Equations in Engineering'' -Academic Press (2nd Edition). (1977) +\bibitem[Schu 92]{Sch92} Sch\"u, J. +Implementing des Cartan-Kuranishi-Theorems in AXIOM'' +Master's diploma thesis (in german), Institut f\"ur Algorithmen und +Kognitive Systeme, Universit\"t Karlsruhe 1992 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Amos 86]{Amos86} Amos D E -Algorithm 644: A Portable Package for Bessel Functions of a Complex -Argument and Nonnegative Order'' -ACM Trans. Math. Softw. 12 265--273. (1986) +\bibitem[Schwarz 88]{Sch88} Schwarz, F. +Programming with abstract data types: the symmetry package SPDE +in Scratchpad'' +In Jan{\ss}en [Jan88], pp167-176, ISBN 3-540-18928-9, +0-387-18928-9 LCCN QA155.7.E4T74 1988 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Anderson 00]{And00} Anderson, Edward -Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem'' -LAPACK Working Note 150, University of Tennessee, UT-CS-00-454, -December 4, 2000. +\bibitem[Schwarz 89]{Sch89} Schwarz, F. +A factorization algorithm for linear ordinary differential equations'' +In ACM [ACM89], pp17-25 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G -DASL - Data Approximation Subroutine Library'' -National Physical Laboratory. (1982) +\bibitem[Schwarz 91]{Sch91} Schwarz, F. +Monomial orderings and Gr{\"o}bner bases'' +SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic +Manipulation) 2591) pp10-23 Jan. 1991 CODEN SIGSBZ ISSN 0163-5824 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M. -On Mechanical Quantifier Elimination for Elementary Algebra and Geometry'' -J. Symbolic Computation 5, 237-259 (1988) -\verb|http://www.sciencedirect.com/science/article/pii/S0747717188800142/| -\verb|pdf?md5=62052077d84e6078cc024bc8e29c23c1&| -\verb|pid=1-s2.0-S0747717188800142-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Arno88.pdf| +\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus +Analysis and Application of the Formal Theory of Partial Differential +Equations'' +PhD thesis, School of Physics and Materials, Lancaster University (1994) +\verb|www.mathematik.uni-kassel.de/~seiler/Papers/Diss/diss.ps.gz| +%\verb|axiom-developer.org/axiom-website/papers/Sei94.pdf| + keywords = "axiomref", abstract = " - We give solutions to two problems of elementary algebra and geometry: - (1) find conditions on real numbers $p$, $q$, and $r$ so that the - polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real - $x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that - the ellipse $\frac{(x-e)^2}{q^2}+\frac{y^2}{b^2}-1=0$ lies inside the - unit circle $y^2+x^2-1=0$. Our solutions are obtained by following the - basic outline of the method of quantifier elimination by cylindrical - algebraic decomposition (Collins, 1975), but we have developed, and - have been considerably aided by, modified versions of certain of its - steps. We have found three equally simple but not obviously equivalent - solutions for the first problem, illustrating the difficulty of - obtaining unique simplest'' solutions to quantifier elimination - problems of elementary algebra and geometry." + An introduction to the formal theory of partial differential equations + is given emphasizing the properties of involutive symbols and + equations. An algorithm to complete any differential equation to an + involutive one is presented. For an involutive equation possible + values for the number of arbitrary functions in its general solution + are determined. The existence and uniqueness of solutions for analytic + equations is proven. Applications of these results include an + analysis of symmetry and reduction methods and a study of gauge + systems. It is show that the Dirac algorithm for systems with + constraints is closely related to the completion of the equation of + motion to an involutive equation. Specific examples treated comprise + the Yang-Mills Equations, Einstein Equations, complete and Jacobian + systems, and some special models in two and three dimensions. To + facilitate the involved tedious computations an environment for + geometric approaches to differential equations has been developed in + the computer algebra system Axiom. The appendices contain among others + brief introductions into Carten-K{\"a}hler Theory and Janet-Riquier + Theory." \end{chunk} -\begin{chunk}{axiom.bib} -@article{Aubr99, - author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc", - title = "On the Theories of Triangular Sets", - year = "1999", - pages = "105-124", - journal = "Journal of Symbolic Computation", - volume = "28", - url = "http://www.csd.uwo.ca/~moreno/Publications/Aubry-Lazard-MorenoMaza-1999-JSC.pdf", - papers = "Aubr99.pdf", - abstract = " - Different notions of triangular sets are presented. The relationship - between these notions are studied. The main result is that four - different existing notions of {\sl good} triangular sets are - equivalent." -} +\begin{chunk}{ignore} +\bibitem[Seiler 94a]{Sei94a} Seiler, W.M. +Completion to involution in AXIOM'' +in Calmet [Cal94] pp103-104 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno -Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods'' -\verb|www.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz| -%\verb|axiom-developer.org/axiom-website/papers/Aub96.ps| +\bibitem[Sieler 94b]{Sei94b} Seiler, W.M. +Pseudo differential operators and integrable systems in AXIOM'' +Computer Physics Communications, 79(2) pp329-340 April 1994 CODEN CPHCBZ +ISSN 0010-4655 +%\verb|axiom-developer.org/axiom-website/papers/Sei94b.pdf| + keywords = "axiomref", abstract = " - Four methods for solving polynomial systems by means of triangular - sets are presented and implemented in a unified way. These methods are - those of Wu, Lazard, Kalkbrener, and Wang. They are compared on - various examples with emphasis on efficiency, conciseness and - legibility of the outputs." + An implementation of the algebra of pseudo differential operators in + the computer algebra system Axiom is described. In several exmaples + the application of the package to typical computations in the theory + of integrable systems is demonstrated." \end{chunk} -\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Bailey 66]{Bai66} Bailey P B -Sturm-Liouville Eigenvalues via a Phase Function'' -SIAM J. Appl. Math . 14 242--249. (1966) +\bibitem[Seiler 95]{Sei95} Seiler, W.M. +Applying AXIOM to partial differential equations'' +Internal Report 95-17, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik +1995 +%\verb|axiom-developer.org/axiom-website/papers/Sei95.pdf| + keywords = "axiomref", + abstract = " + We present an Axiom environment called JET for geometric computations + with partial differential equations within the framework of the jet + bundle formalism. This comprises expecially the completion of a given + differential equation to an involutive one according to the + Cartan-Kuranishi Theorem and the setting up of the determining system + for the generators of classical and non-classical Lie + symmetries. Details of the implementations are described and + applications are given. An appendix contains tables of all exported + functions." \end{chunk} \begin{chunk}{ignore} -\bibitem[Baker 96]{BGM96} Baker, George A.; Graves-Morris, Peter -Pade Approximants'' -Cambridge University Press, March 1996 ISBN 9870521450072 +\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J. +JET -- An Axiom Environment for Geometric Computations with Differential +Equations'' +%\verb|axiom-developer.org/axiom-website/papers/SC95.pdf| + keywords = "axiomref", + abstract = " + JET is an environment within the computer algebra system Axiom to + perform such computations. The current implementation emphasises the + two key concepts involution and symmetry. It provides some packages + for the completion of a given system of differential equations to an + equivalent involutive one based on the Cartan-Kuranishi theorem and + for setting up the determining equations for classical and + non-classical point symmetries." \end{chunk} \begin{chunk}{ignore} -\bibitem[Baker 10]{Ba10} Baker, Martin -3D World Simulation'' -\verb|www.euclideanspace.com| - -\end{chunk} - -\begin{chunk}{axiom.bib} -@misc{Bake14, - author = "Baker, Martin", - title = "Axiom Architecture", - year = "2014", - url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode" -} +\bibitem[Seiler 97]{Sei97} Seiler, Werner M. +Computer Algebra and Differential Equations: An Overview'' +\verb|www.mathematik.uni-kassel.di/~seiler/Papers/Postscript/CADERep.ps.gz| + keywords = "axiomref", + abstract = " + We present an informal overview of a number of approaches to + differential equations which are popular in computer algebra. This + includes symmetry and completion theory, local analysis, differential + ideal and Galois theory, dynamical systems and numerical analysis. A + large bibliography is provided." \end{chunk} \begin{chunk}{ignore} -\bibitem[Banks 68]{BK68} Banks D O; Kurowski I -Computation of Eigenvalues of Singular Sturm-Liouville Systems'' -Math. Computing. 22 304--310. (1968) +\bibitem[Seiler (a)]{Seixx} Seiler, W.M. +DETools: A Library for Differential Equations'' +\verb|iaks-www.ira.uka.de/iaks-calmet/werner/werner.html| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Bard 74]{Bard74} Bard Y -Nonlinear Parameter Estimation'' -Academic Press. 1974 +\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M. +Using Gr{\"o}bner bases to determine algebra +membership, split surjective algebra homomorphisms determine birational +equivalence'' +Journal of Symbolic Computation 6(2-3) pp267-273 +Oct.-Dec. 1988 CODEN JSYCEH ISSN 0747-7171 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K -An Improved Algorithm for Discrete $ll_1$ Linear Approximation'' -SIAM J. Numer. Anal. 10 839--848. (1973) +\bibitem[Sit 89]{Sit89} Sit, W.Y. +On Goldman's algorithm for solving first-order multinomial +autonomous systems'' In Mora [Mor89], pp386-395 ISBN 3-540-51083-4 +LCCN QA268.A35 1998 Conference held jointly with ISSAC '88 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K -Solution of an Overdetermined System of Equations in the $ll_1-norm$.'' -Comm. ACM. 17, 6 319--320. (1974) +\bibitem[Sit 92]{Sit92} Sit, W.Y. +An algorithm for solving parametric linear systems'' +Journal of Symbolic Computations, 13(4) pp353-394, April 1992 CODEN JSYCEH +ISSN 0747-7171 +\verb|www.sciencedirect.com/science/article/pii/S0747717108801046/pdf| +\verb|?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&| +\verb|pid=1-s2.0-S0747717108801046-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Sit92.pdf| + keywords = "axiomref", + abstract = " + We present a theoretical foundation for studying parametric systesm of + linear equations and prove an efficient algorithm for identifying all + parametric values (including degnerate cases) for which the system is + consistent. The algorithm gives a small set of regimes where for each + regime, the solutions of the specialized systems may be given + uniformly. For homogeneous linear systems, or for systems were the + right hand side is arbitrary, this small set is irredunant. We discuss + in detail practical issues concerning implementations, with particular + emphasis on simplification of results. Examples are given based on a + close implementation of the algorithm in SCRATCHPAD II. We also give a + complexity analysis of the Gaussian elimination method and compare + that with our algorithm." \end{chunk} \begin{chunk}{ignore} -\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard -Products of polynomials and a priori estimates for -coefficients in polynomial decompositions: a sharp result'' -J. Symbolic Computation (1992) 13, 463-472 -%\verb|axiom-developer.org/axiom-website/papers/Bea92.pdf| +\bibitem[Sit 06]{Sit06} Sit, Emil +Tools for Repeatable Research'' +\verb|www.emilsit.net/blog/archives/tools-for-repeatable-research| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar; -Wang, Paul S. -Polynomial Factorization: Sharp Bounds, Efficient Algorithms'' -J. Symbolic Computation (1993) 15, 393-413 -%\verb|axiom-developer.org/axiom-website/papers/Bea93.pdf| +\bibitem[Smedley 92]{Sme92} Smedley, Trevor J. +Using pictorial and object oriented programming for computer algebra'' +In Hal Berghel et al., editors. Applied computing -- +technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP +Symposium on Applied Computing, Kansas City Convention Center, March 1-3, 1992 +pp1243-1247. ACM Press, New York, NY 10036, USA, 1992. ISBN 0-89791-502-X +LCCN QA76.76.A65 S95 1992 + keywords = "axiomref", \end{chunk} -\begin{chunk}{axiom.bib} -@article{Bert95, - author = "Bertrand, Laurent", - title = "Computing a hyperelliptic integral using arithmetic in the - jacobian of the curve", - journal = "Applicable Algebra in Engineering, Communication and Computing", - volume = "6", - pages = "275-298", - year = "1995", +\begin{chunk}{ignore} +\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko +Algorithmic differentiation in Axiom'' +ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347-354 +ISBN 978-1-59593-743-8 +%\verb|axiom-developer.org/axiom-website/papers/SDJ07.pdf| + keywords = "axiomref", abstract = " - In this paper, we describe an efficient algorithm for computing an - elementary antiderivative of an algebraic function defined on a - hyperelliptic curve. Our algorithm combines B.M. Trager's integration - algorithm and a technique for computing in the Jacobian of a - hyperelliptic curve introduced by D.G. Cantor. Our method has been - implemented and successfully compared to Trager's general algorithm." -} + This paper describes the design and implementation of an algorithmic + differentiation framework in the Axiom computer algebra system. Our + implementation works by transformations on Spad programs at the level + of the typed abstract syntax tree." \end{chunk} \begin{chunk}{ignore} -\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I. -Design of the Stiff Integrators in the NAG Library'' -Technical Report. TR14/87 NAG. (1987) +\bibitem[SSC92]{SSC92}. +Algorithmic Methods For Lie Pseudogroups'' +In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group +Analysis: Advanced Analytical and Computational Methods in Mathematical +Physics, pp337-344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993 +\verb|iaks-www.ira.uka.de/iaks-calmet/werner/Papers/Acireale92.ps.gz| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Berzins 90]{Ber90} Berzins M -Developments in the NAG Library Software for Parabolic Equations'' -Scientific Software Systems. (ed J C Mason and M G Cox) -Chapman and Hall. 59--72. (1990) +\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G. +Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel'' +Technical Report 640-M, TIM 3 (IMAG), Grenoble, France, Feb 1987 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C -Ordinary Differential Equations'' -Ginn \& Co., Boston and New York. (1962) +\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P. +The Evolution of Lisp'' +\verb|www.dreamsongs.com/Files/HOPL2-Uncut.pdf| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W. -Bounds for the Height of a Factor of a Polynomial in -Terms of Bombieri's Norms: I. The Largest Factor'' -J. Symbolic Computation (1993) 16, 115-130 -%\verb|axiom-developer.org/axiom-website/Boyd93a.pdf| +\bibitem[Sutor 85]{Sut85} Sutor, R.S. +The Scratchpad II computer algebra language and system'' +In Buchberger and Caviness [BC85], pp32-33 ISBN 0-387-15983-5 (vol. 1), +0-387-15984-3 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes. + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Boyd 93b]{Boyd93b} Boyd, David W. -Bounds for the Height of a Factor of a Polynomial in -Terms of Bombieri's Norms: II. The Smallest Factor'' -J. Symbolic Computation (1993) 16, 131-145 -%\verb|axiom-developer.org/axiom-website/Boyd93b.pdf| +\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D. +The type inference and coercion facilities in +the Scratchpad II interpreter'' In Wexelblat [Wex87], pp56-63 +ISBN 0-89791-235-7 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987) +%\verb|axiom-developer.org/axiom-website/papers/SJ87a.pdf| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R. -The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, -and Level 3 Performance'' -SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. +\bibitem[Sutor 87b]{Su87} Sutor, Robert S. +The Scratchpad II Computer Algebra System. Using and +Programming the Interpreter'' +IBM Course presentation slide deck Spring 1987 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R. -The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation'' -SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002. +\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard +The type inference and coercion facilities +in the Scratchpad II interpreter'' +Research report RC 12595 (\#56575), +IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp +%\verb|axiom-developer.org/axiom-website/papers/SJ87c.pdf| + keywords = "axiomref", + abstract = " + The Scratchpad II system is an abstract datatype programming language, + a compiler for the language, a library of packages of polymorphic + functions and parameterized abstract datatypes, and an interpreter + that provides sophisticated type inference and coercion facilities. + Although originally designed for the implementation of symbolic + mathematical algorithms, Scratchpad II is a general purpose + programming language. This paper discusses aspects of the + implementation of the intepreter and how it attempts to provide a user + friendly and relatively weakly typed front end for the strongly typed + programming language." \end{chunk} \begin{chunk}{ignore} -\bibitem[Brent 75]{Bre75} Brent, R. P. -Multiple-Precision Zero-Finding Methods and the Complexity -of Elementary Function Evaluation, Analytic Computational Complexity'' -J. F. Traub, Ed., Academic Press, New York 1975, 151-176 +\bibitem[Sutor 88]{Su88} Sutor, Robert S. +A guide to programming in the scratchpad 2 interpreter'' +IBM Manual, March 1988 + keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T. -Fast Algorithms for Manipulating Formal Power Series'' -Journal of the Association for Computing Machinery, -Vol. 25, No. 4, October 1978, 581-595 - -\end{chunk} +\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Brigham 73]{Bri73} Brigham E O -The Fast Fourier Transform'' -Prentice-Hall. (1973) +\bibitem[Thompson 00]{Tho00} Thompson, Simon +Logic and dependent types in the Aldor Computer Algebra System'' +%\verb|axiom-developer.org/axiom-website/papers/Tho00.pdf| + keywords = "axiomref", + abstract = " + We show how the Aldor type system can represent propositions of + first-order logic, by means of the 'propositions as types' + correspondence. The representation relies on type casts (using + pretend) but can be viewed as a prototype implementation of a modified + type system with {\sl type evaluation} reported elsewhere. The logic + is used to provide an axiomatisation of a number of familiar Aldor + categories as well as a type of vectors." \end{chunk} \begin{chunk}{ignore} -\bibitem[Brillhart 69]{Bri69} Brillhart, John -On the Euler and Bernoulli polynomials'' -J. Reine Angew. Math., v. 234, (1969), pp. 45-64 +\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid +The Aldor\-\- language'' +%\verb|axiom-developer.org/axiom-website/papers/TTxx.pdf| + keywords = "axiomref", + abstract = " + This paper introduces the \verb|Aldor--| language, which is a + functional programming language with dependent types and a powerful, + type-based, overloading mechanism. The language is built on a subset + of Aldor, the 'library compiler' language for the Axiom computer + algebra system. \verb|Aldor--| is designed with the intention of + incorporating logical reasoning into computer algebra computations. + + The paper contains a formal account of the semantics and type system + of \verb|Aldor--|; a general discussion of overloading and how the + overloading in \verb|Aldor--| fits into the general scheme; examples + of logic within \verb|Aldor--| and notes on the implementation of the + system." \end{chunk} \begin{chunk}{ignore} -\bibitem[Brillhart 90]{Bri90} Brillhart, John -Note on Irreducibility Testing'' -Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381 +\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel +Etude du typage dans le syst\eme de calcul scientifique Aldor'' +Universit\'e de Limoges 1998 +%\verb|axiom-developer.org/axiom-website/papers/Tou98.pdf| + keywords = "axiomref", \end{chunk} -\begin{chunk}{ignore} -\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds) -Symbolic Rewriting Techniques'' -Progress in Computer Science and Applied Logic 15, Birkhauser-Verlag, Basel -ISBN 3-7643-5901-3 (1998) - -\end{chunk} +\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Bronstein 88]{Bro88} Bronstein, Manual -The Transcendental Risch Differential Equation'' -J. Symbolic Computation (1990) 9, pp49-60 Feb 1988 -IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY -\verb|www.sciencedirect.com/science/article/pii/S0747717108800065| -%\verb|axiom-developer.org/axiom-website/papers/Bro88.pdf| - abstract = " - We present a new rational algorithm for solving Risch differential - equations in towers of transcendental elementary extensions. In - contrast to a recent algorithm by Davenport we do not require a - progressive reduction of the denominators involved, but use weak - normality to obtain a formula for the denominator of a possible - solution. Implementation timings show this approach to be faster than - a Hermite-like reduction." +\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris +Computer algebra systems and TeXmacs'' +\verb|www.texmacs.org/tmweb/plugins/cas.en.html| + keywords = "axiomref", \end{chunk} \begin{chunk}{axiom.bib} -@techreport{Bron98, - author = "Bronstein, Manuel", - title = "The lazy hermite reduction", - type = "Rapport de Recherche", - number = "RR-3562", - year = "1998", - institution = "French Institute for Research in Computer Science", - paper = "Bron98.pdf", +@article{Hoei94, + author = "{van Hoeij}, M.", + title = "An algorithm for computing an integral basis in an algebraic + function field", + journal = "Journal of Symbolic Computation", + volume = "18", + number = "4", + year = "1994", + pages = "353-363", + issn = "0747-7171", + keywords = "axiomref", + paper = "Hoei94.pdf", abstract = " - The Hermite reduction is a symbolic integration technique that reduces - algebraic functions to integrands having only simple affine - poles. While it is very effective in the case of simple radical - extensions, its use in more general algebraic extensions requires the - precomputation of an integral basis, which makes the reduction - impractical for either multiple algebraic extensions or complicated - ground fields. In this paper, we show that the Hermite reduction can - be performed without {\sl a priori} computation of either a primitive - element or integral basis, computing the smallest order necessary for - a particular integrand along the way." + Algorithms for computing integral bases of an algebraic function field + are implemented in some computer algebra systems. They are used e.g. + for the integration of algebraic functions. The method used by Maple + 5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm + of Ford and Zassenhaus [Ford, 1978], that computes the ring of + integers in an algebraic number field, to the case of a function field. + + It turns out that using algebraic geometry one can write a faster + algorithm. The method we will give is based on Puiseux expansions. + One cas see this as a variant on the Coates' algorithm as it is + described in [Davenport, 1981]. Some difficulties in computing with + Puiseux expansions can be avoided using a sharp bound for the number + of terms required which will be given in Section 3. In Section 5 we + derive which denominator is needed in the integral basis. Using this + result 'intermediate expression swell' can be avoided. + + The Puiseux expansions generally introduce algebraic extensions. These + extensions will not appear in the resulting integral basis." } \end{chunk} \begin{chunk}{axiom.bib} -@misc{Bro98b, - author = "Bronstein, Manuel", - title = "Symbolic Integration Tutorial", - series = "ISSAC'98", - year = "1998", - address = "INRIA Sophia Antipolis", - url = - "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf", - paper = "Bro98b.pdf" +@misc{Hoei08, + author = "{van Hoeij}, Mark and Novocin, Andrew", + title = "A Reduction Algorithm for Algebraic Function Fields", + year = "2008", + month = "April", + url = "http://andy.novocin.com/pro/algext.pdf", + paper = "Hoei08.pdf", + abstract = " + Computer algebra systesm often produce large expressions involving + complicated algebraic numbers. In this paper we study variations of + the {\tt polred} algorithm that can often be used to find better + representations for algebraic numbers. The main new algorithm + presented here is an algorithm that treats the same problem for the + function field case." } \end{chunk} \begin{chunk}{ignore} -\bibitem[Brown 99]{Brow99} Brown, Christopher W. -Solution Formula Construction for Truth Invariant CADs'' -Ph.D Thesis, Univ. Delaware (1999) -\verb|www.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz| -%\verb|axiom-developer.org/axiom-website/papers/Brow99.pdf| - abstract = " - The CAD-based quantifier elimination algorithm takes a formula from - the elementary theory of real closed fields as input, and constructs a - CAD of the space of the formula's unquantified variables. This - decomposition is truth invariant with respect to the input formula, - meaning that the formula is either identically true or identically - false in each cell of the decomposition. The method determines the - truth of the input formula for each cell of the CAD, and then uses the - CAD to construct a solution formula -- a quantifier free formula that - is equivalent to the input formula. This final phase of the algorithm, - the solution formula construction phase, is the focus of this thesis. +\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer +Computational Methods in Commutative Algebra and Algebraic Geometry'' +Springer, Algorithms and Computation in Mathematics, Vol 2 1999 +ISBN 3-540-21311-2 + keywords = "axiomref", - An optimal solution formula construction algorithm would be {\sl - complete} -- i.e. applicable to any truth-invariant CAD, would be {\sl - efficient}, and would produce {\sl simple} solution formulas. Prior to - this thesis, no method was available with even two of these three - properties. Several algorithms are presented, all addressing problems - related to solution formula construction. In combination, these - provide an efficient and complete method for constructing solution - formulas that are simple in a variety of ways. +\end{chunk} - Algorithms presented in this thesis have been implemented using the - SACLIB library, and integrated into QEPCAD, a SACLIB-based - implementation of quantifier elimination by CAD. Example computations - based on these implementations are discussed." +\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Wang 89]{Wan89} Wang, D. +A program for computing the Liapunov functions and Liapunov +constants in Scratchpad II'' +SIGSAM Bulletin (ACM Special Interest Group +on Symbolic and Algebraic Manipulation), 23(4) pp25-31, Oct. 1989, +CODEN SIGSBZ ISSN 0163-5824 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Brown 02]{Brow02} Brown, Christopher W. -QEPCAD B -- A program for computing with semi-algebraic sets using CADs'' -%\verb|axiom-developer.org/axiom-website/papers/Brow02.pdf| - abstract = " - This report introduces QEPCAD B, a program for computing with real - algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD - B both extends and improves upon the QEPCAD system for quantifier - elimination by partial cylindrical algebraic decomposition written by - Hoon Hong in the early 1990s. This paper briefly discusses some of the - improvements in the implementation of CAD and quantifier elimination - vis CAD, and provides somewhat more detail on extensions to the system - that go beyond quantifier elimination. The author is responsible for - most of the extended features of QEPCAD B, but improvements to the - basic CAD implementation and to the SACLIB library on which QEPCAD is - based are the results of many people's work." +\bibitem[Wang 91]{Wan91} Wang, Dongming +Mechanical manipulation for a class of differential systems'' +Journal of Symbolic Computation, 12(2) pp233-254 Aug. 1991 +CODEN JSYCEH ISSN 0747-7171 + keywords = "axiomref", \end{chunk} -\begin{chunk}{axiom.bib} -@article{Burg74, - author = "William H. Burge", - title = "Stream Processing Functions", - year = "1974", - month = "January", - journal = "IBM Journal of Research and Development", - volume = "19", - issue = "1", - pages = "12-25", - papers = "Burg74.pdf", - abstract = " - One principle of structured programming is that a program should be - separated into meaningful independent subprograms, which are then - combined so that the relation of the parts to the whole can be clearly - established. This paper describes several alternative ways to compose - programs. The main method used is to permit the programmer to denote - by an expression the sequence of values taken on by a variable. The - sequence is represented by a function called a stream, which is a - functional analog of a coroutine. The conventional while and for loops - of structured programming may be composed by a technique of stream - processing (analogous to list processing), which results in more - structured programs than the orignals. This technique makes it - possible to structure a program in a natural way into its logically - separate parts, which can then be considered independently." -} +\begin{chunk}{ignore} +\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed) +International System Symposium on Symbolic and +Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992 +ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover), +LCCN QA76.95.I59 1992 + keywords = "axiomref", \end{chunk} -\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{ignore} +\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed) +ISSAC '90 Proceedings of the +International Symposium on Symbolic and Algebraic Computation ACM Press, +New York, NY, 10036, USA. 1990 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990 + keywords = "axiomref", + +\end{chunk} \begin{chunk}{ignore} -\bibitem[Carlson 65]{Car65} Carlson B C -On Computing Elliptic Integrals and Functions'' -J Math Phys. 44 36--51. (1965) +\bibitem[Watt 85]{Wat85} Watt, Stephen +Bounded Parallelism in Computer Algebra'' +PhD Thesis, University of Waterloo +\verb|www.csd.uwo.ca/~watt/pub/reprints/1985-smw-phd.pdf| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Carlson 77a]{Car77a} Carlson B C -Elliptic Integrals of the First Kind'' -SIAM J Math Anal. 8 231--242. (1977) +\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J. +Algebra Snapshot: Linear Ordinary Differential Operators'' +Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986) +\verb|www.csd.uwo.ca/~watt/pub/reprints/1986-snews-lodo.pdf| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Carlson 77b]{Car77b} Carlson B C -Special Functions of Applied Mathematics'' -Academic Press. (1977) +\bibitem[Watt 87]{Wat87} Watt, Stephen +Domains and Subdomains in Scratchpad II'' +in [Wit87], pp3-5 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Carlson 78]{Car78} Carlson B C, -Computing Elliptic Integrals by Duplication'' -(Preprint) Department of Physics, Iowa State University. (1978) +\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H. +Mapping as First Class Objects'' +in [Wit87], pp13-17 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Carlson 88]{Car88} Carlson B C, -A Table of Elliptic Integrals of the Third Kind'' -Math. Comput. 51 267--280. (1988) +\bibitem[Watt 89]{Wat89} Watt, S. M. +A fixed point method for power series computation'' +In Gianni [Gia89], pp206-217 ISBN 3-540-51084-2 LCCN QA76.95.I57 +1988 Conference held jointly with AAECC-6 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Cauchy 1829]{Cau1829} Augustin-Lux Cauchy -Exercices de Math\'ematiques Quatri\eme Ann\'ee. De Bure Fr\eres'' -Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX, -Gauthier-Villars, Paris, 1891). +\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M. +The Scratchpad II type system: Domains and subdomains'' +in A.M. Miola, editor Computing Tools +for Scientific Problem Solving, Academic Press, New York, 1990 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Ch\eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire -Lifting and recombination techniques for absolute factorization'' -Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380-420 -\verb|www.sciencedirect.com/science/article/pii/S0885064X07000465| -%\verb|axiom-developer.org/axiom-website/papers/Chez07.pdf| - abstract = " - In the vein of recent algorithmic advances in polynomial factorization - based on lifting and recombination techniques, we present new faster - algorithms for computing the absolute factorization of a bivariate - polynomial. The running time of our probabilistic algorithm is less - than quadratic in the dense size of the polynomial to be factored." +\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed) +Proceedings of the 1991 International Symposium on +Symbolic and Algebraic Computation, ISSAC'91, July 15-17, 1991, Bonn, Germany, +ACM Press, New York, NY 10036, USA, 1991 ISBN 0-89791-437-6 +LCCN QA76.95.I59 1991 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E; -Nelson P (eds) -Codes for Boundary-value Problems in Ordinary Differential Equations'' -Lecture Notes in Computer Science. 76 (1979) Springer-Verlag +\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.; +Steinback, J.M.; Sutor, R.S. +A\# User's Guide'' +Version 1.0.0 O($\epsilon{}^1$) June 8, 1994 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A. -Efficient Solution of Linear Diophantine Equations'' -JSC (1989) 8, 201-216 +\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.; +Dooley, Samuel S.; Iglio, Pietro +A First Report on the A\# Compiler (including benchmarks)'' +IBM Research Report RC19529 (85075) May 12, 1994 +%\verb|axiom-developer.org/axiom-website/papers/Wat94.pdf| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Clenshaw 55]{Cle55} Clenshaw C W, -A Note on the Summation of Chebyshev Series'' -Math. Tables Aids Comput. 9 118--120. (1955) +\bibitem[Watt 94c]{Wat94c} Watt, Stephen M. +A\# Language Reference Version 0.35'' +IBM Research Division Technical Report RC19530 May 1994 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Clenshaw 60]{Cle60} Clenshaw C W -Curve Fitting with a Digital Computer'' -Comput. J. 2 170--173. (1960) +\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P. +Steinbach, J.M.; Morrison, S.C.; Sutor, R.S. +AXIOM Library Compiler Users Guide'' +The Numerical Algorithms Group (NAG) Ltd, 1994 + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro; +Morrison, Scott C.; Steinbach, Jonathan M. +FOAM: A First Order Abstract Machine Version 0.35'' +IBM T. J. Watson Research Center (2001) +%\verb|axiom-developer.org/axiom-website/papers/Wat01.pdf| + keywords = "axiomref", + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Weber 92]{Webe92} Weber, Andreas +Type Systems for Computer Algebra'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe92.pdf| + keywords = "axiomref", + abstract = " + An important feature of modern computer algebra systems is the support + of a rich type system with the possibility of type inference. Basic + features of such a type system are polymorphism and coercion between + types. Recently the use of order-sorted rewrite systems was proposed + as a general framework. We will give a quite simple example of a + family of types arising in computer algebra whose coercion relations + cannot be captured by a finite set of first-order rewrite rules." \end{chunk} \begin{chunk}{ignore} -\bibitem[Clenshaw 62]{Cle62} Clenshaw C W -Mathematical Tables. Chebyshev Series for Mathematical Functions'' -HMSO. (1962) +\bibitem[Weber 92b]{Webe92b} Weber, Andreas +Structuring the Type System of a Computer Algebra System'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe92b.pdf| + keywords = "axiomref", + abstract = " + Most existing computer algebra systems are pure symbol manipulating + systems without language support for the occuring types. This is + mainly due to the fact taht the occurring types are much more + complicated than in traditional programming languages. In the last + decade the study of type systems has become an active area of + research. We will give a proposal for a type system showing that + several problems for a type system of a symbolic computation system + can be solved by using results of this research. We will also provide + a variety of examples which will show some of the problems that remain + and that will require further research." \end{chunk} \begin{chunk}{ignore} -\bibitem[Cline 84]{CR84} Cline A K; Renka R L, -A Storage-efficient Method for Construction of a Thiessen Triangulation'' -Rocky Mountain J. Math. 14 119--139. (1984) +\bibitem[Weber 93b]{Webe93b} Weber, Andreas +Type Systems for Computer Algebra'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber93b.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe93b.pdf| + keywords = "axiomref", + abstract = " + We study type systems for computer algebra systems, which frequently + correspond to the pragmatically developed'' typing constructs used + in AXIOM. A central concept is that of {\sl type classes} which + correspond to AXIOM categories. We will show that types can be + syntactically described as terms of a regular order-sorted signature + if no type parameters are allowed. Using results obtained for the + functional programming language Haskell we will show that the problem + of {\sl type inference} is decidable. This result still holds if + higher-order functions are present and {\sl parametric polymorphism} + is used. These additional typing constructs are useful for further + extensions of existing computer algebra systems: These typing concepts + can be used to implement category theoretic constructs and there are + many well known constructive interactions between category theory and + algebra." \end{chunk} \begin{chunk}{ignore} -\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.; -Wilson, R. -Atlas of Finite Groups'' -Oxford, Clarendon Press, 1987 - -\end{chunk} +\bibitem[Weber 94]{Web94} Weber, Andreas +Algorithms for Type Inference with Coercions'' +ISSAC 94 ACM 0-89791-638-7/94/0007 +%\verb|axiom-developer.org/axiom-website/papers/Web94.pdf| + keywords = "axiomref", + abstract = " + This paper presents algorithms that perform a type inference for a + type system occurring in the context of computer algebra. The type + system permits various classes of coercions between types and the + algorithms are complete for the precisely defined system, which can be + seen as a formal description of an important subset of the type system + supported by the computer algebra program Axiom. -\begin{chunk}{ignore} -\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A. -On Quaternions and Octonions'' -A.K Peters, Natick, MA. (2003) ISBN 1-56881-134-9 + Previously only algorithms for much more restricted cases of coercions + have been described or the frameworks used have been so general that + the corresponding type inference problems were known to be + undecidable." \end{chunk} \begin{chunk}{ignore} -\bibitem[Cox 72]{Cox72} Cox M G -The Numerical Evaluation of B-splines'' -J. Inst. Math. Appl. 10 134--149. (1972) +\bibitem[Weber 95]{Webe95} Weber, A. +On coherence in computer algebra'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber94e.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe95.pdf| + keywords = "axiomref", + abstract = " + Modern computer algebra systems (e.g. AXIOM) support a rich type + system including parameterized data types and the possibility of + implicit coercions between types. In such a type system it will be + frequently the case that there are different ways of building + coercions between types. An important requirement is that all + coercions between two types coincide, a property which is called {\sl + coherence}. We will prove a coherence theorem for a formal type system + having several possibilities of coercions covering many important + examples. Moreover, we will give some informal reasoning why the + formally defined restrictions can be satisfied by an actual system." \end{chunk} \begin{chunk}{ignore} -\bibitem[CH 73]{CH73} Cox M G; Hayes J G -Curve fitting: a guide and suite of algorithms for the -non-specialist user'' -Report NAC26. National Physical Laboratory. (1973) +\bibitem[Weber 96]{Webe96} Weber, Andreas +Computing Radical Expressions for Roots of Unity'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber96a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe96.pdf| + keywords = "axiomref", + abstract = " + We present an improvement of an algorithm given by Gauss to compute a + radical expression for a $p$-th root of unity. The time complexity of + the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime + factor of $p-1$." \end{chunk} + \begin{chunk}{ignore} -\bibitem[Cox 74a]{Cox74a} Cox M G -A Data-fitting Package for the Non-specialist User'' -Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974) +\bibitem[Weber 99]{Webe99} Weber, Andreas +Solving Cyclotomic Polynomials by Radical Expressions'' +\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/| +\verb|WeberA/WeberKeckeisen99a.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Webe99.pdf| + keywords = "axiomref", + abstract = " + We describe a Maple package that allows the solution of cyclotomic + polynomials by radical expressions. We provide a function that is an + extension of the Maple {\sl solve} command. The major algorithmic + ingredient of the package is an improvement of a method due to Gauss + which gives radical expressions for roots of unity. We will give a + summary for computations up to degree 100, which could be done within + a few hours of cpu time on a standard workstation." \end{chunk} \begin{chunk}{ignore} -\bibitem[Cox 74b]{Cox74b} Cox M G -Numerical methods for the interpolation and approximation of data -by spline functions'' -PhD Thesis. City University, London. (1975) +\bibitem[Wei-Jiang 12]{WJ12} Wei-Jiang +Top free algebra System'' +\verb|wei-jiang.com/it/software/top-free-algebra-system-bye-mathematica-bye-maple| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Cox 75]{Cox75} Cox M G -An Algorithm for Spline Interpolation'' -J. Inst. Math. Appl. 15 95--108. (1975) +\bibitem[Wester 99]{Wes99} Wester, Michael J. +Computer Algebra Systems'' +John Wiley and Sons 1999 ISBN 0-471-98353-5 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Cox 77]{Cox77} Cox M G -A Survey of Numerical Methods for Data and Function Approximation'' -The State of the Art in Numerical Analysis. (ed D A H Jacobs) -Academic Press. 627--668. (1977) - keywords = "survey", +\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed) +Proceedings of the SIGPLAN '87 Symposium on +Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 24-26, 1987 +ACM Press, New York, NY 10036, USA, 1987 ISBN 0-89791-235-7 +LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987) + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Cox 78]{Cox78} Cox M G -The Numerical Evaluation of a Spline from its B-spline Representation'' -J. Inst. Math. Appl. 21 135--143. (1978) +\bibitem[Wityak 87]{Wit87} Wityak, Sandra +Scratchpad II Newsletter'' +Volume 2, Number 1, Nov 1987 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K -On the Estimation of Sparse Jacobian Matrices'' -J. Inst. Maths Applics. 13 117--119. (1974) +\bibitem[WWW1]{WWW1}. +Software Preservation Group +\verb|www.softwarepresentation.org/projects/LISP/common_lisp_family| + keywords = "axiomref", \end{chunk} -\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A -Numerical Methods'' -Prentice- Hall. (1974) +\bibitem[Yap 00]{Yap00} Yap, Chee Keng +Fundamental Problems of Algorithmic Algebra'' +Oxford University Press (2000) ISBN0-19-512516-9 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier -The INRIA OpenMath Library'' -Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998 +\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai +Literate Programming Tools Implemented in ANSI Common Lisp'' +\verb|brlcad.org/~starseeker/cl-web-v0.8.lisp.pamphlet| + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Dantzig 63]{Dan63} Dantzig G B -Linear Programming and Extensions'' -Princeton University Press. (1963) +\bibitem[Yun 83]{Yun83} Yun, David Y.Y. +Computer Algebra and Complex Analysis'' +Computational Aspects of Complex Analysis pp379-393 +D. Reidel Publishing Company H. Werner et. al. (eds.) + keywords = "axiomref", \end{chunk} +\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Davenport]{Dav} Davenport, James -On Brillhart Irreducibility.'' -To appear. +\bibitem[Zen92]{Zen92} Zenger, Ch. +Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre +Implementierung in AXIOM'' +Diplomarbeit, Universit{\"a}t Karlsruhe, +Karlsruhe, Germany, 1992 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Davenport 93]{Ref-Dav93} Davenport, J.H. -Primality testing revisited'' -Technical Report TR2/93 -(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA -and Oxford, UK, August 1993 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +\bibitem[Zip92]{Zip92} Zippel, Richard +Algebraic Computation'' +(unpublished) Cornell University Ithaca, NY Sept 1992 + keywords = "axiomref", \end{chunk} \begin{chunk}{ignore} -\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P -Numerical Integration'' -Blaisdell Publishing Company. 33--52. (1967) +\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel +Handbook of Integration'' +Jones and Bartlett, 1992, ISBN 0-86720-293-9 + keywords = "axiomref", \end{chunk} +\section{Axiom Citations of External Sources} -\begin{chunk}{ignore} -\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P -Methods of Numerical Integration'' -Academic Press. (1975) +\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@article{Abla98, + author = "Ablamowicz, Rafal", + title = "Spinor Representations of Clifford Algebras: A Symbolic Approach", + journal = "Computer Physics Communications", + volume = "115", + number = "2-3", + month = "December", + year = "1998", + pages = "510-535" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[DeBoor 72]{DeB72} De Boor C -On Calculating with B-splines'' -J. Approx. Theory. 6 50--62. (1972) +\begin{chunk}{axiom.bib} +@article{Abra06, + author = "Abramov, Sergey A.", + title = "In Memory of Manuel Bronstein", + journal = "Programming and Computer Software", + volume = "32", + number = "1", + pages = "56-58", + publisher = "Pleiades Publishing Inc", + year = "2006", + paper = "Abra06.pdf" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[De Doncker 78]{DeD78} De Doncker E, -An Adaptive Extrapolation Algorithm for Automatic Integration'' -Signum Newsletter. 13 (2) 12--18. (1978) +\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A. +Handbook of Mathematical Functions'' +(1964) Dover Publications, NY ISBN 0-486-61272-4 \end{chunk} \begin{chunk}{ignore} -\bibitem[Demmel 89]{Dem89} Demmel J W -On Floating-point Errors in Cholesky'' -LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989 +\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A +Handbook of Mathematical Functions'' +Dover Publications. (1968) \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J -Quasi-Newton Methods, Motivation and Theory'' -SIAM Review. 19 46--89. 1977 +\begin{chunk}{axiom.bib} +@book{Altm05, + author = "Altmann, Simon L.", + title = "Rotations, Quaternions, and Double Groups", + publisher = "Dover Publications, Inc.", + year = "2005", + isbn = "0-486-44518-6" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B -A New Derivation of Symmetric Positive-Definite Secant Updates'' -Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson) -Academic Press. 167--199. (1981) +\bibitem[Ames 77]{Ames77} Ames W F +Nonlinear Partial Differential Equations in Engineering'' +Academic Press (2nd Edition). (1977) \end{chunk} \begin{chunk}{ignore} -\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B -Numerical Methods for Unconstrained Optimixation and Nonlinear Equations'' -Prentice-Hall.(1983) +\bibitem[Amos 86]{Amos86} Amos D E +Algorithm 644: A Portable Package for Bessel Functions of a Complex +Argument and Nonnegative Order'' +ACM Trans. Math. Softw. 12 265--273. (1986) \end{chunk} \begin{chunk}{ignore} -\bibitem[Dierckx 75]{Die75} Dierckx P -An Algorithm for Smoothing, Differentiating and Integration of -Experimental Data Using Spline Functions'' -J. Comput. Appl. Math. 1 165--184. (1975) +\bibitem[Anderson 00]{And00} Anderson, Edward +Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem'' +LAPACK Working Note 150, University of Tennessee, UT-CS-00-454, +December 4, 2000. \end{chunk} \begin{chunk}{ignore} -\bibitem[Dierckx 81]{Die81} Dierckx P -An Improved Algorithm for Curve Fitting with Spline Functions'' -Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981 +\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G +DASL - Data Approximation Subroutine Library'' +National Physical Laboratory. (1982) \end{chunk} \begin{chunk}{ignore} -\bibitem[Dierckx 82]{Die82} Dierckx P -A Fast Algorithm for Smoothing Data on a Rectangular Grid while using -Spline Functions'' -SIAM J. Numer. Anal. 19 1286--1304. (1982) +\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M. +On Mechanical Quantifier Elimination for Elementary Algebra and Geometry'' +J. Symbolic Computation 5, 237-259 (1988) +\verb|http://www.sciencedirect.com/science/article/pii/S0747717188800142/| +\verb|pdf?md5=62052077d84e6078cc024bc8e29c23c1&| +\verb|pid=1-s2.0-S0747717188800142-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Arno88.pdf| + abstract = " + We give solutions to two problems of elementary algebra and geometry: + (1) find conditions on real numbers $p$, $q$, and $r$ so that the + polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real + $x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that + the ellipse $\frac{(x-e)^2}{q^2}+\frac{y^2}{b^2}-1=0$ lies inside the + unit circle $y^2+x^2-1=0$. Our solutions are obtained by following the + basic outline of the method of quantifier elimination by cylindrical + algebraic decomposition (Collins, 1975), but we have developed, and + have been considerably aided by, modified versions of certain of its + steps. We have found three equally simple but not obviously equivalent + solutions for the first problem, illustrating the difficulty of + obtaining unique simplest'' solutions to quantifier elimination + problems of elementary algebra and geometry." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R; -Stewart G W -LINPACK Users' Guide'' -SIAM, Philadelphia. (1979) +\begin{chunk}{axiom.bib} +@article{Aubr99, + author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc", + title = "On the Theories of Triangular Sets", + year = "1999", + pages = "105-124", + journal = "Journal of Symbolic Computation", + volume = "28", + url = "http://www.csd.uwo.ca/~moreno/Publications/Aubry-Lazard-MorenoMaza-1999-JSC.pdf", + papers = "Aubr99.pdf", + abstract = " + Different notions of triangular sets are presented. The relationship + between these notions are studied. The main result is that four + different existing notions of {\sl good} triangular sets are + equivalent." +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S; -Hanson R J -A Proposal for an Extended set of Fortran Basic Linear -Algebra Subprograms'' -SIGNUM Newsletter. 20 (1) 2--18. (1985) +\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno +Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods'' +\verb|www.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz| +%\verb|axiom-developer.org/axiom-website/papers/Aub96.ps| + abstract = " + Four methods for solving polynomial systems by means of triangular + sets are presented and implemented in a unified way. These methods are + those of Wu, Lazard, Kalkbrener, and Wang. They are compared on + various examples with emphasis on efficiency, conciseness and + legibility of the outputs." \end{chunk} +\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Dongarra 88]{REF-DON88} Dongarra, Jack J.; Du Croz, Jeremy; -Hammarling, Sven; Hanson, Richard J. -An Extended Set of FORTRAN Basic Linear Algebra Subroutines'' -ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988, -pp 1-17 +\bibitem[Bailey 66]{Bai66} Bailey P B +Sturm-Liouville Eigenvalues via a Phase Function'' +SIAM J. Appl. Math . 14 242--249. (1966) \end{chunk} \begin{chunk}{ignore} -\bibitem[Dongarra 88a]{REF-DON88a} Dongarra, Jack J.; Du Croz, Jeremy; -Hammarling, Sven; Hanson, Richard J. -ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms: -Model Implementation and Test Programs'' -ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988, -pp 18-32 +\bibitem[Baker 96]{BGM96} Baker, George A.; Graves-Morris, Peter +Pade Approximants'' +Cambridge University Press, March 1996 ISBN 9870521450072 \end{chunk} \begin{chunk}{ignore} -\bibitem[Dongarra 90]{REF-DON90} Dongarra, Jack J.; Du Croz, Jeremy; -Hammarling, Sven; Duff, Iain S. -A Set of Level 3 Basic Linear Algebra Subprograms'' -ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990, -pp 1-17 +\bibitem[Baker 10]{Ba10} Baker, Martin +3D World Simulation'' +\verb|www.euclideanspace.com| \end{chunk} -\begin{chunk}{ignore} -\bibitem[Dongarra 90a]{REF-DON90a} Dongarra, Jack J.; Du Croz, Jeremy; -Hammarling, Sven; Duff, Iain S. -ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms: -Model Implementation and Test Programs'' -ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990, -pp 18-28 +\begin{chunk}{axiom.bib} +@misc{Bake14, + author = "Baker, Martin", + title = "Axiom Architecture", + year = "2014", + url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Ducos 00]{Duc00} Ducos, Lionel -Optimizations of the subresultant algorithm'' -Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149-163 +\bibitem[Banks 68]{BK68} Banks D O; Kurowski I +Computation of Eigenvalues of Singular Sturm-Liouville Systems'' +Math. Computing. 22 304--310. (1968) \end{chunk} \begin{chunk}{ignore} -\bibitem[Duff 77]{Duff77} Duff I S, -MA28 -- a set of Fortran subroutines for sparse unsymmetric linear -equations'' -A.E.R.E. Report R.8730. HMSO. (1977) +\bibitem[Bard 74]{Bard74} Bard Y +Nonlinear Parameter Estimation'' +Academic Press. 1974 \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alez-Vega, L. -Dynamic Evaluation and Real Closure'' -Mathematics and Computers in Simulation 42 pp 551-560 (1996) -%\verb|axiom-developer.org/axiom-website/papers/Duva96a.pdf| - abstract = " - The aim of this paper is to present how the dynamic evaluation method - can be used to deal with the real closure of an ordered field. Two - kinds of questions, or tests, may be asked in an ordered field: - equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are - handled through splittings, exactly as in the algebraic closure of a - field. Sign tests are handled throug a structure called Tarski data - type''." +\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K +An Improved Algorithm for Discrete $ll_1$ Linear Approximation'' +SIAM J. Numer. Anal. 10 839--848. (1973) \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C. -Sketches and Computations over Fields'' -Mathematics and Computers in Simulation 42 pp 363-373 (1996) -%\verb|axiom-developer.org/axiom-website/papers/Duva96.pdf| - abstract = " - The goal of this short paper is to describe one possible use of - sketches in computer algebra. We show that sketches are a powerful - tool for the description of mathematical structures and for the - description of computations." +\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K +Solution of an Overdetermined System of Equations in the $ll_1-norm$.'' +Comm. ACM. 17, 6 319--320. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C. -Sketches and Computation (Part I): Basic Definitions and Static Evaluation'' -Mathematical Structures in Computer Science, 4, p 185-238 Cambridge University Press (1994) -\verb|journals.cambridge.org/abstract_S0960129500000438| -%\verb|axiom-developer.org/axiom-website/papers/Duva94a.pdf| - abstract = " - We define a categorical framework, based on the notion of {\sl - sketch}, for specification and evaluation in the sense of algebraic - specifications and algebraic programming. This framework goes far - beyond our initial motivations, which was to specify computation with - algebraic numbers. We begin by redefining sketches in order to deal - explicitly with programs. Expressions and terms are carefully defined - and studied, then {\sl quasi-projective sketches} are introduced. We - describe {\sl static evaluation} in these sketches: we propose a - rigorous basis for evaluation in the corresponding structures. These - structures admit an initial model, but are not necessarily - equational. In Part II (Duval and Reynaud 1994), we study a more - general process, called {\sl dynamic evaluation}, for structures that - may have no initial model." +\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard +Products of polynomials and a priori estimates for +coefficients in polynomial decompositions: a sharp result'' +J. Symbolic Computation (1992) 13, 463-472 +%\verb|axiom-developer.org/axiom-website/papers/Bea92.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C. -Sketches and Computation (Part II): Dynamic Evaluation and Applications'' -Mathematical Structures in Computer Science, 4, p 239-271. Cambridge University Press (1994) -\verb|journals.cambridge.org/abstract_S096012950000044X| -%\verb|axiom-developer.org/axiom-website/papers/Duva94b.pdf| - abstract = " - In the first part of this paper (Duval and Reynaud 1994), we defined a - categorical framework, based on the notion of {\sl sketch}, for - specification and evaluation in the senses of algebraic specification - and algebraic programming. {\sl Static evaluation} in {\sl - quasi-projective sketches} was defined in Part I; in this paper, {\sl - dynamic evaluation} is introduced. It deals with more general - structures, which may have no initial model. Until now, this process - has not been used in algebraic specification systems, but computer - algebra systems are beginning to use it as a basic tool. Finally, we - give some applications of dynamic evaluation to computation in field - extensions." +\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar; +Wang, Paul S. +Polynomial Factorization: Sharp Bounds, Efficient Algorithms'' +J. Symbolic Computation (1993) 15, 393-413 +%\verb|axiom-developer.org/axiom-website/papers/Bea93.pdf| \end{chunk} -\begin{chunk}{ignore} -\bibitem[Duval 94c]{Duva94c} Duval, Dominique -Algebraic Numbers: An Example of Dynamic Evaluation'' -J. Symbolic Computation 18, 429-445 (1994) -\verb|www.sciencedirect.com/science/article/pii/S0747717106000551| -%\verb|axiom-developer.org/axiom-website/papers/Duva94c.pdf| +\begin{chunk}{axiom.bib} +@article{Bert95, + author = "Bertrand, Laurent", + title = "Computing a hyperelliptic integral using arithmetic in the + jacobian of the curve", + journal = "Applicable Algebra in Engineering, Communication and Computing", + volume = "6", + pages = "275-298", + year = "1995", abstract = " - Dynamic evaluation is presented through examples: computations - involving algebraic numbers, automatic case discussion according to - the characteristic of a field. Implementation questions are addressed - too. Finally, branches are presented as dual'' to binary functions, - according to the approach of sketch theory." + In this paper, we describe an efficient algorithm for computing an + elementary antiderivative of an algebraic function defined on a + hyperelliptic curve. Our algorithm combines B.M. Trager's integration + algorithm and a technique for computing in the Jacobian of a + hyperelliptic curve introduced by D.G. Cantor. Our method has been + implemented and successfully compared to Trager's general algorithm." +} \end{chunk} -\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Fateman 08]{Fat08} Fateman, Richard -Revisiting numeric/symbolic indefinite integration of rational functions, and extensions'' -\verb|www.eecs.berkeley.edu/~fateman/papers/integ.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Fat08.pdf| - abstract = " - We know we can solve this problem: Given any rational function - $f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over - the rationals, compute its {\sl indefinite} integral, using if - necessary, algebraic numbers. But in many circumstances an approximate - result is more likely to be of use. Furthermore, it is plausible that - it would be more useful to solve the problem to allow definite - integration, or introduce additional parameters so that we can solve - multiple definite integrations. How can a computer algebra system - best answer the more useful questions? Finally, what if the integrand - is not a ratio of polynomials, but something more challenging?" +\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I. +Design of the Stiff Integrators in the NAG Library'' +Technical Report. TR14/87 NAG. (1987) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Flet01, - author = "Fletcher, John P.", - title = "Symbolic processing of Clifford Numbers in C++", - year = "2001", - journal = "Paper 25, AGACSE 2001." -} +\begin{chunk}{ignore} +\bibitem[Berzins 90]{Ber90} Berzins M +Developments in the NAG Library Software for Parabolic Equations'' +Scientific Software Systems. (ed J C Mason and M G Cox) +Chapman and Hall. 59--72. (1990) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Flet09, - author = "Fletcher, John P.", - title = "Clifford Numbers and their inverses calculated using the matrix - representation", - publisher = "Chemical Engineering and Applied Chemistry, School of - Engineering and Applied Science, Aston University, Aston Triangle, - Birmingham B4 7 ET, U. K.", - url = - "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php" -} +\begin{chunk}{ignore} +\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C +Ordinary Differential Equations'' +Ginn \& Co., Boston and New York. (1962) \end{chunk} \begin{chunk}{ignore} -\bibitem[Fletcher 81]{Fle81} Fletcher R -Practical Methods of Optimization'' -Vol 2. Constrained Optimization. Wiley. (1981) +\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W. +Bounds for the Height of a Factor of a Polynomial in +Terms of Bombieri's Norms: I. The Largest Factor'' +J. Symbolic Computation (1993) 16, 115-130 +%\verb|axiom-developer.org/axiom-website/Boyd93a.pdf| \end{chunk} -\begin{chunk}{axiom.bib} -@article{Floy63, - author = "Floyd, R. W.", - title = "Semantic Analysis and Operator Precedence", - journal = "JACM", - volume = "10", - number = "3", - pages = "316-333", - year = "1963" -} +\begin{chunk}{ignore} +\bibitem[Boyd 93b]{Boyd93b} Boyd, David W. +Bounds for the Height of a Factor of a Polynomial in +Terms of Bombieri's Norms: II. The Smallest Factor'' +J. Symbolic Computation (1993) 16, 131-145 +%\verb|axiom-developer.org/axiom-website/Boyd93b.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Forsythe 57]{For57} Forsythe G E, -Generation and use of orthogonal polynomials for data fitting -with a digital computer'' -J. Soc. Indust. Appl. Math. 5 74--88. (1957) +\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R. +The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, +and Level 3 Performance'' +SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. \end{chunk} \begin{chunk}{ignore} -\bibitem[Fortenbacher 90]{REF-For90} Fortenbacher, A. -Efficient type inference and coercion in computer algebra'' -Design and Implementation of Symbolic Computation Systems (DISCO 90) -A. Miola, (ed) vol 429 of Lecture Notes in Computer Science -Springer-Verlag, pp56-60 - abstract = " - Computer algebra systems of the new generation, like Scratchpad, are - characterized by a very rich type concept, which models the - relationship between mathematical domains of computation. To use these - systems interactively, however, the user should be freed of type - information. A type inference mechanism determines the appropriate - function to call. All known models which allow to define a semantics - for type inference cannot express the rich mathematical'' type - structure, so presently type inference is done heuristically. The - following paper defines a semantics for a subproblem thereof, namely - coercion, which is based on rewrite rules. From this definition, and - efficient coercion algorith for Scratchpad is constructed using graph - techniques." +\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R. +The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation'' +SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002. \end{chunk} \begin{chunk}{ignore} -\bibitem[Fox 68]{Fox68} Fox L.; Parker I B. -Chebyshev Polynomials in Numerical Analysis'' -Oxford University Press. (1968) +\bibitem[Brent 75]{Bre75} Brent, R. P. +Multiple-Precision Zero-Finding Methods and the Complexity +of Elementary Function Evaluation, Analytic Computational Complexity'' +J. F. Traub, Ed., Academic Press, New York 1975, 151-176 \end{chunk} \begin{chunk}{ignore} -\bibitem[Franke 80]{FN80} Franke R.; Nielson G -Smooth Interpolation of Large Sets of Scattered Data'' -Internat. J. Num. Methods Engrg. 15 1691--1704. (1980) +\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T. +Fast Algorithms for Manipulating Formal Power Series'' +Journal of the Association for Computing Machinery, +Vol. 25, No. 4, October 1978, 581-595 \end{chunk} \begin{chunk}{ignore} -\bibitem[Fritsch 82]{Fri82} Fritsch F N -PCHIP Final Specifications'' -Report UCID-30194. Lawrence Livermore National Laboratory. (1982) +\bibitem[Brigham 73]{Bri73} Brigham E O +The Fast Fourier Transform'' +Prentice-Hall. (1973) \end{chunk} \begin{chunk}{ignore} -\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J. -A Method for Constructing Local Monotone Piecewise Cubic Interpolants'' -SIAM J. Sci. Statist. Comput. 5 300--304. (1984) +\bibitem[Brillhart 69]{Bri69} Brillhart, John +On the Euler and Bernoulli polynomials'' +J. Reine Angew. Math., v. 234, (1969), pp. 45-64 \end{chunk} \begin{chunk}{ignore} -\bibitem[Froberg 65]{Fro65} Froberg C E. -Introduction to Numerical Analysis'' -Addison-Wesley. 181--187. (1965) +\bibitem[Brillhart 90]{Bri90} Brillhart, John +Note on Irreducibility Testing'' +Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381 \end{chunk} -\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H. -A tower of Artin-Schreier extensions of function fields attaining the -Drinfeld-Vladut bound'' -Invent. Math., vol. 121, 1995, pp. 211--222. +\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds) +Symbolic Rewriting Techniques'' +Progress in Computer Science and Applied Logic 15, Birkhauser-Verlag, Basel +ISBN 3-7643-5901-3 (1998) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark -Constructing Normal Bases in Finite Fields'' -J. Symbolic Computation pp 547-570 (1990) -%\verb|axiom-developer.org/axiom-website/papers/Gat90a.pdf| +\bibitem[Bronstein 88]{Bro88} Bronstein, Manual +The Transcendental Risch Differential Equation'' +J. Symbolic Computation (1990) 9, pp49-60 Feb 1988 +IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY +\verb|www.sciencedirect.com/science/article/pii/S0747717108800065| +%\verb|axiom-developer.org/axiom-website/papers/Bro88.pdf| abstract = " - An efficient probabilistic algorithm to find a normal basis in a - finite field is presented. It can, in fact, find an element of - arbitrary prescribed additive order. It is based on a density estimate - for normal elements. A similar estimate yields a probabilistic - polynomial-time reduction from finding primitive normal elements to - finding primitive elements." + We present a new rational algorithm for solving Risch differential + equations in towers of transcendental elementary extensions. In + contrast to a recent algorithm by Davenport we do not require a + progressive reduction of the denominators involved, but use weak + normality to obtain a formula for the denominator of a possible + solution. Implementation timings show this approach to be faster than + a Hermite-like reduction." \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur -Functional Decomposition Polynomials: the Tame Case'' -Journal of Symbolic Computation (1990) 9, 281-299 +\begin{chunk}{axiom.bib} +@techreport{Bron98, + author = "Bronstein, Manuel", + title = "The lazy hermite reduction", + type = "Rapport de Recherche", + number = "RR-3562", + year = "1998", + institution = "French Institute for Research in Computer Science", + paper = "Bron98.pdf", + abstract = " + The Hermite reduction is a symbolic integration technique that reduces + algebraic functions to integrands having only simple affine + poles. While it is very effective in the case of simple radical + extensions, its use in more general algebraic extensions requires the + precomputation of an integral basis, which makes the reduction + impractical for either multiple algebraic extensions or complicated + ground fields. In this paper, we show that the Hermite reduction can + be performed without {\sl a priori} computation of either a primitive + element or integral basis, computing the smallest order necessary for + a particular integrand along the way." +} \end{chunk} \begin{chunk}{axiom.bib} -@book{Gath99, - author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen}, - title = "Modern Computer Algebra", - publisher = "Cambridge University Press", - year = "1999", - isbn = "0-521-64176-4" +@misc{Bro98b, + author = "Bronstein, Manuel", + title = "Symbolic Integration Tutorial", + series = "ISSAC'98", + year = "1998", + address = "INRIA Sophia Antipolis", + url = + "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf", + paper = "Bro98b.pdf" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gautschi 79a]{Gau79a} Gautschi W. -A Computational Procedure for Incomplete Gamma Functions'' -ACM Trans. Math. Softw. 5 466--481. (1979) +\begin{chunk}{ignore} +\bibitem[Brown 99]{Brow99} Brown, Christopher W. +Solution Formula Construction for Truth Invariant CADs'' +Ph.D Thesis, Univ. Delaware (1999) +\verb|www.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz| +%\verb|axiom-developer.org/axiom-website/papers/Brow99.pdf| + abstract = " + The CAD-based quantifier elimination algorithm takes a formula from + the elementary theory of real closed fields as input, and constructs a + CAD of the space of the formula's unquantified variables. This + decomposition is truth invariant with respect to the input formula, + meaning that the formula is either identically true or identically + false in each cell of the decomposition. The method determines the + truth of the input formula for each cell of the CAD, and then uses the + CAD to construct a solution formula -- a quantifier free formula that + is equivalent to the input formula. This final phase of the algorithm, + the solution formula construction phase, is the focus of this thesis. + + An optimal solution formula construction algorithm would be {\sl + complete} -- i.e. applicable to any truth-invariant CAD, would be {\sl + efficient}, and would produce {\sl simple} solution formulas. Prior to + this thesis, no method was available with even two of these three + properties. Several algorithms are presented, all addressing problems + related to solution formula construction. In combination, these + provide an efficient and complete method for constructing solution + formulas that are simple in a variety of ways. + + Algorithms presented in this thesis have been implemented using the + SACLIB library, and integrated into QEPCAD, a SACLIB-based + implementation of quantifier elimination by CAD. Example computations + based on these implementations are discussed." + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Brown 02]{Brow02} Brown, Christopher W. +QEPCAD B -- A program for computing with semi-algebraic sets using CADs'' +%\verb|axiom-developer.org/axiom-website/papers/Brow02.pdf| + abstract = " + This report introduces QEPCAD B, a program for computing with real + algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD + B both extends and improves upon the QEPCAD system for quantifier + elimination by partial cylindrical algebraic decomposition written by + Hoon Hong in the early 1990s. This paper briefly discusses some of the + improvements in the implementation of CAD and quantifier elimination + vis CAD, and provides somewhat more detail on extensions to the system + that go beyond quantifier elimination. The author is responsible for + most of the extended features of QEPCAD B, but improvements to the + basic CAD implementation and to the SACLIB library on which QEPCAD is + based are the results of many people's work." + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Burg74, + author = "William H. Burge", + title = "Stream Processing Functions", + year = "1974", + month = "January", + journal = "IBM Journal of Research and Development", + volume = "19", + issue = "1", + pages = "12-25", + papers = "Burg74.pdf", + abstract = " + One principle of structured programming is that a program should be + separated into meaningful independent subprograms, which are then + combined so that the relation of the parts to the whole can be clearly + established. This paper describes several alternative ways to compose + programs. The main method used is to permit the programmer to denote + by an expression the sequence of values taken on by a variable. The + sequence is represented by a function called a stream, which is a + functional analog of a coroutine. The conventional while and for loops + of structured programming may be composed by a technique of stream + processing (analogous to list processing), which results in more + structured programs than the orignals. This technique makes it + possible to structure a program in a natural way into its logically + separate parts, which can then be considered independently." +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Gautschi 79b]{Gau79b} Gautschi W. -Algorithm 542: Incomplete Gamma Functions'' -ACM Trans. Math. Softw. 5 482--489. (1979) - -\end{chunk} +\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Gentlemen 69]{Gen69} Gentlemen W M -An Error Analysis of Goertzel's (Watt's) Method for Computing -Fourier Coefficients'' -Comput. J. 12 160--165. (1969) +\bibitem[Carlson 65]{Car65} Carlson B C +On Computing Elliptic Integrals and Functions'' +J Math Phys. 44 36--51. (1965) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gentleman 73]{Gen73} Gentleman W M. -Least-squares Computations by Givens Transformations without Square Roots'' -J. Inst. Math. Applic. 12 329--336. (1973) +\bibitem[Carlson 77a]{Car77a} Carlson B C +Elliptic Integrals of the First Kind'' +SIAM J Math Anal. 8 231--242. (1977) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gentleman 74]{Gen74} Gentleman W M. -Algorithm AS 75. Basic Procedures for Large Sparse or -Weighted Linear Least-squares Problems'' -Appl. Statist. 23 448--454. (1974) +\bibitem[Carlson 77b]{Car77b} Carlson B C +Special Functions of Applied Mathematics'' +Academic Press. (1977) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B. -More on algorithms that reveal properties of floating point -arithmetic units'' -Comms. of the ACM, 17, 276-277. (1974) +\bibitem[Carlson 78]{Car78} Carlson B C, +Computing Elliptic Integrals by Duplication'' +(Preprint) Department of Physics, Iowa State University. (1978) \end{chunk} \begin{chunk}{ignore} -\bibitem[Genz 80]{GM80} Genz A C.; Malik A A. -An Adaptive Algorithm for Numerical Integration over an N-dimensional -Rectangular Region'' -J. Comput. Appl. Math. 6 295--302. (1980) +\bibitem[Carlson 88]{Car88} Carlson B C, +A Table of Elliptic Integrals of the Third Kind'' +Math. Comput. 51 267--280. (1988) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 72]{GM72} Gill P E.; Miller G F. -An Algorithm for the Integration of Unequally Spaced Data'' -Comput. J. 15 80--83. (1972) +\bibitem[Cauchy 1829]{Cau1829} Augustin-Lux Cauchy +Exercices de Math\'ematiques Quatri\eme Ann\'ee. De Bure Fr\eres'' +Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX, +Gauthier-Villars, Paris, 1891). \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds) -Numerical Methods for Constrained Optimization'' -Academic Press. (1974) +\bibitem[Ch\eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire +Lifting and recombination techniques for absolute factorization'' +Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380-420 +\verb|www.sciencedirect.com/science/article/pii/S0885064X07000465| +%\verb|axiom-developer.org/axiom-website/papers/Chez07.pdf| + abstract = " + In the vein of recent algorithmic advances in polynomial factorization + based on lifting and recombination techniques, we present new faster + algorithms for computing the absolute factorization of a bivariate + polynomial. The running time of our probabilistic algorithm is less + than quadratic in the dense size of the polynomial to be factored." \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W. -Minimization subject to bounds on the variables'' -Report NAC 72. National Physical Laboratory. (1976) +\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E; +Nelson P (eds) +Codes for Boundary-value Problems in Ordinary Differential Equations'' +Lecture Notes in Computer Science. 76 (1979) Springer-Verlag \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W. -Algorithms for the Solution of the Nonlinear Least-squares Problem'' -NAC 71 National Physical Laboratory. (1976) +\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A. +Efficient Solution of Linear Diophantine Equations'' +JSC (1989) 8, 201-216 \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 78]{GM78} Gill P E.; Murray W. -Algorithms for the Solution of the Nonlinear Least-squares Problem'' -SIAM J. Numer. Anal. 15 977--992. (1978) +\bibitem[Clenshaw 55]{Cle55} Clenshaw C W, +A Note on the Summation of Chebyshev Series'' +Math. Tables Aids Comput. 9 118--120. (1955) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 79]{GM79} Gill P E.; Murray W; -Conjugate-gradient Methods for Large-scale Nonlinear Optimization'' -Technical Report SOL 79-15. Department of Operations Research, -Stanford University. (1979) +\bibitem[Clenshaw 60]{Cle60} Clenshaw C W +Curve Fitting with a Digital Computer'' +Comput. J. 2 170--173. (1960) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H. -Practical Optimization'' -Academic Press. 1981 +\bibitem[Clenshaw 62]{Cle62} Clenshaw C W +Mathematical Tables. Chebyshev Series for Mathematical Functions'' +HMSO. (1962) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H. -The design and implementation of a quadratic programming algorithm'' -Report SOL 82-7. Department of Operations Research, -Stanford University. (1982) +\bibitem[Cline 84]{CR84} Cline A K; Renka R L, +A Storage-efficient Method for Construction of a Thiessen Triangulation'' +Rocky Mountain J. Math. 14 119--139. (1984) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H -User's Guide for SOL/QPSOL Version 3.2'' -Report SOL 84-5. Department of Operations Research, Stanford University. 1984 +\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.; +Wilson, R. +Atlas of Finite Groups'' +Oxford, Clarendon Press, 1987 \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H -Procedures for Optimization Problems with a Mixture of -Bounds and General Linear Constraints'' -ACM Trans. Math. Softw. 10 282--298. 1984 +\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A. +On Quaternions and Octonions'' +A.K Peters, Natick, MA. (2003) ISBN 1-56881-134-9 \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.; -Saunders M A.; Wright M H. -User's Guide for LSSOL (Version 1.0)'' -Report SOL 86-1. Department of Operations Research, Stanford University. 1986 +\bibitem[Cox 72]{Cox72} Cox M G +The Numerical Evaluation of B-splines'' +J. Inst. Math. Appl. 10 134--149. (1972) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H. -Some Theoretical Properties of an Augmented Lagrangian Merit Function'' -Report SOL 86-6R. Department of Operations Research, Stanford University. 1986 +\bibitem[CH 73]{CH73} Cox M G; Hayes J G +Curve fitting: a guide and suite of algorithms for the +non-specialist user'' +Report NAC26. National Physical Laboratory. (1973) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gladwell 79]{Gla79} Gladwell I -Initial Value Routines in the NAG Library'' -ACM Trans Math Softw. 5 386--400. (1979) +\bibitem[Cox 74a]{Cox74a} Cox M G +A Data-fitting Package for the Non-specialist User'' +Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K -Computational Techniques for Ordinary Differential Equations'' -Academic Press. 1980 +\bibitem[Cox 74b]{Cox74b} Cox M G +Numerical methods for the interpolation and approximation of data +by spline functions'' +PhD Thesis. City University, London. (1975) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gladwell 86]{Gla86} Gladwell I -Vectorisation of one dimensional quadrature codes'' -Techincal Report. TR7/86 NAG. (1986) +\bibitem[Cox 75]{Cox75} Cox M G +An Algorithm for Spline Interpolation'' +J. Inst. Math. Appl. 15 95--108. (1975) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gladwell 87]{Gla87} Gladwell I -The NAG Library Boundary Value Codes'' -Numerical Analysis Report. 134 Manchester University. (1987) +\bibitem[Cox 77]{Cox77} Cox M G +A Survey of Numerical Methods for Data and Function Approximation'' +The State of the Art in Numerical Analysis. (ed D A H Jacobs) +Academic Press. 627--668. (1977) + keywords = "survey", \end{chunk} \begin{chunk}{ignore} -\bibitem[Goedel 40]{God40} Goedel -The consistency of the continuum hypothesis'' -Ann. Math. Studies, Princeton Univ. Press, 1940 +\bibitem[Cox 78]{Cox78} Cox M G +The Numerical Evaluation of a Spline from its B-spline Representation'' +J. Inst. Math. Appl. 21 135--143. (1978) \end{chunk} \begin{chunk}{ignore} -\bibitem[Goldman 87]{Gold87} Goldman, L. -Integrals of multinomial systems of ordinary differential equations'' -J. of Pure and Applied Algebra, 45, 225-240 (1987) -\verb|www.sciencedirect.com/science/article/pii/0022404987900727/pdf| -\verb|?md5=5a0c70643eab514ccf47d80e4fc6ec5a&| -\verb|pid=1-s2.0-0022404987900727-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Gold87.pdf| +\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K +On the Estimation of Sparse Jacobian Matrices'' +J. Inst. Maths Applics. 13 117--119. (1974) \end{chunk} +\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier -Algorithms in Representation Theory and -their Realization in the Computer Algebra System Scratchpad'' -Bayreuther Mathematische Schriften, Heft 33, 1990, 1-23 +\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A +Numerical Methods'' +Prentice- Hall. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F. -Matrix Computations'' -Johns Hopkins University Press ISBN 0-8018-3772-3 (1989) +\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier +The INRIA OpenMath Library'' +Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998 \end{chunk} \begin{chunk}{ignore} -\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F. -Matrix Computations'' -Johns Hopkins University Press ISBN 978-0-8018-5414-9 (1996) +\bibitem[Dantzig 63]{Dan63} Dantzig G B +Linear Programming and Extensions'' +Princeton University Press. (1963) \end{chunk} \begin{chunk}{ignore} -\bibitem[Grabmeier]{Grab} Grabmeier, J. -On Plesken's root finding algorithm'' -in preparation +\bibitem[Davenport]{Dav} Davenport, James +On Brillhart Irreducibility.'' +To appear. \end{chunk} \begin{chunk}{ignore} -\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.; -The Evaluation of Irreducible Polynomial Representations of the General -Linear Groups and of the Unitary Groups over Fields of Characteristic 0'' -Acta Appl. Math. 8 (1987), 271-291 +\bibitem[Davenport 93]{Ref-Dav93} Davenport, J.H. +Primality testing revisited'' +Technical Report TR2/93 +(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA +and Oxford, UK, August 1993 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| \end{chunk} \begin{chunk}{ignore} -\bibitem[Grabmeier 92]{REF-GS92} Grabmeier, J.; Scheerhorn, A. -Finite fields in Axiom'' -AXIOM Technical Report TR7/92 (ATR/5)(NP2522), -Numerical Algorithms Group, Inc., Downer's -Grove, IL, USA and Oxford, UK, 1992 -\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| +\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P +Numerical Integration'' +Blaisdell Publishing Company. 33--52. (1967) \end{chunk} \begin{chunk}{ignore} -\bibitem[Granville 1911]{Gran1911} Granville, William Anthony -Elements of the Differential and Integral Calculus'' -\verb|djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Gran1911.pdf| +\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P +Methods of Numerical Integration'' +Academic Press. (1975) \end{chunk} \begin{chunk}{ignore} -\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik -Limit computation in computer algebra'' -\verb|algo.inria.fr/seminars/sem92-93/gruntz.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Gru93.pdf| - abstract = " - The automatic computation of limits can be reduced to two main - sub-problems. The first one is asymptotic comparison where one must - decide automatically which one of two functions in a specified class - dominates the other one asymptotically. The second one is asymptotic - cancellation and is usually exemplified by - $e^x[exp(1/x+e^{-x})-exp(1/x)],\quad{}x \leftarrow \infty$ +\bibitem[DeBoor 72]{DeB72} De Boor C +On Calculating with B-splines'' +J. Approx. Theory. 6 50--62. (1972) - In this example, if the sum is expanded in powers of $1/x$, the - expansion always yields $O(x^{-k})$, and this is not enough to - conclude. +\end{chunk} - In 1990, J.Shackell found an algorithm that solved both these problems - for the case of $exp-log$ functions, i.e. functions build by recursive - application of exponential, logarithm, algebraic extension and field - operations to one variable and the rational numbers. D. Gruntz and - G. Gonnet propose a slightly different algorithm for exp-log - functions. Extensions to larger classes of functions are also - discussed." +\begin{chunk}{ignore} +\bibitem[De Doncker 78]{DeD78} De Doncker E, +An Adaptive Extrapolation Algorithm for Automatic Integration'' +Signum Newsletter. 13 (2) 12--18. (1978) \end{chunk} -\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{axiom.bib} -@article{Hach95, - author = "Hach\'e, G. and Le Brigand, D.", - title = "Effective construction of algebraic geometry codes", - journal = "IEEE Transaction on Information Theory", - volume = "41", - month = "November", - year = "1995", - pages = "1615--1628" -} +\begin{chunk}{ignore} +\bibitem[Demmel 89]{Dem89} Demmel J W +On Floating-point Errors in Cholesky'' +LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989 \end{chunk} -\begin{chunk}{axiom.bib} -@article{Hach95a, - author = "Hach\'e, G.", - title = "Computation in algebraic function fields for effective - construction of algebraic-geometric codes", - journal = "Lecture Notes in Computer Science", - volume = "948", - year = "1995", - pages = "262--278" -} +\begin{chunk}{ignore} +\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J +Quasi-Newton Methods, Motivation and Theory'' +SIAM Review. 19 46--89. 1977 \end{chunk} -\begin{chunk}{axiom.bib} -@phdthesis{Hach96, - author = "Hach\'e, G.", - title = "Construction effective des codes g\'eom\'etriques", - school = "l'Universit\'e Pierre et Marie Curie (Paris 6)", - year = "1996", - month = "Septembre" -} +\begin{chunk}{ignore} +\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B +A New Derivation of Symmetric Positive-Definite Secant Updates'' +Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson) +Academic Press. 167--199. (1981) \end{chunk} \begin{chunk}{ignore} -\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds), -Modern Numerical Methods for Ordinary Differential Equations'' -Clarendon Press. (1976) +\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B +Numerical Methods for Unconstrained Optimixation and Nonlinear Equations'' +Prentice-Hall.(1983) \end{chunk} \begin{chunk}{ignore} -\bibitem[Hamdy 04]{Ham04} Hamdy, S. -LiDIA A library for computational number theory'' -Reference manual Edition 2.1.1 May 2004 -\verb|www.cdc.informatik.tu-darmstadt.de/TI/LiDIA| +\bibitem[Dierckx 75]{Die75} Dierckx P +An Algorithm for Smoothing, Differentiating and Integration of +Experimental Data Using Spline Functions'' +J. Comput. Appl. Math. 1 165--184. (1975) \end{chunk} \begin{chunk}{ignore} -\bibitem[Hammarling 85]{Ham85} Hammarling S. - The Singular Value Decomposition in Multivariate Statistics'' -ACM Signum Newsletter. 20, 3 2--25. (1985) +\bibitem[Dierckx 81]{Die81} Dierckx P +An Improved Algorithm for Curve Fitting with Spline Functions'' +Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981 \end{chunk} \begin{chunk}{ignore} -\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C. -Monte-Carlo Methods'' -Methuen. (1967) +\bibitem[Dierckx 82]{Die82} Dierckx P +A Fast Algorithm for Smoothing Data on a Rectangular Grid while using +Spline Functions'' +SIAM J. Numer. Anal. 19 1286--1304. (1982) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Hath1896, - author = "Hathway, Arthur S.", - title = "A Primer Of Quaternions", - year = "1896" -} +\begin{chunk}{ignore} +\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R; +Stewart G W +LINPACK Users' Guide'' +SIAM, Philadelphia. (1979) \end{chunk} -\begin{chunk}{axiom.bib} -@book{Haya05, - author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and - Schreitmueller, S. and Tarquinio, T. and Thompson, J.", - title = "AIX 5L Practical Performance Tools and Tuning Guide", - publisher = "IBM", - year = "2005", - url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf", - paper = "Haya05.pdf" -} +\begin{chunk}{ignore} +\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S; +Hanson R J +A Proposal for an Extended set of Fortran Basic Linear +Algebra Subprograms'' +SIGNUM Newsletter. 20 (1) 2--18. (1985) \end{chunk} + \begin{chunk}{ignore} -\bibitem[Hayes 70]{Hay70} Hayes J G. -Curve Fitting by Polynomials in One Variable'' -Numerical Approximation to Functions and Data. -(ed J G Hayes) Athlone Press, London. (1970) +\bibitem[Dongarra 88]{REF-DON88} Dongarra, Jack J.; Du Croz, Jeremy; +Hammarling, Sven; Hanson, Richard J. +An Extended Set of FORTRAN Basic Linear Algebra Subroutines'' +ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988, +pp 1-17 \end{chunk} \begin{chunk}{ignore} -\bibitem[Hayes 74]{Hay74} Hayes J G. -Numerical Methods for Curve and Surface Fitting'' -Bull Inst Math Appl. 10 144--152. (1974) +\bibitem[Dongarra 88a]{REF-DON88a} Dongarra, Jack J.; Du Croz, Jeremy; +Hammarling, Sven; Hanson, Richard J. +ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms: +Model Implementation and Test Programs'' +ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988, +pp 18-32 \end{chunk} \begin{chunk}{ignore} -\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J, -The Least-squares Fitting of Cubic Spline Surfaces to General Data Sets'' -J. Inst. Math. Appl. 14 89--103. (1974) +\bibitem[Dongarra 90]{REF-DON90} Dongarra, Jack J.; Du Croz, Jeremy; +Hammarling, Sven; Duff, Iain S. +A Set of Level 3 Basic Linear Algebra Subprograms'' +ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990, +pp 1-17 \end{chunk} \begin{chunk}{ignore} -\bibitem[Henrici 56]{Hen56} Henrici, Peter -Automatic Computations with Power Series'' -Journal of the Association for Computing Machinery, Volume 3, No. 1, -January 1956, 10-15 +\bibitem[Dongarra 90a]{REF-DON90a} Dongarra, Jack J.; Du Croz, Jeremy; +Hammarling, Sven; Duff, Iain S. +ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms: +Model Implementation and Test Programs'' +ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990, +pp 18-28 \end{chunk} \begin{chunk}{ignore} -\bibitem[Higham 88]{Hig88} Higham, N.J. -FORTRAN codes for estimating the one-norm of a -real or complex matrix, with applications to condition estimation'' -ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. +\bibitem[Ducos 00]{Duc00} Ducos, Lionel +Optimizations of the subresultant algorithm'' +Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149-163 \end{chunk} \begin{chunk}{ignore} -\bibitem[Higham 02]{Hig02} Higham, Nicholas J. -Accuracy and stability of numerical algorithms'' -SIAM Philadelphia, PA ISBN 0-89871-521-0 (2002) +\bibitem[Duff 77]{Duff77} Duff I S, +MA28 -- a set of Fortran subroutines for sparse unsymmetric linear +equations'' +A.E.R.E. Report R.8730. HMSO. (1977) \end{chunk} \begin{chunk}{ignore} -\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K. -Test Examples for Nonlinear Programming Codes'' -Lecture Notes in Economics and Mathematical Systems. 187 Springer-Verlag. 1981 +\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alez-Vega, L. +Dynamic Evaluation and Real Closure'' +Mathematics and Computers in Simulation 42 pp 551-560 (1996) +%\verb|axiom-developer.org/axiom-website/papers/Duva96a.pdf| + abstract = " + The aim of this paper is to present how the dynamic evaluation method + can be used to deal with the real closure of an ordered field. Two + kinds of questions, or tests, may be asked in an ordered field: + equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are + handled through splittings, exactly as in the algebraic closure of a + field. Sign tests are handled throug a structure called Tarski data + type''." \end{chunk} \begin{chunk}{ignore} -\bibitem[Householder 70]{Hou70} Householder A S. -The Numerical Treatment of a Single Nonlinear Equation'' -McGraw-Hill. (1970) +\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C. +Sketches and Computations over Fields'' +Mathematics and Computers in Simulation 42 pp 363-373 (1996) +%\verb|axiom-developer.org/axiom-website/papers/Duva96.pdf| + abstract = " + The goal of this short paper is to describe one possible use of + sketches in computer algebra. We show that sketches are a powerful + tool for the description of mathematical structures and for the + description of computations." \end{chunk} -\begin{chunk}{axiom.bib} -@book{Hous81, - author = "Householder, Alston S.", - title = "Principles of Numerical Analysis", - publisher = "Dover Publications, Mineola, NY", - year = "1981", - isbn = "0-486-45312-X" -} +\begin{chunk}{ignore} +\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C. +Sketches and Computation (Part I): Basic Definitions and Static Evaluation'' +Mathematical Structures in Computer Science, 4, p 185-238 Cambridge University Press (1994) +\verb|journals.cambridge.org/abstract_S0960129500000438| +%\verb|axiom-developer.org/axiom-website/papers/Duva94a.pdf| + abstract = " + We define a categorical framework, based on the notion of {\sl + sketch}, for specification and evaluation in the sense of algebraic + specifications and algebraic programming. This framework goes far + beyond our initial motivations, which was to specify computation with + algebraic numbers. We begin by redefining sketches in order to deal + explicitly with programs. Expressions and terms are carefully defined + and studied, then {\sl quasi-projective sketches} are introduced. We + describe {\sl static evaluation} in these sketches: we propose a + rigorous basis for evaluation in the corresponding structures. These + structures admit an initial model, but are not necessarily + equational. In Part II (Duval and Reynaud 1994), we study a more + general process, called {\sl dynamic evaluation}, for structures that + may have no initial model." \end{chunk} \begin{chunk}{ignore} -\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D. -Efficient algorithms for Riemann-Roch problem and for addition in the -jacobian of a curve'' -Proceedings 32nd Annual Symposium on Foundations of Computer Sciences. -IEEE Comput. Soc. Press, pp. 678--687. +\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C. +Sketches and Computation (Part II): Dynamic Evaluation and Applications'' +Mathematical Structures in Computer Science, 4, p 239-271. Cambridge University Press (1994) +\verb|journals.cambridge.org/abstract_S096012950000044X| +%\verb|axiom-developer.org/axiom-website/papers/Duva94b.pdf| + abstract = " + In the first part of this paper (Duval and Reynaud 1994), we defined a + categorical framework, based on the notion of {\sl sketch}, for + specification and evaluation in the senses of algebraic specification + and algebraic programming. {\sl Static evaluation} in {\sl + quasi-projective sketches} was defined in Part I; in this paper, {\sl + dynamic evaluation} is introduced. It deals with more general + structures, which may have no initial model. Until now, this process + has not been used in algebraic specification systems, but computer + algebra systems are beginning to use it as a basic tool. Finally, we + give some applications of dynamic evaluation to computation in field + extensions." \end{chunk} -\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[IBM]{IBM}. -SCRIPT Mathematical Formula Formatter User's Guide, SH20-6453, -IBM Corporation, Publishing Systems Information Development, -Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 80301-9191. +\bibitem[Duval 94c]{Duva94c} Duval, Dominique +Algebraic Numbers: An Example of Dynamic Evaluation'' +J. Symbolic Computation 18, 429-445 (1994) +\verb|www.sciencedirect.com/science/article/pii/S0747717106000551| +%\verb|axiom-developer.org/axiom-website/papers/Duva94c.pdf| + abstract = " + Dynamic evaluation is presented through examples: computations + involving algebraic numbers, automatic case discussion according to + the characteristic of a field. Implementation questions are addressed + too. Finally, branches are presented as dual'' to binary functions, + according to the approach of sketch theory." \end{chunk} +\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S. -A fast algorithm for computing multiplicative inverses -in $GF(2^m)$ using normal bases'' -Inf. and Comp. 78, pp.171-177, 1988 -%\verb|axiom-developer.org/axiom-website/Itoh88.pdf| +\bibitem[Fateman 08]{Fat08} Fateman, Richard +Revisiting numeric/symbolic indefinite integration of rational functions, and extensions'' +\verb|www.eecs.berkeley.edu/~fateman/papers/integ.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Fat08.pdf| abstract = " - This paper proposes a fast algorithm for computing multiplicative - inverses in $GF(2^m)$ using normal bases. Normal bases have the - following useful property: In the case that an element $x$ in - $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an - element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift - of its vector representation. C.C. Wang et al. proposed an algorithm - for computing multiplicative inverses using normal bases, which - requires $(m-2)$ multiplications in $GF(2^m)$ and $(m-1)$ cyclic - shifts. The fast algorithm proposed in this paper also uses normal - bases, and computes multiplicative inverses iterating multiplications - in $GF(2^m)$. It requires at most $2[log_2(m-1)]$ multiplications in - $GF(2^m)$ and $(m-1)$ cyclic shifts, which are much less than those - required in Wang's method. The same idea of the proposed fast - algorithm is applicable to the general power operation in $GF(2^m)$ - and the computation of multiplicative inverses in $GF(q^m)$ - $(q=2^n)$." + We know we can solve this problem: Given any rational function + $f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over + the rationals, compute its {\sl indefinite} integral, using if + necessary, algebraic numbers. But in many circumstances an approximate + result is more likely to be of use. Furthermore, it is plausible that + it would be more useful to solve the problem to allow definite + integration, or introduce additional parameters so that we can solve + multiple definite integrations. How can a computer algebra system + best answer the more useful questions? Finally, what if the integrand + is not a ratio of polynomials, but something more challenging?" \end{chunk} -\begin{chunk}{ignore} -\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada -Encyclopedic Dictionary of Mathematics'' -1977 +\begin{chunk}{axiom.bib} +@misc{Flet01, + author = "Fletcher, John P.", + title = "Symbolic processing of Clifford Numbers in C++", + year = "2001", + journal = "Paper 25, AGACSE 2001." +} \end{chunk} -\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{axiom.bib} +@misc{Flet09, + author = "Fletcher, John P.", + title = "Clifford Numbers and their inverses calculated using the matrix + representation", + publisher = "Chemical Engineering and Applied Chemistry, School of + Engineering and Applied Science, Aston University, Aston Triangle, + Birmingham B4 7 ET, U. K.", + url = + "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php" +} + +\end{chunk} \begin{chunk}{ignore} -\bibitem[Jacobson 68]{Jac68} Jacobson, N. -Structure and Representations of Jordan Algebras'' -AMS, Colloquium Publications Volume 39 +\bibitem[Fletcher 81]{Fle81} Fletcher R +Practical Methods of Optimization'' +Vol 2. Constrained Optimization. Wiley. (1981) \end{chunk} -\begin{chunk}{ignore} -\bibitem[James 81]{JK81} James, Gordon; Kerber, Adalbert -The Representation Theory of the Symmetric Group'' -Encyclopedia of Mathematics and its Applications Vol. 16 -Addison-Wesley, 1981 +\begin{chunk}{axiom.bib} +@article{Floy63, + author = "Floyd, R. W.", + title = "Semantic Analysis and Operator Precedence", + journal = "JACM", + volume = "10", + number = "3", + pages = "316-333", + year = "1963" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Jaswon 77]{JS77} Jaswon, M A.; Symm G T. -Integral Equation Methods in Potential Theory and Elastostatics'' -Academic Press. (1977) +\bibitem[Forsythe 57]{For57} Forsythe G E, +Generation and use of orthogonal polynomials for data fitting +with a digital computer'' +J. Soc. Indust. Appl. Math. 5 74--88. (1957) \end{chunk} \begin{chunk}{ignore} -\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan -Handbook of Mathematical Formulas and Integrals'' -Third Edition, Elsevier Academic Press ISBN 0-12-382256-4 +\bibitem[Fortenbacher 90]{REF-For90} Fortenbacher, A. +Efficient type inference and coercion in computer algebra'' +Design and Implementation of Symbolic Computation Systems (DISCO 90) +A. Miola, (ed) vol 429 of Lecture Notes in Computer Science +Springer-Verlag, pp56-60 + abstract = " + Computer algebra systems of the new generation, like Scratchpad, are + characterized by a very rich type concept, which models the + relationship between mathematical domains of computation. To use these + systems interactively, however, the user should be freed of type + information. A type inference mechanism determines the appropriate + function to call. All known models which allow to define a semantics + for type inference cannot express the rich mathematical'' type + structure, so presently type inference is done heuristically. The + following paper defines a semantics for a subproblem thereof, namely + coercion, which is based on rewrite rules. From this definition, and + efficient coercion algorith for Scratchpad is constructed using graph + techniques." \end{chunk} \begin{chunk}{ignore} -\bibitem[Jenning 66]{Jen66} Jennings A -A Compact Storage Scheme for the Solution of Symmetric Linear -Simultaneous Equations'' -Comput. J. 9 281--285. (1966) +\bibitem[Fox 68]{Fox68} Fox L.; Parker I B. +Chebyshev Polynomials in Numerical Analysis'' +Oxford University Press. (1968) \end{chunk} -\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{chunk}{ignore} +\bibitem[Franke 80]{FN80} Franke R.; Nielson G +Smooth Interpolation of Large Sets of Scattered Data'' +Internat. J. Num. Methods Engrg. 15 1691--1704. (1980) + +\end{chunk} \begin{chunk}{ignore} -\bibitem[Kalkbrener 91]{Kal91} Kalkbrener, M. -Three contributions to elimination theory'' -Ph. D. Thesis, University of Linz, Austria, 1991 +\bibitem[Fritsch 82]{Fri82} Fritsch F N +PCHIP Final Specifications'' +Report UCID-30194. Lawrence Livermore National Laboratory. (1982) \end{chunk} \begin{chunk}{ignore} -\bibitem[Kalkbrener 98]{Kal98} Kalkbrener, M. -Algorithmic properties of polynomial rings'' -Journal of Symbolic Computation 1998 +\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J. +A Method for Constructing Local Monotone Piecewise Cubic Interpolants'' +SIAM J. Sci. Statist. Comput. 5 300--304. (1984) \end{chunk} \begin{chunk}{ignore} -\bibitem[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S. -Hypercomplex Numbers'' -Springer Verlag Heidelberg, 1989, ISBN 0-387-96980-2 +\bibitem[Froberg 65]{Fro65} Froberg C E. +Introduction to Numerical Analysis'' +Addison-Wesley. 181--187. (1965) \end{chunk} +\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis; -Moore J Strother -Computer-Aided Reasoning: An Approach'' -Springer, July 31. 2000 ISBN 0792377443 +\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H. +A tower of Artin-Schreier extensions of function fields attaining the +Drinfeld-Vladut bound'' +Invent. Math., vol. 121, 1995, pp. 211--222. \end{chunk} \begin{chunk}{ignore} -\bibitem[Knuth 71]{Knu71} Knuth, Donald -The Art of Computer Programming'' -2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing, -Addison-Wesley 1971, p. 397-398 +\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark +Constructing Normal Bases in Finite Fields'' +J. Symbolic Computation pp 547-570 (1990) +%\verb|axiom-developer.org/axiom-website/papers/Gat90a.pdf| + abstract = " + An efficient probabilistic algorithm to find a normal basis in a + finite field is presented. It can, in fact, find an element of + arbitrary prescribed additive order. It is based on a density estimate + for normal elements. A similar estimate yields a probabilistic + polynomial-time reduction from finding primitive normal elements to + finding primitive elements." \end{chunk} \begin{chunk}{ignore} -\bibitem[Knuth 84]{Knu84} Knuth, Donald -{\it The \TeX{}book}. -Reading, Massachusetts, Addison-Wesley Publishing Company, Inc., -1984. ISBN 0-201-13448-9 +\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur +Functional Decomposition Polynomials: the Tame Case'' +Journal of Symbolic Computation (1990) 9, 281-299 \end{chunk} \begin{chunk}{axiom.bib} -@book{Knut92, - author = "Knuth, Donald E.", - title = "Literate Programming", - publisher = "Center for the Study of Language and Information, Stanford CA", - year = "1992", - isbn = "0-937073-81-4" -} +@book{Gath99, + author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen}, + title = "Modern Computer Algebra", + publisher = "Cambridge University Press", + year = "1999", + isbn = "0-521-64176-4" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Knu98]{Knu98} Donald Knuth -The Art of Computer Programming'' Vol. 3 -(Sorting and Searching) -Addison-Wesley 1998 +\bibitem[Gautschi 79a]{Gau79a} Gautschi W. +A Computational Procedure for Incomplete Gamma Functions'' +ACM Trans. Math. Softw. 5 466--481. (1979) \end{chunk} \begin{chunk}{ignore} -\bibitem[Kobayashi 89]{Koba89} Kobayashi, H.; Moritsugu, S.; Hogan, R.W. -On Radical Zero-Dimensional Ideals'' -J. Symbolic Computations 8, 545-552 (1989) -\verb|www.sciencedirect.com/science/article/pii/S0747717189800604/pdf| -\verb|?md5=f06dc6269514c90dcae57f0184bcbe65&| -\verb|pid=1-s2.0-S0747717189800604-main.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Koba88.pdf| +\bibitem[Gautschi 79b]{Gau79b} Gautschi W. +Algorithm 542: Incomplete Gamma Functions'' +ACM Trans. Math. Softw. 5 482--489. (1979) \end{chunk} \begin{chunk}{ignore} -\bibitem[Kolchin 73]{Kol73} Kolchin, E.R. -Differential Algebra and Algebraic Groups'' -(Academic Press, 1973). +\bibitem[Gentlemen 69]{Gen69} Gentlemen W M +An Error Analysis of Goertzel's (Watt's) Method for Computing +Fourier Coefficients'' +Comput. J. 12 160--165. (1969) \end{chunk} \begin{chunk}{ignore} -\bibitem[Koutschan 10]{Kou10} Koutschan, Christoph -Axiom / FriCAS'' -\verb|www.risc.jku.at/education/courses/ws2010/cas/axiom.pdf| +\bibitem[Gentleman 73]{Gen73} Gentleman W M. +Least-squares Computations by Givens Transformations without Square Roots'' +J. Inst. Math. Applic. 12 329--336. (1973) \end{chunk} \begin{chunk}{ignore} -\bibitem[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan -Polynomial Decomposition Algorithms'' -Journal of Symbolic Computation (1989) 7, 445-456 +\bibitem[Gentleman 74]{Gen74} Gentleman W M. +Algorithm AS 75. Basic Procedures for Large Sparse or +Weighted Linear Least-squares Problems'' +Appl. Statist. 23 448--454. (1974) \end{chunk} -\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -\begin{chunk}{axiom.bib} -@book{Lamp86, - author = "Lamport, Leslie", - title = "LaTeX: A Document Preparation System", - publisher = "Addison-Wesley Publishing Company, Reading, Massachusetts", - year = "1986", - isbn = "0-201-15790-X" -} +\begin{chunk}{ignore} +\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B. +More on algorithms that reveal properties of floating point +arithmetic units'' +Comms. of the ACM, 17, 276-277. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Lautrup 71]{Lau71} Lautrup B. -An Adaptive Multi-dimensional Integration Procedure'' -Proc. 2nd Coll. on Advanced Methods in Theoretical Physics, Marseille. (1971) +\bibitem[Genz 80]{GM80} Genz A C.; Malik A A. +An Adaptive Algorithm for Numerical Integration over an N-dimensional +Rectangular Region'' +J. Comput. Appl. Math. 6 295--302. 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(1974) \end{chunk} -\begin{chunk}{axiom.bib} -@article{Laws79, - author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.", - title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage", - journal = "ACM Transactions on Mathematical Software", - volume = "5", - number = "3", - month = "September", - year = "1979", - pages = "308-323" -} +\begin{chunk}{ignore} +\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W. +Minimization subject to bounds on the variables'' +Report NAC 72. National Physical Laboratory. (1976) \end{chunk} \begin{chunk}{ignore} -\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R; - Krogh F T -Basic Linear Algebra Subprograms for Fortran Usage'' -ACM Trans. Math. Softw. 5 308--325. (1979) +\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W. +Algorithms for the Solution of the Nonlinear Least-squares Problem'' +NAC 71 National Physical Laboratory. (1976) \end{chunk} \begin{chunk}{ignore} -\bibitem[Lazard 91]{Laz91} Lazard, D. -A new method for solving algebraic systems of positive dimension'' -Discr. App. Math. 33:147-160,1991 +\bibitem[Gill 78]{GM78} Gill P E.; Murray W. +Algorithms for the Solution of the Nonlinear Least-squares Problem'' +SIAM J. Numer. Anal. 15 977--992. (1978) \end{chunk} \begin{chunk}{ignore} -\bibitem[Lazard92]{Laz92} Lazard, D. -Solving Zero-dimensional Algebraic Systems'' -Journal of Symbolic Computation, 1992, 13, 117-131 +\bibitem[Gill 79]{GM79} Gill P E.; Murray W; +Conjugate-gradient Methods for Large-scale Nonlinear Optimization'' +Technical Report SOL 79-15. Department of Operations Research, +Stanford University. (1979) \end{chunk} -\begin{chunk}{axiom.bib} -@article{Laza90, - author = "Lazard, Daniel and Rioboo, Renaud", - title = "Integration of rational functions: Rational computation of the - logarithmic part", - journal = "Journal of Symbolic Computation", - volume = "9", - number = "2", - year = "1990", - month = "February", - pages = "113-115", - keywords = "axiomref", - paper = "Laza90.pdf", - abstract = " - A new formula is given for the logarithmic part of the integral of a - rational function, one that strongly improves previous algorithms and - does not need any computation in an algebraic extension of the field - of constants, nor any factorisation since only polynomial arithmetic - and GCD computations are used. This formula was independently found - and implemented in SCRATCHPAD by B.M. Trager." -} +\begin{chunk}{ignore} +\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H. +Practical Optimization'' +Academic Press. 1981 \end{chunk} -\begin{chunk}{axiom.bib} -@article{LeBr88, - author = "Le Brigand, D.; Risler, J.J.", - title = "Algorithme de Brill-Noether et codes de Goppa", - journal = "Bull. Soc. Math. France", - volume = "116", - year = "1988", - pages = "231--253" -} +\begin{chunk}{ignore} +\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H. +The design and implementation of a quadratic programming algorithm'' +Report SOL 82-7. Department of Operations Research, +Stanford University. (1982) \end{chunk} -\begin{chunk}{axiom.bib} -@book{Lege11, - author = "Legendre, George L. and Grazini, Stefano", - title = "Pasta by Design", - publisher = "Thames and Hudson", - isbn = "978-0-500-51580-8", - year = "2011" -} +\begin{chunk}{ignore} +\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H +User's Guide for SOL/QPSOL Version 3.2'' +Report SOL 84-5. Department of Operations Research, Stanford University. 1984 \end{chunk} \begin{chunk}{ignore} -\bibitem[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J. -Primitivive Normal Bases for Finite Fields'' -Math. Comp. 48, 1987, pp. 217-231 +\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H +Procedures for Optimization Problems with a Mixture of +Bounds and General Linear Constraints'' +ACM Trans. Math. Softw. 10 282--298. 1984 \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Leop03, - author = "Leopardi, Paul", - title = "A quick introduction to Clifford Algebras", - publisher = "School of Mathematics, University of New South Wales", - year = "2003", - paper = "Leop03.pdf" -} +\begin{chunk}{ignore} +\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.; +Saunders M A.; Wright M H. +User's Guide for LSSOL (Version 1.0)'' +Report SOL 86-1. Department of Operations Research, Stanford University. 1986 \end{chunk} \begin{chunk}{ignore} -\bibitem[Lewis 77]{Lew77} Lewis J G, -Algorithms for sparse matrix eigenvalue problems'' -Technical Report STAN-CS-77-595. Computer Science Department, -Stanford University. (1977) +\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H. +Some Theoretical Properties of an Augmented Lagrangian Merit Function'' +Report SOL 86-6R. Department of Operations Research, Stanford University. 1986 \end{chunk} \begin{chunk}{ignore} -\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H. -Finite Field, Encycoldia of Mathematics and Its Applications'' -Vol. 20, Cambridge Univ. Press, 1983 ISBN 0-521-30240-4 +\bibitem[Gladwell 79]{Gla79} Gladwell I +Initial Value Routines in the NAG Library'' +ACM Trans Math Softw. 5 386--400. (1979) \end{chunk} \begin{chunk}{ignore} -\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.; -Witt, Bernard I. -Structured Programming: Theory and Practice'' -Addison-Wesley (March 1979) ISBN 0201144611 +\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K +Computational Techniques for Ordinary Differential Equations'' +Academic Press. 1980 \end{chunk} \begin{chunk}{ignore} -\bibitem[Lipson 81]{Lip81} Lipson, D. -Elements of Algebra and Algebraic Computing'' -The Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981. +\bibitem[Gladwell 86]{Gla86} Gladwell I +Vectorisation of one dimensional quadrature codes'' +Techincal Report. TR7/86 NAG. (1986) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Loet09, - author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter", - title = "Understanding the Dynamics of Complex Lisp Programs", - year = "2009", - url = "http://www.martin-loetzsch.de/papers/loetzsch09understanding.pdf", - paper = "Loet09.pdf" -} +\begin{chunk}{ignore} +\bibitem[Gladwell 87]{Gla87} Gladwell I +The NAG Library Boundary Value Codes'' +Numerical Analysis Report. 134 Manchester University. (1987) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Loet00, - author = "Loetzsch, M.", - title = "GTFL - A graphical terminal for Lisp", - year = "2000", - url = "http://martin-loetzsch.de/gtfl" -} +\begin{chunk}{ignore} +\bibitem[Goedel 40]{God40} Goedel +The consistency of the continuum hypothesis'' +Ann. Math. Studies, Princeton Univ. Press, 1940 \end{chunk} -\begin{chunk}{axiom.bib} -@book{Losc60, - author = {L\"osch, Friedrich}, - title = "Tables of Higher Functions", - publisher = "McGraw-Hill Book Company", - year = "1960" -} +\begin{chunk}{ignore} +\bibitem[Goldman 87]{Gold87} Goldman, L. +Integrals of multinomial systems of ordinary differential equations'' +J. of Pure and Applied Algebra, 45, 225-240 (1987) +\verb|www.sciencedirect.com/science/article/pii/0022404987900727/pdf| +\verb|?md5=5a0c70643eab514ccf47d80e4fc6ec5a&| +\verb|pid=1-s2.0-0022404987900727-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Gold87.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[LTU10]{LTU10}. -Lambda the Ultimate'' -\verb|lambda-the-ultimate.org/node/3663#comment-62440| +\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier +Algorithms in Representation Theory and +their Realization in the Computer Algebra System Scratchpad'' +Bayreuther Mathematische Schriften, Heft 33, 1990, 1-23 \end{chunk} -\begin{chunk}{axiom.bib} -@book{Luke69a, - author = "Luke, Yudell L.", - title = "The Special Functions and their Approximations", - volume = "1", - publisher = "Academic Press", - year = "1969", - booktitle = "Mathematics in Science and Engineering Volume 53-I" -} +\begin{chunk}{ignore} +\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F. +Matrix Computations'' +Johns Hopkins University Press ISBN 0-8018-3772-3 (1989) \end{chunk} -\begin{chunk}{axiom.bib} -@book{Luke69b, - author = "Luke, Yudell L.", - title = "The Special Functions and their Approximations", - volume = "2", - publisher = "Academic Press", - year = "1969", - booktitle = "Mathematics in Science and Engineering Volume 53-I" -} +\begin{chunk}{ignore} +\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F. +Matrix Computations'' +Johns Hopkins University Press ISBN 978-0-8018-5414-9 (1996) \end{chunk} \begin{chunk}{ignore} -\bibitem[Lyness 83]{Lyn83} Lyness J N. -When not to use an automatic quadrature routine'' -SIAM Review. 25 63--87. (1983) +\bibitem[Grabmeier]{Grab} Grabmeier, J. +On Plesken's root finding algorithm'' +in preparation \end{chunk} -\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret -Algebra'' -AMS Chelsea Publishing ISBN 0821816462 +\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.; +The Evaluation of Irreducible Polynomial Representations of the General +Linear Groups and of the Unitary Groups over Fields of Characteristic 0'' +Acta Appl. Math. 8 (1987), 271-291 \end{chunk} \begin{chunk}{ignore} -\bibitem[Malcolm 72]{Mal72} Malcolm M. A. -Algorithms to reveal properties of floating-point arithmetic'' -Comms. of the ACM, 15, 949-951. (1972) +\bibitem[Grabmeier 92]{REF-GS92} Grabmeier, J.; Scheerhorn, A. +Finite fields in Axiom'' +AXIOM Technical Report TR7/92 (ATR/5)(NP2522), +Numerical Algorithms Group, Inc., Downer's +Grove, IL, USA and Oxford, UK, 1992 +\verb|www.nag.co.uk/doc/TechRep/axiomtr.html| \end{chunk} \begin{chunk}{ignore} -\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B. -Local Versus Global Strategies for Adaptive Quadrature'' -ACM Trans. Math. Softw. 1 129--146. (1976) +\bibitem[Granville 1911]{Gran1911} Granville, William Anthony +Elements of the Differential and Integral Calculus'' +\verb|djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Gran1911.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Marden 66]{Mar66} Marden M. -Geometry of Polynomials'' -Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966) +\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik +Limit computation in computer algebra'' +\verb|algo.inria.fr/seminars/sem92-93/gruntz.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Gru93.pdf| + abstract = " + The automatic computation of limits can be reduced to two main + sub-problems. The first one is asymptotic comparison where one must + decide automatically which one of two functions in a specified class + dominates the other one asymptotically. The second one is asymptotic + cancellation and is usually exemplified by + $e^x[exp(1/x+e^{-x})-exp(1/x)],\quad{}x \leftarrow \infty$ + + In this example, if the sum is expanded in powers of $1/x$, the + expansion always yields $O(x^{-k})$, and this is not enough to + conclude. + + In 1990, J.Shackell found an algorithm that solved both these problems + for the case of $exp-log$ functions, i.e. functions build by recursive + application of exponential, logarithm, algebraic extension and field + operations to one variable and the rational numbers. D. Gruntz and + G. Gonnet propose a slightly different algorithm for exp-log + functions. Extensions to larger classes of functions are also + discussed." \end{chunk} +\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@misc{Mars07, - author = "Marshak, U.", - title = "HT-AJAX - AJAX framework for Hunchentoot", - year = "2007", - url = "http://common-lisp.net/project/ht-ajax/ht-ajax.html" +@article{Hach95, + author = "Hach\'e, G. and Le Brigand, D.", + title = "Effective construction of algebraic geometry codes", + journal = "IEEE Transaction on Information Theory", + volume = "41", + month = "November", + year = "1995", + pages = "1615--1628" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R. -Computations of gcd over algebraic towers of simple extensions'' -In proceedings of AAECC11 Paris, 1995. +\begin{chunk}{axiom.bib} +@article{Hach95a, + author = "Hach\'e, G.", + title = "Computation in algebraic function fields for effective + construction of algebraic-geometric codes", + journal = "Lecture Notes in Computer Science", + volume = "948", + year = "1995", + pages = "262--278" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Maza 97]{Maz97} Maza, M. Moreno -Calculs de pgcd au-dessus des tours -d'extensions simples et resolution des systemes d'equations algebriques'' -These, Universite P.etM. Curie, Paris, 1997. +\begin{chunk}{axiom.bib} +@phdthesis{Hach96, + author = "Hach\'e, G.", + title = "Construction effective des codes g\'eom\'etriques", + school = "l'Universit\'e Pierre et Marie Curie (Paris 6)", + year = "1996", + month = "Septembre" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Maza 98]{Maz98} Maza, M. Moreno -A new algorithm for computing triangular -decomposition of algebraic varieties'' - NAG Tech. Rep. 4/98. +\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds), +Modern Numerical Methods for Ordinary Differential Equations'' +Clarendon Press. (1976) \end{chunk} \begin{chunk}{ignore} -\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice -Some Useful Bounds'' -Computing, Suppl. 4, 259-263 (1982), Springer-Verlag +\bibitem[Hamdy 04]{Ham04} Hamdy, S. +LiDIA A library for computational number theory'' +Reference manual Edition 2.1.1 May 2004 +\verb|www.cdc.informatik.tu-darmstadt.de/TI/LiDIA| \end{chunk} \begin{chunk}{ignore} -\bibitem[McCarthy 83]{McC83} McCarthy G J. -Investigation into the Multigrid Code MGD1'' -Report AERE-R 10889. Harwell. (1983) +\bibitem[Hammarling 85]{Ham85} Hammarling S. + The Singular Value Decomposition in Multivariate Statistics'' +ACM Signum Newsletter. 20, 3 2--25. (1985) \end{chunk} \begin{chunk}{ignore} -\bibitem[Mie97]{Mie97} Mielenz, Klaus D. -Computation of Fresnel Integrals'' -J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 May-June 1997 pp363-365 +\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C. +Monte-Carlo Methods'' +Methuen. (1967) \end{chunk} -\begin{chunk}{ignore} -\bibitem[Mie00]{Mie00} Mielenz, Klaus D. -Computation of Fresnel Integrals II'' -J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 July-Aug 2000 pp589-590 +\begin{chunk}{axiom.bib} +@misc{Hath1896, + author = "Hathway, Arthur S.", + title = "A Primer Of Quaternions", + year = "1896" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Millen 68]{Mil68} Millen, J. K. -CHARYBDIS: A LISP program to display mathematical expressions on -typewriter-like devices'' -Interactive Systems for Experimental and Applied Mathematics -M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp79-90 -%\verb|axiom-developer.org/axiom-website/papers/Mil68.pdf| +\begin{chunk}{axiom.bib} +@book{Haya05, + author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and + Schreitmueller, S. and Tarquinio, T. and Thompson, J.", + title = "AIX 5L Practical Performance Tools and Tuning Guide", + publisher = "IBM", + year = "2005", + url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf", + paper = "Haya05.pdf" +} \end{chunk} - \begin{chunk}{ignore} -\bibitem[Minc 79]{Min79} Henryk Minc -Evaluation of Permanents'' -Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32. +\bibitem[Hayes 70]{Hay70} Hayes J G. +Curve Fitting by Polynomials in One Variable'' +Numerical Approximation to Functions and Data. +(ed J G Hayes) Athlone Press, London. (1970) \end{chunk} \begin{chunk}{ignore} -\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E. -User Guide for Minpack-1'' -ANL-80-74 Argonne National Laboratory. (1974) +\bibitem[Hayes 74]{Hay74} Hayes J G. +Numerical Methods for Curve and Surface Fitting'' +Bull Inst Math Appl. 10 144--152. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L. -Approximate Methods for the Solution of Differential and -Integral Equations'' -Elsevier. (1967) +\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J, +The Least-squares Fitting of Cubic Spline Surfaces to General Data Sets'' +J. Inst. Math. Appl. 14 89--103. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F. -The Finite Difference Method in Partial Differential Equations'' -Wiley. (1980) +\bibitem[Henrici 56]{Hen56} Henrici, Peter +Automatic Computations with Power Series'' +Journal of the Association for Computing Machinery, Volume 3, No. 1, +January 1956, 10-15 \end{chunk} \begin{chunk}{ignore} -\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W. -An Algorithm for Generalized Matrix Eigenproblems'' -SIAM J. Numer. Anal. 10 241--256. 1973 +\bibitem[Higham 88]{Hig88} Higham, N.J. +FORTRAN codes for estimating the one-norm of a +real or complex matrix, with applications to condition estimation'' +ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. \end{chunk} -\begin{chunk}{axiom.bib} -@article{Muld97, - author = "Mulders, Thom", - title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in - Rational Function Integration", - journal = "Journal of Symbolic Computation", - year = "1997", - volume = "24", - number = "1", - month = "July", - pages = "45-50", - paper = "Muld97.pdf", - abstract = " - An ambiguity in a formula of Lazard, Rioboo and Trager, connecting - subresultants and rational function integration, is indicated and - examples of incorrect interpretations are given." -} +\begin{chunk}{ignore} +\bibitem[Higham 02]{Hig02} Higham, Nicholas J. +Accuracy and stability of numerical algorithms'' +SIAM Philadelphia, PA ISBN 0-89871-521-0 (2002) \end{chunk} \begin{chunk}{ignore} -\bibitem[Munksgaard 80]{Mun80} Munksgaard N. -Solving Sparse Symmetric Sets of Linear Equations by Pre-conditioned -Conjugate Gradients'' -ACM Trans. Math. Softw. 6 206--219. (1980) +\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K. +Test Examples for Nonlinear Programming Codes'' +Lecture Notes in Economics and Mathematical Systems. 187 Springer-Verlag. 1981 \end{chunk} \begin{chunk}{ignore} -\bibitem[Murray 72]{Mur72} Murray W, (ed) -Numerical Methods for Unconstrained Optimization'' -Academic Press. (1972) +\bibitem[Householder 70]{Hou70} Householder A S. +The Numerical Treatment of a Single Nonlinear Equation'' +McGraw-Hill. (1970) \end{chunk} -\begin{chunk}{ignore} -\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A -MINOS 5.0 User's Guide'' -Report SOL 83-20. Department of Operations Research, Stanford University 1983 +\begin{chunk}{axiom.bib} +@book{Hous81, + author = "Householder, Alston S.", + title = "Principles of Numerical Analysis", + publisher = "Dover Publications, Mineola, NY", + year = "1981", + isbn = "0-486-45312-X" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Musser 78]{Mus78} Musser, David R. -On the Efficiency of a Polynomial Irreducibility Test'' -Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271-282 +\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D. +Efficient algorithms for Riemann-Roch problem and for addition in the +jacobian of a curve'' +Proceedings 32nd Annual Symposium on Foundations of Computer Sciences. +IEEE Comput. Soc. Press, pp. 678--687. \end{chunk} -\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf -Combinatorical Algorithms'' -Academic Press, New York 1978. +\bibitem[IBM]{IBM}. +SCRIPT Mathematical Formula Formatter User's Guide, SH20-6453, +IBM Corporation, Publishing Systems Information Development, +Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 80301-9191. \end{chunk} \begin{chunk}{ignore} -\bibitem[Nikolai 79]{Nik79} Nikolai P J. -Algorithm 538: Eigenvectors and eigenvalues of real generalized -symmetric matrices by simultaneous iteration'' -ACM Trans. Math. Softw. 5 118--125. (1979) - -\end{chunk} - -\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{axiom.bib} -@misc{OCAM14, - author = "unknown", - title = "The OCAML website", - url = "http://ocaml.org" -} +\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S. +A fast algorithm for computing multiplicative inverses +in $GF(2^m)$ using normal bases'' +Inf. and Comp. 78, pp.171-177, 1988 +%\verb|axiom-developer.org/axiom-website/Itoh88.pdf| + abstract = " + This paper proposes a fast algorithm for computing multiplicative + inverses in $GF(2^m)$ using normal bases. Normal bases have the + following useful property: In the case that an element $x$ in + $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an + element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift + of its vector representation. C.C. Wang et al. proposed an algorithm + for computing multiplicative inverses using normal bases, which + requires $(m-2)$ multiplications in $GF(2^m)$ and $(m-1)$ cyclic + shifts. The fast algorithm proposed in this paper also uses normal + bases, and computes multiplicative inverses iterating multiplications + in $GF(2^m)$. It requires at most $2[log_2(m-1)]$ multiplications in + $GF(2^m)$ and $(m-1)$ cyclic shifts, which are much less than those + required in Wang's method. The same idea of the proposed fast + algorithm is applicable to the general power operation in $GF(2^m)$ + and the computation of multiplicative inverses in $GF(q^m)$ + $(q=2^n)$." \end{chunk} \begin{chunk}{ignore} -\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin -Algorithms and methods in differential algebra'' -\verb|www.lix.polytechnique.fr/~moulin/papiers/atelier.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Olla94.pdf| +\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada +Encyclopedic Dictionary of Mathematics'' +1977 \end{chunk} -\begin{chunk}{ignore} -\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.; -Boisvert, Ronald F.; Clark, Charles W. 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M.W.", - title = "De l'int\'{e}gration des fractions rationelles.", - journal = "Bulletin de la Classe Physico-Math\'{e}matiques de - l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,", - volume = "IV", - pages = "145-167,286-300", - year = "1845" -} +\begin{chunk}{ignore} +\bibitem[Jaswon 77]{JS77} Jaswon, M A.; Symm G T. +Integral Equation Methods in Potential Theory and Elastostatics'' +Academic Press. (1977) \end{chunk} -\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Paige 75]{PS75} Paige C C.; Saunders M A. -Solution of Sparse Indefinite Systems of Linear Equations'' -SIAM J. Numer. Anal. 12 617--629. 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Symbolic Computations 8, 545-552 (1989) +\verb|www.sciencedirect.com/science/article/pii/S0747717189800604/pdf| +\verb|?md5=f06dc6269514c90dcae57f0184bcbe65&| +\verb|pid=1-s2.0-S0747717189800604-main.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Koba88.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Peters 67b]{Pet67b} Peters G. -NPL Algorithms Library'' -Document No.F1/04/A (1967) +\bibitem[Kolchin 73]{Kol73} Kolchin, E.R. +Differential Algebra and Algebraic Groups'' +(Academic Press, 1973). \end{chunk} \begin{chunk}{ignore} -\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H. -The Least-squares Problem and Pseudo-inverses'' -Comput. J. 13 309--316. (1970) +\bibitem[Koutschan 10]{Kou10} Koutschan, Christoph +Axiom / FriCAS'' +\verb|www.risc.jku.at/education/courses/ws2010/cas/axiom.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Peters 71]{PW71} Peters G.; Wilkinson J H. -Practical Problems Arising in the Solution of Polynomial Equations'' -J. Inst. Maths Applics. 8 16--35. 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(1976) +\begin{chunk}{axiom.bib} +@article{Laws79, + author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.", + title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage", + journal = "ACM Transactions on Mathematical Software", + volume = "5", + number = "3", + month = "September", + year = "1979", + pages = "308-323" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Piessens 83]{PDUK83} Piessens R.; De Doncker-Kapenga E.; -Uberhuber C.; Kahaner D. -QUADPACK, A Subroutine Package for Automatic Integration'' -Springer-Verlag.(1983) +\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R; + Krogh F T +Basic Linear Algebra Subprograms for Fortran Usage'' +ACM Trans. Math. Softw. 5 308--325. 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(1970) +\bibitem[Lazard92]{Laz92} Lazard, D. +Solving Zero-dimensional Algebraic Systems'' +Journal of Symbolic Computation, 1992, 13, 117-131 + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{Laza90, + author = "Lazard, Daniel and Rioboo, Renaud", + title = "Integration of rational functions: Rational computation of the + logarithmic part", + journal = "Journal of Symbolic Computation", + volume = "9", + number = "2", + year = "1990", + month = "February", + pages = "113-115", + keywords = "axiomref", + paper = "Laza90.pdf", + abstract = " + A new formula is given for the logarithmic part of the integral of a + rational function, one that strongly improves previous algorithms and + does not need any computation in an algebraic extension of the field + of constants, nor any factorisation since only polynomial arithmetic + and GCD computations are used. This formula was independently found + and implemented in SCRATCHPAD by B.M. Trager." +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@article{LeBr88, + author = "Le Brigand, D.; Risler, J.J.", + title = "Algorithme de Brill-Noether et codes de Goppa", + journal = "Bull. Soc. Math. France", + volume = "116", + year = "1988", + pages = "231--253" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Powell 74]{Pow74} Powell M J D. -Introduction to Constrained Optimization'' -Numerical Methods for Constrained Optimization. -(ed P E Gill and W Murray) Academic Press. pp1-28. 1974 +\begin{chunk}{axiom.bib} +@book{Lege11, + author = "Legendre, George L. and Grazini, Stefano", + title = "Pasta by Design", + publisher = "Thames and Hudson", + isbn = "978-0-500-51580-8", + year = "2011" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Powell 83]{Pow83} Powell M J D. -Variable Metric Methods in Constrained Optimization'' -Mathematical Programming: The State of the Art. -(ed A Bachem, M Groetschel and B Korte) Springer-Verlag. pp288--311. 1983 +\bibitem[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J. +Primitivive Normal Bases for Finite Fields'' +Math. Comp. 48, 1987, pp. 217-231 \end{chunk} \begin{chunk}{axiom.bib} -@inproceedings{Prat73, - author = "Pratt, Vaughan R.", - title = "Top down operator precedence", - booktitle = "Proc. 1st annual ACM SIGACT-SIGPLAN Symposium on Principles - of Programming Languages", - series = "POPL'73", - pages = "41-51", - year = "1973", - url = "http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf", - keywords = "axiomref", - paper = "Prat73.pdf" +@misc{Leop03, + author = "Leopardi, Paul", + title = "A quick introduction to Clifford Algebras", + publisher = "School of Mathematics, University of New South Wales", + year = "2003", + paper = "Leop03.pdf" } \end{chunk} \begin{chunk}{ignore} -\bibitem[Press 95]{PTVF95} Press, William H.; Teukolsky, Saul A.; -Vetterling, William T.; Flannery, Brian P. -Numerical Recipes in C'' -Cambridge University Press (1995) ISBN 0-521-43108-5 +\bibitem[Lewis 77]{Lew77} Lewis J G, +Algorithms for sparse matrix eigenvalue problems'' +Technical Report STAN-CS-77-595. Computer Science Department, +Stanford University. (1977) \end{chunk} \begin{chunk}{ignore} -\bibitem[Pryce 77]{PH77} Pryce J D.; Hargrave B A. -The Scale Pruefer Method for one-parameter and multi-parameter eigenvalue -problems in ODEs'' -Inst. Math. Appl., Numerical Analysis Newsletter. 1(3) (1977) +\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H. +Finite Field, Encycoldia of Mathematics and Its Applications'' +Vol. 20, Cambridge Univ. Press, 1983 ISBN 0-521-30240-4 \end{chunk} \begin{chunk}{ignore} -\bibitem[Pryce 81]{Pry81} Pryce J D. -Two codes for Sturm-Liouville problems'' -Technical Report CS-81-01. Dept of Computer Science, Bristol University (1981) +\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.; +Witt, Bernard I. +Structured Programming: Theory and Practice'' +Addison-Wesley (March 1979) ISBN 0201144611 \end{chunk} \begin{chunk}{ignore} -\bibitem[Pryce 86]{Pry86} Pryce J D. -Error Estimation for Phase-function Shooting Methods for -Sturm-Liouville Problems'' -J. Num. Anal. 6 103--123. (1986) +\bibitem[Lipson 81]{Lip81} Lipson, D. +Elements of Algebra and Algebraic Computing'' +The Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981. \end{chunk} \begin{chunk}{axiom.bib} -@misc{Puff09, - author = "Puffinware LLC", - title = "Singular Value Decomposition (SVD) Tutorial", - url = "http://www.puffinwarellc.com/p3a.htm" +@misc{Loet09, + author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter", + title = "Understanding the Dynamics of Complex Lisp Programs", + year = "2009", + url = "http://www.martin-loetzsch.de/papers/loetzsch09understanding.pdf", + paper = "Loet09.pdf" } \end{chunk} -\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Quintana-Orti 06]{QG06} Quintana-Orti, Gregorio; -van de Geijn, Robert -Improving the performance of reduction to Hessenberg form'' -ACM Transactions on Mathematical Software, 32(2):180-194, June 2006. - -\end{chunk} - -\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{ignore} -\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P. -Numerical Methods for Nonlinear Algebraic Equations'' -Gordon and Breach. (1970) +\begin{chunk}{axiom.bib} +@misc{Loet00, + author = "Loetzsch, M.", + title = "GTFL - A graphical terminal for Lisp", + year = "2000", + url = "http://martin-loetzsch.de/gtfl" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Ralston 65]{Ral65} Ralston A. -A First Course in Numerical Analysis'' -McGraw-Hill. 87--90. (1965) +\begin{chunk}{axiom.bib} +@book{Losc60, + author = {L\"osch, Friedrich}, + title = "Tables of Higher Functions", + publisher = "McGraw-Hill Book Company", + year = "1960" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya -A Gentle Introduction to Lyapunov Functions'' -ORSUM August 2003 -\verb|www.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf| +\bibitem[LTU10]{LTU10}. +Lambda the Ultimate'' +\verb|lambda-the-ultimate.org/node/3663#comment-62440| \end{chunk} -\begin{chunk}{ignore} -\bibitem[Ramsey 03]{Ra03} Ramsey, Norman -Noweb--A Simple, Extensible Tool for Literate Programming'' -\verb|www.eecs.harvard.edu/~nr/noweb| +\begin{chunk}{axiom.bib} +@book{Luke69a, + author = "Luke, Yudell L.", + title = "The Special Functions and their Approximations", + volume = "1", + publisher = "Academic Press", + year = "1969", + booktitle = "Mathematics in Science and Engineering Volume 53-I" +} \end{chunk} -\begin{chunk}{ignore} -\bibitem[Redfield 27]{Red27} Redfield, J.H. -The Theory of Group-Reduced Distributions'' -American J. Math., 49 (1927) 433-455. +\begin{chunk}{axiom.bib} +@book{Luke69b, + author = "Luke, Yudell L.", + title = "The Special Functions and their Approximations", + volume = "2", + publisher = "Academic Press", + year = "1969", + booktitle = "Mathematics in Science and Engineering Volume 53-I" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Reinsch 67]{Rei67} Reinsch C H. -Smoothing by Spline Functions'' -Num. Math. 10 177--183. (1967) +\bibitem[Lyness 83]{Lyn83} Lyness J N. +When not to use an automatic quadrature routine'' +SIAM Review. 25 63--87. (1983) \end{chunk} -\begin{chunk}{ignore} -\bibitem[Renka 84]{Ren84} Renka R L. -Algorithm 624: Triangulation and Interpolation of Arbitrarily Distributed -Points in the Plane'' -ACM Trans. Math. Softw. 10 440--442. (1984) - -\end{chunk} +\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Renka 84]{RC84} Renka R L.; Cline A K. -A Triangle-based C Interpolation Method'' -Rocky Mountain J. Math. 14 223--237. (1984) +\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret +Algebra'' +AMS Chelsea Publishing ISBN 0821816462 \end{chunk} \begin{chunk}{ignore} -\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe -Free Lie Algebras'' -Oxford University Press, June 1993 ISBN 0198536798 +\bibitem[Malcolm 72]{Mal72} Malcolm M. A. +Algorithms to reveal properties of floating-point arithmetic'' +Comms. of the ACM, 15, 949-951. (1972) \end{chunk} \begin{chunk}{ignore} -\bibitem[Reznick 93]{Rezn93} Reznick, Bruce -An Inequality for Products of Polynomials'' -Proc. AMS Vol 117 No 4 April 1993 -%\verb|axiom-developer.org/axiom-website/papers/Rezn93.pdf| +\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B. +Local Versus Global Strategies for Adaptive Quadrature'' +ACM Trans. Math. Softw. 1 129--146. (1976) \end{chunk} \begin{chunk}{ignore} -\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J. -Crafting a Repository of Knowledge Based on Transformation'' -\verb|www.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Rixx.pdf| - abstract = " - We describe the development of a repository of mathematical knowledge - based on transformation rules. The specific mathematical problem is - indefinite integration. It is important that the repository be not - confused with a look-up table. The database of transformation rules is - at present encoded in Mathematica, but this is only one convenient - form of the repository, and it could be readily translated into other - formats. The principles upon which the set of rules is compiled is - described. One important principle is minimality. The benefits of the - approach are illustrated with examples, and with the results of - comparisons with other approaches." +\bibitem[Marden 66]{Mar66} Marden M. +Geometry of Polynomials'' +Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966) \end{chunk} -\begin{chunk}{ignore} -\bibitem[Rich 10]{Ri10} Rich, Albert D. -Rule-based Mathematics'' -\verb|www.apmaths.uwo.ca/~arich| +\begin{chunk}{axiom.bib} +@misc{Mars07, + author = "Marshak, U.", + title = "HT-AJAX - AJAX framework for Hunchentoot", + year = "2007", + url = "http://common-lisp.net/project/ht-ajax/ht-ajax.html" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John -The identity problem for elementary functions and constants'' -ACM Proc. of ISSAC 94 pp285-290 ISBN 0-89791-638-7 +\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R. +Computations of gcd over algebraic towers of simple extensions'' +In proceedings of AAECC11 Paris, 1995. \end{chunk} \begin{chunk}{ignore} -\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W. -Difference Methods for Initial-value Problems'' -Interscience (2nd Edition). (1967) +\bibitem[Maza 97]{Maz97} Maza, M. Moreno +Calculs de pgcd au-dessus des tours +d'extensions simples et resolution des systemes d'equations algebriques'' +These, Universite P.etM. Curie, Paris, 1997. \end{chunk} \begin{chunk}{ignore} -\bibitem[Rioboo 92]{REF-Rio92} Rioboo, R. -Real algebraic closure of an ordered field, implementation in Axiom'' -In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover) -In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215. -0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992 +\bibitem[Maza 98]{Maz98} Maza, M. Moreno +A new algorithm for computing triangular +decomposition of algebraic varieties'' + NAG Tech. Rep. 4/98. \end{chunk} \begin{chunk}{ignore} -\bibitem[Rioboo 96]{Rio96} Rioboo, R. -Generic computation of the real closure of an ordered field'' -In Mathematics and Computers in Simulation Volume 42, Issue 4-6, -November 1996. +\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice +Some Useful Bounds'' +Computing, Suppl. 4, 259-263 (1982), Springer-Verlag \end{chunk} \begin{chunk}{ignore} -\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels -Differential Algebra'' -AMS Colloquium Publications Volume 33 ISBN 978-0-8218-4638-4 +\bibitem[McCarthy 83]{McC83} McCarthy G J. +Investigation into the Multigrid Code MGD1'' +Report AERE-R 10889. Harwell. (1983) \end{chunk} \begin{chunk}{ignore} -\bibitem[Rote 01]{Rote01} Rote, G\"unter -Division-free algorithms for the determinant and the Pfaffian'' -in Computational Discrete Mathematics ISBN 3-540-42775-9 pp119-135 -\verb|page.mi.fu-berlin.de/rote/Papers/pdf/Division-free+algorithms.pdf| +\bibitem[Mie97]{Mie97} Mielenz, Klaus D. +Computation of Fresnel Integrals'' +J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 May-June 1997 pp363-365 \end{chunk} \begin{chunk}{ignore} -\bibitem[Rubey 07]{Rub07} Rubey, Martin -Formula Guessing with Axiom'' -April 2007 +\bibitem[Mie00]{Mie00} Mielenz, Klaus D. +Computation of Fresnel Integrals II'' +J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 July-Aug 2000 pp589-590 \end{chunk} \begin{chunk}{ignore} -\bibitem[Rutishauser 69]{Rut69} Rutishauser H. -Computational aspects of F L Bauer's simultaneous iteration method'' -Num. Math. 13 4--13. (1969) +\bibitem[Millen 68]{Mil68} Millen, J. K. +CHARYBDIS: A LISP program to display mathematical expressions on +typewriter-like devices'' +Interactive Systems for Experimental and Applied Mathematics +M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp79-90 +%\verb|axiom-developer.org/axiom-website/papers/Mil68.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Rutishauser 70]{Rut70} Rutishauser H. -Simultaneous iteration method for symmetric matrices'' -Num. Math. 16 205--223. (1970) +\bibitem[Minc 79]{Min79} Henryk Minc +Evaluation of Permanents'' +Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32. \end{chunk} -\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Schafer 66]{Sch66} Schafer, R.D. -An Introduction to Nonassociative Algebras'' -Academic Press, New York, 1966 +\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E. +User Guide for Minpack-1'' +ANL-80-74 Argonne National Laboratory. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A. -On Polya Frequency Functions III'' -Trans. Amer. Math. Soc. 74 246--259. (1953) +\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L. +Approximate Methods for the Solution of Differential and +Integral Equations'' +Elsevier. (1967) \end{chunk} \begin{chunk}{ignore} -\bibitem[Schoenhage 82]{Sch82} Schoenhage, A. -The fundamental theorem of algebra in terms of computational complexity'' -preliminary report, Univ. Tuebingen, 1982 +\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F. +The Finite Difference Method in Partial Differential Equations'' +Wiley. (1980) \end{chunk} \begin{chunk}{ignore} -\bibitem[Schonfelder 76]{Sch76} Schonfelder J L. -The Production of Special Function Routines for a Multi-Machine Library'' -Software Practice and Experience. 6(1) (1976) +\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W. +An Algorithm for Generalized Matrix Eigenproblems'' +SIAM J. Numer. Anal. 10 241--256. 1973 \end{chunk} \begin{chunk}{axiom.bib} -@book{Segg93, - author = "{von Seggern}, David Henry", - title = "CRC Standard Curves and Surfaces", - publisher = "CRC Press", - year = "1993", - isbn = "0-8493-0196-3" +@article{Muld97, + author = "Mulders, Thom", + title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in + Rational Function Integration", + journal = "Journal of Symbolic Computation", + year = "1997", + volume = "24", + number = "1", + month = "July", + pages = "45-50", + paper = "Muld97.pdf", + abstract = " + An ambiguity in a formula of Lazard, Rioboo and Trager, connecting + subresultants and rational function integration, is indicated and + examples of incorrect interpretations are given." } \end{chunk} \begin{chunk}{ignore} -\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J. -JET -- An Axiom Environment for Geometric Computations with Differential -Equations'' -%\verb|axiom-developer.org/axiom-website/papers/Sei95a.pdf| +\bibitem[Munksgaard 80]{Mun80} Munksgaard N. +Solving Sparse Symmetric Sets of Linear Equations by Pre-conditioned +Conjugate Gradients'' +ACM Trans. Math. Softw. 6 206--219. (1980) \end{chunk} \begin{chunk}{ignore} -\bibitem[Shepard 68]{She68} Shepard D. -A Two-dimensional Interpolation Function for Irregularly Spaced Data'' -Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc., -Princeton. 517--523. 1968 +\bibitem[Murray 72]{Mur72} Murray W, (ed) +Numerical Methods for Unconstrained Optimization'' +Academic Press. (1972) \end{chunk} \begin{chunk}{ignore} -\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi -Floating point Gr\"obner bases'' -Mathematics and Computers in Simulation 42 pp 509-528 (1996) -%\verb|axiom-developer.org/axiom-website/papers/Shir96.pdf| - abstract = " - Bracket coefficients for polynomials are introduced. These are like - specific precision floating point numbers together with error - terms. Working in terms of bracket coefficients, an algorithm that - computes a Gr{\"o}bner basis with floating point coefficients is - presented, and a new criterion for determining whether a bracket - coefficient is zero is proposed. Given a finite set $F$ of polynomials - with real coefficients, let $G_\mu$ be the result of the algorithm for - $F$ and a precision $\mu$, and $G$ be a true Gr{\"o}bner basis of - $F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$ - coefficientwise. Moreover, there is a precision $M$ such that if - $\mu \ge M$, then the sets of monomials with non-zero coefficients of - $G_\mu$ and $G$ are exactly the same. The practical usefulness of the - algorithm is suggested by experimental results." +\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A +MINOS 5.0 User's Guide'' +Report SOL 83-20. Department of Operations Research, Stanford University 1983 \end{chunk} \begin{chunk}{ignore} -\bibitem[Sims 71]{Sims71} Sims, C. -Determining the Conjugacy Classes of a Permutation Group'' -Computers in Algebra and Number Theory, SIAM-AMS Proc., Vol. 4, -American Math. Soc., 1991, pp191-195 +\bibitem[Musser 78]{Mus78} Musser, David R. +On the Efficiency of a Polynomial Irreducibility Test'' +Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271-282 \end{chunk} +\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Singer 89]{Sing89} Singer, M.F. -Formal Solutions of Differential Equations'' -J. Symbolic COmputation 10, No.1 59-94 (1990) -%\verb|axiom-developer.org/axiom-website/papers/Sing89.pdf| - keywords = "survey", - abstract = " - We give a survey of some methods for finding formal solutions of - differential equations. These include methods for finding power series - solutions, elementary and liouvillian solutions, first integrals, Lie - theoretic methods, transform methods, asymptotic methods. A brief - discussion of difference equations is also included." +\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf +Combinatorical Algorithms'' +Academic Press, New York 1978. \end{chunk} \begin{chunk}{ignore} -\bibitem[Sit 92]{REF-Sit92} Sit, William -An Algorithm for Parametric Linear Systems'' -J. Sym. Comp., April 1992 +\bibitem[Nikolai 79]{Nik79} Nikolai P J. +Algorithm 538: Eigenvectors and eigenvalues of real generalized +symmetric matrices by simultaneous iteration'' +ACM Trans. Math. Softw. 5 118--125. (1979) + +\end{chunk} + +\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{axiom.bib} +@misc{OCAM14, + author = "unknown", + title = "The OCAML website", + url = "http://ocaml.org" +} \end{chunk} \begin{chunk}{ignore} -\bibitem[Smith 67]{Smi67} Smith B T. -ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method'' -Technical Report. Department of Computer Science, University of Toronto, -Canada. (1967) +\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin +Algorithms and methods in differential algebra'' +\verb|www.lix.polytechnique.fr/~moulin/papiers/atelier.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Olla94.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Smith 85]{Smi85} Smith G D. -Numerical Solution of Partial Differential Equations: Finite Difference -Methods'' -Oxford University Press (3rd Edition). (1985) +\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.; +Boisvert, Ronald F.; Clark, Charles W. (ed) +NIST Handbook of Mathematical Functions'' +(2010) Cambridge University Press ISBN 978-0-521-19225-5 \end{chunk} \begin{chunk}{ignore} -\bibitem[Sobol 74]{Sob74} Sobol I M. -The Monte Carlo Method'' -The University of Chicago Press. 1974 +\bibitem[OpenM]{OpenM}. +OpenMath Technical Overview'' +\verb|www.openmath.org/overview/technical.html| \end{chunk} \begin{chunk}{ignore} -\bibitem[Steele 90]{Ste90} Steele, Guy L. -Common Lisp The Language'' -Second Edition ISBN 1-55558-041-6 Digital Press (1990) +\bibitem[Ortega 70]{OR70} Ortega J M.; Rheinboldt W C. +Iterative Solution of Nonlinear Equations in Several Variables'' +Academic Press. (1970) \end{chunk} \begin{chunk}{axiom.bib} -@misc{Stic93, - author = "Stichtenoth, H.", - title = "Algebraic function fields and codes", - publisher = "Springer-Verlag", - year = "1993" +@misc{Ostr1845, + author = "Ostrogradsky. M.W.", + title = "De l'int\'{e}gration des fractions rationelles.", + journal = "Bulletin de la Classe Physico-Math\'{e}matiques de + l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,", + volume = "IV", + pages = "145-167,286-300", + year = "1845" } \end{chunk} -\begin{chunk}{ignore} -\bibitem[Stinson 90]{Stin90} Stinson, D.R. -Some observations on parallel Algorithms for fast exponentiation -in $GF(2^n)$'' -Siam J. Comp., Vol.19, No.4, pp.711-717, August 1990 -%\verb|axiom-developer.org/axiom-website/Stin90.pdf| - abstract = " - A normal basis represention in $GF(2^n)$ allows squaring to be - accomplished by a cyclic shift. Algorithms for multiplication in - $GF(2^n)$ using a normal basis have been studied by several - researchers. In this paper, algorithms for performing exponentiation - in $GF(2^n)$ using a normal basis, and how they can be speeded up by - using parallelization, are investigated." - -\end{chunk} +\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{chunk}{ignore} -\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D. -Gaussian Quadrature Formulas'' -Prentice-Hall. (1966) +\bibitem[Paige 75]{PS75} Paige C C.; Saunders M A. +Solution of Sparse Indefinite Systems of Linear Equations'' +SIAM J. Numer. Anal. 12 617--629. (1975) \end{chunk} \begin{chunk}{ignore} -\bibitem[Stroud 71]{Str71} Stroud A H. -Approximate Calculation of Multiple Integrals'' -Prentice-Hall 1971 +\bibitem[Paige 82a]{PS82a} Paige C C.; Saunders M A. +LSQR: An Algorithm for Sparse Linear Equations and Sparse Least-squares'' +ACM Trans. Math. Softw. 8 43--71. (1982) \end{chunk} \begin{chunk}{ignore} -\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A. -Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial -Differential Equations'' -ACM Trans. Math. Softw. 5 352--364. (1979) +\bibitem[Paige 82b]{PS82b} Paige C C.; Saunders M A. +ALGORITHM 583 LSQR: Sparse Linear Equations and Least-squares Problems'' +ACM Trans. Math. Softw. 8 195--209. (1982) \end{chunk} \begin{chunk}{ignore} -\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N. -Fast Poisson Solvers'' -Studies in Numerical Analysis. (ed G H Golub) -Mathematical Association of America. (1984) - -\end{chunk} - -\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{axiom.bib} -@book{Tait1890, - author = "Tait, P.G.", - title = "An Elementary Treatise on Quaternions", - publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse, - Ave Maria Lane", - year = "1890" -} +\bibitem[Parker 84]{Par84} Parker, R. A. +The Computer Calculation of Modular Characters (The Meat-Axe)'' +M. D. Atkinson (Ed.), Computational Group Theory +Academic Press, Inc., London 1984 \end{chunk} \begin{chunk}{ignore} -\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero -On the Notion of Inheritance'' -ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438-479 +\bibitem[Parlett 80]{Par80} Parlett B N. +The Symmetric Eigenvalue Problem'' +Prentice-Hall. 1980 \end{chunk} \begin{chunk}{ignore} -\bibitem[Temme 87]{Tem87} Temme N M. -On the Computation of the Incomplete Gamma Functions for Large Values of -the Parameters'' -Algorithms for Approximation. (ed J C Mason and M G Cox) -Oxford University Press. (1987) +\bibitem[Parnas 10]{PJ10} Parnas, David Lorge; Jin, Ying +Defining the meaning of tabular mathematical expressions'' +Science of Computer Programming V75 No.11 Nov 2010 pp980-1000 Elesevier \end{chunk} \begin{chunk}{ignore} -\bibitem[Temperton 83a]{Tem83a} Temperton C. -Self-sorting Mixed-radix Fast Fourier Transforms'' -J. Comput. Phys. 52 1--23. (1983) +\bibitem[Parnas 95]{PM95} Parnas, David Lorge; Madey, Jan +Functional Documents for Computer Systems'' +Science of Computer Programming V25 No.1 Oct 1995 pp41-61 Elesevier \end{chunk} \begin{chunk}{ignore} -\bibitem[Temperton 83b]{Tem83b} Temperton C. -Fast Mixed-Radix Real Fourier Transforms'' -J. Comput. Phys. 52 340--350. (1983) +\bibitem[Paul 81]{Paul81} Paul, Richard +Robot Manipulators'' +MIT Press 1981 \end{chunk} \begin{chunk}{axiom.bib} -@article{Thur94, - author = "Thurston, William P.", - title = "On Proof and Progress in Mathematics", - journal = "Bulletin AMS", - volume = "30", - number = "2", - month = "April", - year = "1994", - url = "http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf", - paper = "Thur94.pdf" +@book{Pear56, + author = "Pearcey, T.", + title = "Table of the Fresnel Integral", + publisher = "Cambridge University Press", + year = "1956" } \end{chunk} -\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Unknown 61]{Unk61} Unknown -Chebyshev-series'' -Modern Computing Methods -Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961 +\bibitem[Pereyra 79]{Per79} Pereyra V. +PASVA3: An Adaptive Finite-Difference Fortran Program for First Order +Nonlinear, Ordinary Boundary Problems'' +Codes for Boundary Value Problems in Ordinary Differential Equations. +Lecture Notes in Computer Science. +(ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76 +Springer-Verlag. (1979) \end{chunk} -\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L. -An Adaptive Algorithm for Numerical Integration over an N-dimensional -Cube'' -J. Comput. Appl. Math. 2 207--217. (1976) +\bibitem[Peters 67a]{Pet67a} Peters G. +NPL Algorithms Library'' +Document No. F2/03/A. (1967) \end{chunk} \begin{chunk}{ignore} -\bibitem[van Hoeij 94]{REF-vH94} van Hoeij, M. -An algorithm for computing an integral -basis in an algebraic function field'' -{\sl J. Symbolic Computation} -18(4):353-364, October 1994 +\bibitem[Peters 67b]{Pet67b} Peters G. +NPL Algorithms Library'' +Document No.F1/04/A (1967) \end{chunk} \begin{chunk}{ignore} -\bibitem[Van Loan 92]{Van92} Van Loan, C. -Computational Frameworks for the Fast Fourier Transform'' -SIAM Philadelphia. (1992) +\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H. +The Least-squares Problem and Pseudo-inverses'' +Comput. J. 13 309--316. (1970) \end{chunk} -\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R. -Finite Element Analysis and Application'' -Wiley. (1985) +\bibitem[Peters 71]{PW71} Peters G.; Wilkinson J H. +Practical Problems Arising in the Solution of Polynomial Equations'' +J. Inst. Maths Applics. 8 16--35. (1971) \end{chunk} \begin{chunk}{ignore} -\bibitem[Wang 92]{Wang92} Wang, D.M. -An implementation of the characteristic set method in Maple'' -Proc. DISCO'92 Bath, England +\bibitem[Pierce 82]{Pie82} R.S. Pierce +Associative Algebras'' +Graduate Texts in Mathematics 88 +Springer-Verlag, Heidelberg, 1982, ISBN 0-387-90693-2 \end{chunk} \begin{chunk}{ignore} -\bibitem[Ward 75]{War75} Ward, R C. -The Combination Shift QZ Algorithm'' -SIAM J. Numer. Anal. 12 835--853. 1975 - -\end{chunk} - -\begin{chunk}{axiom.bib} -@misc{Watt03, - author = "Watt, Stephen", - title = "Aldor", - url = "http://www.aldor.org", - year = "2003" -} +\bibitem[Piessens 73]{Pie73} Piessens R. +An Algorithm for Automatic Integration'' +Angewandte Informatik. 15 399--401. (1973) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Weil71, - author = "Weil, Andr\'{e}", - title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes", - year = "1971" -} +\begin{chunk}{ignore} +\bibitem[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M. +Integration of Functions having End-point Singularities'' +Angewandte Informatik. 16 65--68. (1974) \end{chunk} \begin{chunk}{ignore} -\bibitem[Weisstein]{Wein} Weisstein, Eric W. -Hypergeometric Function'' -MathWorld - A Wolfram Web Resource -\verb|mathworld.wolfram.com/HypergeometricFunction.html| +\bibitem[Piessens 75]{PB75} Piessens R.; Branders M. +Algorithm 002. Computation of Oscillating Integrals'' +J. Comput. Appl. Math. 1 153--164. (1975) \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Weit03, - author = "Weitz, E.", - title = "CL-WHO -Yet another Lisp markup language", - year = "2003", - url = "http://www.weitz.de/cl-who/" -} +\begin{chunk}{ignore} +\bibitem[Piessens 76]{PVRBM76} Piessens R.; Van Roy-Branders M.; Mertens I. +The Automatic Evaluation of Cauchy Principal Value Integrals'' +Angewandte Informatik. 18 31--35. 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(1982) +\bibitem[Powell 70]{Pow70} Powell M J D. +A Hybrid Method for Nonlinear Algebraic Equations'' +Numerical Methods for Nonlinear Algebraic Equations. +(ed P Rabinowitz) Gordon and Breach. 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(1986) + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{Puff09, + author = "Puffinware LLC", + title = "Singular Value Decomposition (SVD) Tutorial", + url = "http://www.puffinwarellc.com/p3a.htm" +} \end{chunk} +\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Wilkinson 65]{Wil65} Wilkinson J H. -The Algebraic Eigenvalue Problem'' - Oxford University Press. (1965) +\bibitem[Quintana-Orti 06]{QG06} Quintana-Orti, Gregorio; +van de Geijn, Robert +Improving the performance of reduction to Hessenberg form'' +ACM Transactions on Mathematical Software, 32(2):180-194, June 2006. \end{chunk} +\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{ignore} -\bibitem[Wilkinson 78]{Wil78} Wilkinson J H. -Singular Value Decomposition -- Basic Aspects'' -Numerical Software -- Needs and Availability. -(ed D A H Jacobs) Academic Press. (1978) +\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P. +Numerical Methods for Nonlinear Algebraic Equations'' +Gordon and Breach. (1970) \end{chunk} \begin{chunk}{ignore} -\bibitem[Wilkinson 79]{Wil79} Wilkinson J H. -Kronecker's Canonical Form and the QZ Algorithm'' -Linear Algebra and Appl. 28 285--303. 1979 +\bibitem[Ralston 65]{Ral65} Ralston A. +A First Course in Numerical Analysis'' +McGraw-Hill. 87--90. (1965) \end{chunk} \begin{chunk}{ignore} -\bibitem[Wisbauer 91]{Wis91} Wisbauer, R. -Bimodule Structure of Algebra'' -Lecture Notes Univ. Duesseldorf 1991 +\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya +A Gentle Introduction to Lyapunov Functions'' +ORSUM August 2003 +\verb|www.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Woerz-Busekros 80]{Woe80} Woerz-Busekros, A. -Algebra in Genetics'' -Lectures Notes in Biomathematics 36, Springer-Verlag, Heidelberg, 1980 +\bibitem[Ramsey 03]{Ra03} Ramsey, Norman +Noweb--A Simple, Extensible Tool for Literate Programming'' +\verb|www.eecs.harvard.edu/~nr/noweb| \end{chunk} \begin{chunk}{ignore} -\bibitem[Wolberg 67]{Wol67} Wolberg J R. -Prediction Analysis'' -Van Nostrand. (1967) +\bibitem[Redfield 27]{Red27} Redfield, J.H. +The Theory of Group-Reduced Distributions'' +American J. Math., 49 (1927) 433-455. \end{chunk} \begin{chunk}{ignore} -\bibitem[Wolfram 09]{Wo09} Wolfram Research -\verb|mathworld.wolfram.com/Quaternion.html| +\bibitem[Reinsch 67]{Rei67} Reinsch C H. +Smoothing by Spline Functions'' +Num. Math. 10 177--183. (1967) \end{chunk} \begin{chunk}{ignore} -\bibitem[Wu 87]{WU87} Wu, W.T. -A Zero Structure Theorem for polynomial equations solving'' -MM Research Preprints, 1987 +\bibitem[Renka 84]{Ren84} Renka R L. +Algorithm 624: Triangulation and Interpolation of Arbitrarily Distributed +Points in the Plane'' +ACM Trans. Math. Softw. 10 440--442. (1984) \end{chunk} \begin{chunk}{ignore} -\bibitem[Wynn 56]{Wynn56} Wynn P. -On a Device for Computing the $e_m(S_n )$ Transformation'' -Math. Tables Aids Comput. 10 91--96. (1956) +\bibitem[Renka 84]{RC84} Renka R L.; Cline A K. +A Triangle-based C Interpolation Method'' +Rocky Mountain J. Math. 14 223--237. (1984) \end{chunk} -\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{chunk}{ignore} -\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena -Applications of Hermite transform in computer algebra'' -\verb|www.imfm.si/preprinti/PDF/00835.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Zak02.pdf| - abstract = " - let $L$ be a linear differential operator with polynomial - coefficients. We show that there is an isomorphism of differential - operators ${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$ - (called the Hermite transform) on functions for which $({\bf - D_\alpha}{\bf L})f(x)=0$ implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We - present an algorithm that computes the Hermite transform of a rational - function and use it to find $n+1$ linearly independent solutions of - ${\bf L}y=0$ when $({\bf D_\alpha}{\bf L})f(x)=0$ has a rational - solution with $n$ distinct finite poles." +\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe +Free Lie Algebras'' +Oxford University Press, June 1993 ISBN 0198536798 \end{chunk} -\begin{chunk}{axiom.bib} -@misc{Zdan14, - author = "Zdancewic, Steve and Martin, Milo M.K.", - title = "Vellvm: Verifying the LLVM", - url = "http://www.cis.upenn.edu/~stevez/vellvm" -} +\begin{chunk}{ignore} +\bibitem[Reznick 93]{Rezn93} Reznick, Bruce +An Inequality for Products of Polynomials'' +Proc. AMS Vol 117 No 4 April 1993 +%\verb|axiom-developer.org/axiom-website/papers/Rezn93.pdf| \end{chunk} \begin{chunk}{ignore} -\bibitem[Zhi 97]{Zhi97} Zhi, Lihong -Optimal Algorithm for Algebraic Factoring'' -\verb|www.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf| -%\verb|axiom-developer.org/axiom-website/papers/Zhi97.pdf| +\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J. +Crafting a Repository of Knowledge Based on Transformation'' +\verb|www.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Rixx.pdf| abstract = " - This paper presents an optimized method for factoring multivariate - polynomials over algebraic extension fields which defined by an - irreducible ascending set. The basic idea is to convert multivariate - polynomials to univariate polynomials and algebraic extensions fields - to algebraic number fields by suitable integer substitutions, then - factorize the univariate polynomials over the algebraic number fields. - Finally, construct multivariate factors of the original polynomial by - Hensel lemma and TRUEFACTOR test. Some examples with timing are - included." + We describe the development of a repository of mathematical knowledge + based on transformation rules. The specific mathematical problem is + indefinite integration. It is important that the repository be not + confused with a look-up table. The database of transformation rules is + at present encoded in Mathematica, but this is only one convenient + form of the repository, and it could be readily translated into other + formats. The principles upon which the set of rules is compiled is + described. One important principle is minimality. The benefits of the + approach are illustrated with examples, and with the results of + comparisons with other approaches." \end{chunk} -\subsection{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{chunk}{axiom.bib} -@PhdThesis{Kalt82, - author = "Kaltofen, E.", - title = "On the complexity of factoring polynomials with integer - coefficients", - school = "RPI", - address = "Troy, N. Y.", - year = "1982", - month = "December", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf", - paper = "Kalt82.pdf" -} +\begin{chunk}{ignore} +\bibitem[Rich 10]{Ri10} Rich, Albert D. +Rule-based Mathematics'' +\verb|www.apmaths.uwo.ca/~arich| \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt82a, - author = "Kaltofen, E.", - title = "A polynomial-time reduction from bivariate to univariate - integral polynomial factorization", - booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.", - year = "1982", - pages = "57--64", - organization = "IEEE", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf", - paper = "Kalt82a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John +The identity problem for elementary functions and constants'' +ACM Proc. of ISSAC 94 pp285-290 ISBN 0-89791-638-7 \end{chunk} -\begin{chunk}{axiom.bib} -@InCollection{Kalt82b, - author = "Kaltofen, E.", - title = "Polynomial factorization", - editor = "B. Buchberger and G. Collins and R. Loos", - booktitle = "Computer Algebra", - edition = "2", - pages = "95--113", - publisher = "Springer-Verlag", - year = "1982", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz", - keywords = "survey", - paper = "Kalt82b.ps" -} +\begin{chunk}{ignore} +\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W. +Difference Methods for Initial-value Problems'' +Interscience (2nd Edition). (1967) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt83, - author = "Kaltofen, E.", - title = "On the complexity of finding short vectors in integer lattices", - booktitle = "Proc. EUROCAL '83", - series = "Lect. Notes Comput. Sci.", - year = "1983", - volume = "162", - pages = "236--244", - publisher = "Springer-Verlag", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf", - paper = "Kalt83.pdf" -} +\begin{chunk}{ignore} +\bibitem[Rioboo 92]{REF-Rio92} Rioboo, R. +Real algebraic closure of an ordered field, implementation in Axiom'' +In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover) +In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215. +0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt84, - author = "Kaltofen, E.", - title = "A Note on the {Risch} Differential Equation", - booktitle = "Proc. EUROSAM '84", - pages = "359--366", - crossref = "EUROSAM84", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz", - paper = "Kalt84.ps" -} +\begin{chunk}{ignore} +\bibitem[Rioboo 96]{Rio96} Rioboo, R. +Generic computation of the real closure of an ordered field'' +In Mathematics and Computers in Simulation Volume 42, Issue 4-6, +November 1996. \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt84a, - author = "Kaltofen, E. and Yui, N.", - title = "Explicit construction of the {Hilbert} class field of imaginary - quadratic fields with class number 7 and 11", - booktitle = "Proc. EUROSAM '84", - pages = "310--320", - crossref = "EUROSAM84", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz", - paper = "Kalt84a.ps" -} +\begin{chunk}{ignore} +\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels +Differential Algebra'' +AMS Colloquium Publications Volume 33 ISBN 978-0-8218-4638-4 \end{chunk} -\begin{chunk}{axiom.bib} -@TechReport{Kalt84b, - author = "Kaltofen, E.", - title = "The Algebraic Theory of Integration", - institution = "RPI", - address = "Dept. Comput. Sci., Troy, New York", - year = "1984", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf", - paper = "Kalt84b.pdf" -} - -\end{chunk} +\begin{chunk}{ignore} +\bibitem[Rote 01]{Rote01} Rote, G\"unter +Division-free algorithms for the determinant and the Pfaffian'' +in Computational Discrete Mathematics ISBN 3-540-42775-9 pp119-135 +\verb|page.mi.fu-berlin.de/rote/Papers/pdf/Division-free+algorithms.pdf| -\begin{chunk}{axiom.bib} -@InProceedings{Kalt85, - author = "Kaltofen, E.", - title = "Effective {Hilbert} Irreducibility", - booktitle = "Proc. EUROSAM '84", - pages = "275--284", - crossref = "EUROSAM84", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz", - paper = "Kalt85.ps" -} - \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt85a, - author = "Kaltofen, E.", - title = "Fast parallel absolute irreducibility testing", - journal = "Journal of Symbolic Computation", - year = "1985", - volume = "1", - number = "1", - pages = "57--67", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf", - paper = "Kalt85a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Rubey 07]{Rub07} Rubey, Martin +Formula Guessing with Axiom'' +April 2007 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt85b, - author = "Kaltofen, E.", - title = "Computing with polynomials given by straight-line programs {II}; - sparse factorization", - booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.", - year = "1985", - pages = "451--458", - organization = "IEEE", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz", - paper = "Kalt85b.ps" -} +\begin{chunk}{ignore} +\bibitem[Rutishauser 69]{Rut69} Rutishauser H. +Computational aspects of F L Bauer's simultaneous iteration method'' +Num. Math. 13 4--13. (1969) \end{chunk} -\begin{chunk}{axiom.bib} -@TechReport{Kalt85c, - author = "E. Kaltofen", - title = "Sparse Hensel lifting", - institution = "RPI", - address = "Dept. Comput. Sci., Troy, N. Y.", - year = "1985", - number = "85-12", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf", - paper = "Kalt85c.pdf" -} +\begin{chunk}{ignore} +\bibitem[Rutishauser 70]{Rut70} Rutishauser H. +Simultaneous iteration method for symmetric matrices'' +Num. Math. 16 205--223. (1970) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt85d, - author = "Kaltofen, E.", - title = "Sparse Hensel lifting", - booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2", - crossref = "EUROCAL85", - pages = "4--17", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf", - paper = "Kalt85d.pdf" -} +\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Schafer 66]{Sch66} Schafer, R.D. +An Introduction to Nonassociative Algebras'' +Academic Press, New York, 1966 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt85e, - author = "Kaltofen, E.", - title = "Polynomial-time reductions from multivariate to bi- and univariate - integral polynomial factorization", - journal = "{SIAM} J. Comput.", - year = "1985", - volume = "14", - number = "2", - pages = "469--489", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf", - paper = "Kalt85e.pdf" -} +\begin{chunk}{ignore} +\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A. +On Polya Frequency Functions III'' +Trans. Amer. Math. Soc. 74 246--259. (1953) \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Gath85, - author = "Gathen, Joachim von zur; Kaltofen, E.", - title = "Factoring sparse multivariate polynomials", - journal = "J. Comput. System Sci.", - year = "1985", - volume = "31", - pages = "265--287", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz", - paper = "Gath85.ps" -} +\begin{chunk}{ignore} +\bibitem[Schoenhage 82]{Sch82} Schoenhage, A. +The fundamental theorem of algebra in terms of computational complexity'' +preliminary report, Univ. Tuebingen, 1982 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt86, - author = "Kaltofen, E.", - title = "Uniform closure properties of p-computable functions", - booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.", - year = "1986", - pages = "330--337", - organization = "ACM", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf", - paper = "Kalt86.pdf" -} +\begin{chunk}{ignore} +\bibitem[Schonfelder 76]{Sch76} Schonfelder J L. +The Production of Special Function Routines for a Multi-Machine Library'' +Software Practice and Experience. 6(1) (1976) \end{chunk} \begin{chunk}{axiom.bib} -@Article{Kalt87, - author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.", - title = "Fast parallel computation of Hermite and Smith forms of - polynomial matrices", - journal = "SIAM J. Alg. Discrete Math.", - year = "1987", - volume = "8", - pages = "683--690", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf", - paper = "Kalt87.pdf" +@book{Segg93, + author = "{von Seggern}, David Henry", + title = "CRC Standard Curves and Surfaces", + publisher = "CRC Press", + year = "1993", + isbn = "0-8493-0196-3" } \end{chunk} -\begin{chunk}{axiom.bib} -@InCollection{Kalt87a, - author = "Kaltofen, E.", - editor = "J. F. Traub", - title = "Computer algebra algorithms", - booktitle = "Annual Review in Computer Science", - pages = "91--118", - publisher = "Annual Reviews Inc.", - year = "1987", - volume = "2", - address = "Palo Alto, California", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf", - paper = "Kalt87a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J. +JET -- An Axiom Environment for Geometric Computations with Differential +Equations'' +%\verb|axiom-developer.org/axiom-website/papers/Sei95a.pdf| \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt87b, - author = "Kaltofen, E.", - title = "Single-factor Hensel lifting and its application to the - straight-line complexity of certain polynomials", - booktitle = "Proc. 19th Annual ACM Symp. Theory Comput.", - year = "1987", - pages = "443--452", - organization = "ACM", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf", - paper = "Kalt87b.pdf" -} +\begin{chunk}{ignore} +\bibitem[Shepard 68]{She68} Shepard D. +A Two-dimensional Interpolation Function for Irregularly Spaced Data'' +Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc., +Princeton. 517--523. 1968 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt87c, - author = "Kaltofen, E.", - title = "Deterministic irreducibility testing of polynomials over - large finite fields", - journal = "Journal of Symbolic Computation", - year = "1987", - volume = "4", - pages = "77--82", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz", - paper = "Kalt87c.ps" -} +\begin{chunk}{ignore} +\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi +Floating point Gr\"obner bases'' +Mathematics and Computers in Simulation 42 pp 509-528 (1996) +%\verb|axiom-developer.org/axiom-website/papers/Shir96.pdf| + abstract = " + Bracket coefficients for polynomials are introduced. These are like + specific precision floating point numbers together with error + terms. Working in terms of bracket coefficients, an algorithm that + computes a Gr{\"o}bner basis with floating point coefficients is + presented, and a new criterion for determining whether a bracket + coefficient is zero is proposed. Given a finite set $F$ of polynomials + with real coefficients, let $G_\mu$ be the result of the algorithm for + $F$ and a precision $\mu$, and $G$ be a true Gr{\"o}bner basis of + $F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$ + coefficientwise. Moreover, there is a precision $M$ such that if + $\mu \ge M$, then the sets of monomials with non-zero coefficients of + $G_\mu$ and $G$ are exactly the same. The practical usefulness of the + algorithm is suggested by experimental results." \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt88, - author = "Kaltofen, E. and Trager, B.", - title = "Computing with polynomials given by black boxes for their - evaluations: Greatest common divisors, factorization, separation of - numerators and denominators", - booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.", - pages = "296--305", - year = "1988", - organization = "IEEE", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz", - paper = "Kalt88.ps" -} +\begin{chunk}{ignore} +\bibitem[Sims 71]{Sims71} Sims, C. +Determining the Conjugacy Classes of a Permutation Group'' +Computers in Algebra and Number Theory, SIAM-AMS Proc., Vol. 4, +American Math. Soc., 1991, pp191-195 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Mill88, - author = "Miller, G.L. and Ramachandran, V. and Kaltofen, E.", - title = "Efficient parallel evaluation of straight-line code and - arithmetic circuits", - journal = "SIAM J. Comput.", - year = "1988", - volume = "17", - number = "4", - pages = "687--695", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf", - paper = "Mill88.pdf" -} +\begin{chunk}{ignore} +\bibitem[Singer 89]{Sing89} Singer, M.F. +Formal Solutions of Differential Equations'' +J. Symbolic COmputation 10, No.1 59-94 (1990) +%\verb|axiom-developer.org/axiom-website/papers/Sing89.pdf| + keywords = "survey", + abstract = " + We give a survey of some methods for finding formal solutions of + differential equations. These include methods for finding power series + solutions, elementary and liouvillian solutions, first integrals, Lie + theoretic methods, transform methods, asymptotic methods. A brief + discussion of difference equations is also included." \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt88a, - author = "Kaltofen, E. and Yagati, Lakshman", - title = "Improved sparse multivariate polynomial interpolation algorithms", - booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.", - crossref = "ISSAC88", - pages = "467--474", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf", - paper = "Kalt88a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Sit 92]{REF-Sit92} Sit, William +An Algorithm for Parametric Linear Systems'' +J. Sym. Comp., April 1992 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt88b, - author = "Kaltofen, E.", - title = "Greatest common divisors of polynomials given by - straight-line programs", - journal = "J. ACM", - year = "1988", - volume = "35", - number = "1", - pages = "231--264", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf", - paper = "Kalt88b.pdf" -} +\begin{chunk}{ignore} +\bibitem[Smith 67]{Smi67} Smith B T. +ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method'' +Technical Report. Department of Computer Science, University of Toronto, +Canada. (1967) \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Free88, - author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and - Yagati, Lakshman", - title = "DAGWOOD: A system for manipulating polynomials given by - straight-line programs", - journal = "ACM Trans. Math. Software", - year = "1988", - volume = "14", - number = "3", - pages = "218--240", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf", - paper = "Free88.pdf" -} +\begin{chunk}{ignore} +\bibitem[Smith 85]{Smi85} Smith G D. +Numerical Solution of Partial Differential Equations: Finite Difference +Methods'' +Oxford University Press (3rd Edition). (1985) \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Greg88, - author = "Gregory, B.; Kaltofen, E.", - title = "Analysis of the binary complexity of asymptotically fast - algorithms for linear system solving", - journal = "SIGSAM Bulletin", - year = "1988", - month = "April", - volume = "22", - number = "2", - pages = "41--49", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf", - paper = "Grey88.pdf" -} +\begin{chunk}{ignore} +\bibitem[Sobol 74]{Sob74} Sobol I M. +The Monte Carlo Method'' +The University of Chicago Press. 1974 \end{chunk} -\begin{chunk}{axiom.bib} -@InCollection{Kalt89, - author = "Kaltofen, E.", - editor = "S. Micali", - title = "Factorization of polynomials given by straight-line programs", - booktitle = "Randomness and Computation", - pages = "375--412", - publisher = "JAI Press Inc.", - year = "1989", - volume = "5", - series = "Advances in Computing Research", - address = "Greenwhich, Connecticut", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf", - paper = "Kalt89.pdf" -} +\begin{chunk}{ignore} +\bibitem[Steele 90]{Ste90} Steele, Guy L. +Common Lisp The Language'' +Second Edition ISBN 1-55558-041-6 Digital Press (1990) \end{chunk} \begin{chunk}{axiom.bib} -@Article{Kalt89a, - author = "Kaltofen, E.; Rolletschek, H.", - title = "Computing greatest common divisors and factorizations in - quadratic number fields", - journal = "Math. Comput.", - year = "1989", - volume = "53", - number = "188", - pages = "697--720", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf", - paper = "Kalt89a.pdf" +@misc{Stic93, + author = "Stichtenoth, H.", + title = "Algebraic function fields and codes", + publisher = "Springer-Verlag", + year = "1993" } \end{chunk} -\begin{chunk}{axiom.bib} -@Unpublished{Kalt89b, - author = "Kaltofen, E.", - title = "Processor efficient parallel computation of polynomial greatest - common divisors", - year = "1989", - month = "July", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_gcd.ps.gz", - paper = "Kalt89b.ps" -} +\begin{chunk}{ignore} +\bibitem[Stinson 90]{Stin90} Stinson, D.R. +Some observations on parallel Algorithms for fast exponentiation +in $GF(2^n)$'' +Siam J. Comp., Vol.19, No.4, pp.711-717, August 1990 +%\verb|axiom-developer.org/axiom-website/Stin90.pdf| + abstract = " + A normal basis represention in $GF(2^n)$ allows squaring to be + accomplished by a cyclic shift. Algorithms for multiplication in + $GF(2^n)$ using a normal basis have been studied by several + researchers. In this paper, algorithms for performing exponentiation + in $GF(2^n)$ using a normal basis, and how they can be speeded up by + using parallelization, are investigated." \end{chunk} -\begin{chunk}{axiom.bib} -@TechReport{Kalt89c, - author = "Kaltofen, E.", - title = "Parallel Algebraic Algorithm Design", - institution = "RPI", - address = "Dept. Comput. Sci., Troy, New York", - year = "1989", - month = "July", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz", - paper = "Kalt89c.ps" -} +\begin{chunk}{ignore} +\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D. +Gaussian Quadrature Formulas'' +Prentice-Hall. (1966) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Cann89, - author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman", - title = "Solving systems of non-linear polynomial equations faster", - booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC89", - pages = "121--128", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf", - paper = "Cann89.pdf" -} +\begin{chunk}{ignore} +\bibitem[Stroud 71]{Str71} Stroud A H. +Approximate Calculation of Multiple Integrals'' +Prentice-Hall 1971 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt89d, - author = "Kaltofen, E. and Valente, T. and Yui, N.", - title = "An improved {Las Vegas} primality test", - booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC89", - pages = "26--33", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf", - paper = "Kalt89d.pdf" -} +\begin{chunk}{ignore} +\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A. +Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial +Differential Equations'' +ACM Trans. Math. Softw. 5 352--364. (1979) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt90, - author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.", - editor = "S. Watanabe and M. Nagata", - title = "Modular rational sparse multivariate polynomial interpolation", - booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.", - pages = "135--139", - publisher = "ACM Press", - year = "1990", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf", - paper = "Kalt90.pdf" -} +\begin{chunk}{ignore} +\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N. +Fast Poisson Solvers'' +Studies in Numerical Analysis. (ed G H Golub) +Mathematical Association of America. (1984) \end{chunk} +\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{chunk}{axiom.bib} -@Article{Kalt90a, - author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.", - title = "Parallel algorithms for matrix normal forms", - journal = "Linear Algebra and Applications", - year = "1990", - volume = "136", - pages = "189--208", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf", - paper = "Kalt90a.pdf" +@book{Tait1890, + author = "Tait, P.G.", + title = "An Elementary Treatise on Quaternions", + publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse, + Ave Maria Lane", + year = "1890" } \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt90b, - author = "Kaltofen, E.", - title = "Computing the irreducible real factors and components of an - algebraic curve", - journal = "Applic. Algebra Engin. Commun. Comput.", - year = "1990", - volume = "1", - number = "2", - pages = "135--148", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf", - paper = "Kalt90b.pdf" -} +\begin{chunk}{ignore} +\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero +On the Notion of Inheritance'' +ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438-479 \end{chunk} -\begin{chunk}{axiom.bib} -@InCollection{Kalt90c, - author = "Kaltofen, E.", - editor = "D. V. Chudnovsky and R. D. Jenks", - title = "Polynomial Factorization 1982-1986", - booktitle = "Computers in Mathematics", - pages = "285--309", - publisher = "Marcel Dekker, Inc.", - year = "1990", - volume = "125", - series = "Lecture Notes in Pure and Applied Mathematics", - address = "New York, N. Y.", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz", - keywords = "survey", - paper = "Kalt90c.ps" -} +\begin{chunk}{ignore} +\bibitem[Temme 87]{Tem87} Temme N M. +On the Computation of the Incomplete Gamma Functions for Large Values of +the Parameters'' +Algorithms for Approximation. (ed J C Mason and M G Cox) +Oxford University Press. (1987) \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt90d, - author = "Kaltofen, E.; Trager, B.", - title = "Computing with polynomials given by black boxes for their - evaluations: Greatest common divisors, factorization, separation of - numerators and denominators", - journal = "J. Symbolic Comput.", - year = "1990", - volume = "9", - number = "3", - pages = "301--320", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf", - paper = "Kalt90d.pdf" -} +\begin{chunk}{ignore} +\bibitem[Temperton 83a]{Tem83a} Temperton C. +Self-sorting Mixed-radix Fast Fourier Transforms'' +J. Comput. Phys. 52 1--23. (1983) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt91, - author = "Kaltofen, E. and Saunders, B.D.", - editor = "H. F. Mattson and T. Mora and T. R. N. Rao", - title = "On {Wiedemann's} method of solving sparse linear systems", - booktitle = "Proc. AAECC-9", - series = "Lect. Notes Comput. Sci.", - volume = "539", - pages = "29--38", - publisher = "Springer-Verlag", - year = "1991", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf", - paper = "Kalt91.pdf" -} +\begin{chunk}{ignore} +\bibitem[Temperton 83b]{Tem83b} Temperton C. +Fast Mixed-Radix Real Fourier Transforms'' +J. Comput. Phys. 52 340--350. (1983) \end{chunk} \begin{chunk}{axiom.bib} -@InProceedings{Kalt91a, - author = "Kaltofen, E. and Singer, M.F.", - editor = "D. V. Shirkov and V. A. Rostovtsev and V. P. Gerdt", - title = "Size efficient parallel algebraic circuits for partial derivatives", - booktitle = - "IV International Conference on Computer Algebra in Physical Research", - pages = "133--145", - publisher = "World Scientific Publ. Co.", - year = "1991", - address = "Singapore", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf", - paper = "Kalt91a.pdf" +@article{Thur94, + author = "Thurston, William P.", + title = "On Proof and Progress in Mathematics", + journal = "Bulletin AMS", + volume = "30", + number = "2", + month = "April", + year = "1994", + url = "http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf", + paper = "Thur94.pdf" } \end{chunk} -\begin{chunk}{axiom.bib} -@InCollection{Kalt91b, - author = "Kaltofen, E. and Yui, N.", - editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and - M. B. Nathanson", - title = "Explicit construction of {Hilbert} class fields of imaginary - quadratic fields by integer lattice reduction", - booktitle = "Number Theory New York Seminar 1989--1990", - pages = "150--202", - publisher = "Springer-Verlag", - year = "1991", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf", - paper = "Kalt91b.pdf" -} +\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Unknown 61]{Unk61} Unknown +Chebyshev-series'' +Modern Computing Methods +Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Diaz91, - author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.", - title = "DSC A System for Distributed Symbolic Computation", - booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC91", - pages = "323--332", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf", - paper = "Diaz91.pdf" -} +\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L. +An Adaptive Algorithm for Numerical Integration over an N-dimensional +Cube'' +J. Comput. Appl. Math. 2 207--217. (1976) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt91c, - author = "Kaltofen, E. and Pan, V.", - title = "Processor efficient parallel solution of linear systems over - an abstract field", - booktitle = "Proc. SPAA '91 3rd Ann. ACM Symp. Parallel Algor. Architecture", - pages = "180--191", - publisher = "ACM Press", - year = "1991", - address = "New York, N.Y.", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf", - paper = "Kalt91c.pdf" -} +\begin{chunk}{ignore} +\bibitem[van Hoeij 94]{REF-vH94} van Hoeij, M. +An algorithm for computing an integral +basis in an algebraic function field'' +{\sl J. Symbolic Computation} +18(4):353-364, October 1994 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Cant91, - author = "Cantor, D.G. and Kaltofen, E.", - title = "On fast multiplication of polynomials over arbitrary algebras", - journal = "Acta Inform.", - year = "1991", - volume = "28", - number = "7", - pages = "693--701", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf", - paper = "Cant91.pdf" -} +\begin{chunk}{ignore} +\bibitem[Van Loan 92]{Van92} Van Loan, C. +Computational Frameworks for the Fast Fourier Transform'' +SIAM Philadelphia. (1992) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt92, - author = "Kaltofen, E. and Pan, V.", - title = "Processor-efficient parallel solution of linear systems {II}: - the positive characteristic and singular cases", - booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.", - year = "1992", - pages = "714--723", - publisher = "IEEE Computer Society Press", - address = "Los Alamitos, California", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf", - paper = "Kalt92.pdf" -} +\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R. +Finite Element Analysis and Application'' +Wiley. (1985) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt92a, - author = "Kaltofen, E.", - title = "On computing determinants of matrices without divisions", - booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC92", - pages = "342--349", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf", - paper = "Kalt92a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Wang 92]{Wang92} Wang, D.M. +An implementation of the characteristic set method in Maple'' +Proc. DISCO'92 Bath, England \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt92b, - author = "Kaltofen, E.", - title = "Polynomial factorization 1987-1991", - booktitle = "Proc. LATIN '92", - editor = "I. Simon", - series = "Lect. Notes Comput. Sci.", - volume = "583", - pages = "294--313", - publisher = "Springer-Verlag", - year = "1992", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf", - keywords = "survey", - paper = "Kalt92b.pdf" -} +\begin{chunk}{ignore} +\bibitem[Ward 75]{War75} Ward, R C. +The Combination Shift QZ Algorithm'' +SIAM J. Numer. Anal. 12 835--853. 1975 \end{chunk} \begin{chunk}{axiom.bib} -@InProceedings{Kalt93, - author = "Kaltofen, E.", - title = "Computational Differentiation and Algebraic Complexity Theory", - booktitle = "Workshop Report on First Theory Institute on Computational - Differentiation", - editor = "C. H. Bischof and A. Griewank and P. M. Khademi", - publisher = "Argonne National Laboratory", - address = "Argonne, Illinois", - series = "Tech. Rep.", - volume = "ANL/MCS-TM-183", - month = "December", - year = "1993", - pages = "28--30", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf", - paper = "Kalt93.pdf" +@misc{Watt03, + author = "Watt, Stephen", + title = "Aldor", + url = "http://www.aldor.org", + year = "2003" } \end{chunk} \begin{chunk}{axiom.bib} -@InCollection{Kalt93a, - author = "Kaltofen, E.", - editor = "J. Reif", - title = "Dynamic parallel evaluation of computation {DAG}s", - booktitle = "Synthesis of Parallel Algorithms", - pages = "723--758", - publisher = "Morgan Kaufmann Publ.", - year = "1993", - address = "San Mateo, California", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz", - paper = "Kalt93a.ps" +@misc{Weil71, + author = "Weil, Andr\'{e}", + title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes", + year = "1971" } \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Diaz93, - author = "Diaz, A. and Kaltofen, E. and Lobo, A. and Valente, T.", - editor = "A. Miola", - title = "Process scheduling in {DSC} and the large sparse linear - systems challenge", - booktitle = "Proc. DISCO '93", - series = "Lect. Notes Comput. Sci.", - pages = "66--80", - year = "1993", - volume = "722", - publisher = "Springer-Verlag", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf", - paper = "Diaz93.pdf" -} +\begin{chunk}{ignore} +\bibitem[Weisstein]{Wein} Weisstein, Eric W. +Hypergeometric Function'' +MathWorld - A Wolfram Web Resource +\verb|mathworld.wolfram.com/HypergeometricFunction.html| \end{chunk} \begin{chunk}{axiom.bib} -@Article{Kalt93b, - author = "Kaltofen, E.", - title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor", - journal = "SIGSAM Bulletin", - year = "1993", - volume = "27", - number = "4", - pages = "2", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz", - paper = "Kalt93b.ps" +@misc{Weit03, + author = "Weitz, E.", + title = "CL-WHO -Yet another Lisp markup language", + year = "2003", + url = "http://www.weitz.de/cl-who/" } \end{chunk} \begin{chunk}{axiom.bib} -@InProceedings{Kalt94, - author = "Kaltofen, E. and Pan, V.", - title = "Parallel solution of Toeplitz and Toeplitz-like linear - systems over fields of small positive characteristic", - booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.", - crossref = "PASCO94", - pages = "225--233", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf", - paper = "Kalt94.pdf" +@misc{Weit06, + author = "Weitz, E.", + title = "HUNCHENTOOT - The Common Lisp web server formerly known as TBNL", + year = "2006", + url = "http://www.weitz.de/hunchentoot" } \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Chan94, - author = "Chan, K.C. and Diaz, A. and Kaltofen, E.", - editor = "R. J. Lopez", - title = "A distributed approach to problem solving in Maple", - booktitle = "Maple V: Mathematics and its Application", - pages = "13--21", - publisher = {Birkh\"auser}, - year = "1994", - series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)", - address = "Boston", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz", - paper = "Chan94.ps" -} +\begin{chunk}{ignore} +\bibitem[Wesseling 82a]{Wes82a} Wesseling, P. +MGD1 - A Robust and Efficient Multigrid Method'' +Multigrid Methods. Lecture Notes in Mathematics. 960 +Springer-Verlag. 614--630. (1982) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt94a, - author = "Kaltofen, E. and Lobo, A.", - title = "Factoring high-degree polynomials by the black box - Berlekamp algorithm", - booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC94", - pages = "90--98", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz", - paper = "Kalt94a.ps" -} +\begin{chunk}{ignore} +\bibitem[Wesseling 82b]{Wes82b} Wesseling, P. +Theoretical Aspects of a Multigrid Method'' +SIAM J. Sci. Statist. Comput. 3 387--407. (1982) \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt94b, - author = "Kaltofen, E.", - title = "Asymptotically fast solution of {Toeplitz}-like singular - linear systems", - booktitle = "Proc. 1994 Internat. Symp. 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Tentner", - pages = "244--247", - organization = "Society for Computer Simulation", - publisher = "Simulation Councils, Inc.", - address = "San Diego, CA", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf", - paper = "Kalt96a.pdf" -} +\begin{chunk}{ignore} +\bibitem[Wisbauer 91]{Wis91} Wisbauer, R. +Bimodule Structure of Algebra'' +Lecture Notes Univ. Duesseldorf 1991 \end{chunk} -\begin{chunk}{axiom.bib} -@InProceedings{Kalt96b, - author = "Kaltofen, E.", - title = "Blocked iterative sparse linear system solvers for finite fields", - booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular - Applic. (Stratagem '96)", - editor = "C. Roucairol", - publisher = "INRIA", - address = "Sophia Antipolis, France", - pages = "91--95", - year = "1996", - url = - "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz", - paper = "Kalt96b.ps" -} +\begin{chunk}{ignore} +\bibitem[Woerz-Busekros 80]{Woe80} Woerz-Busekros, A. +Algebra in Genetics'' +Lectures Notes in Biomathematics 36, Springer-Verlag, Heidelberg, 1980 \end{chunk} -\begin{chunk}{axiom.bib} -@Article{Kalt97, - author = "E. Kaltofen", - title = "Teaching Computational Abstract Algebra", - journal = "Journal of Symbolic Computation", - volume = "23", - number = "5-6", - pages = "503--515", - year = "1997", - note = "Special issue on education, L. Lambe, editor.", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/Ka97_jsc.pdf", - keywords = "axiomref,read", - paper = "Kalt97.pdf", +\begin{chunk}{ignore} +\bibitem[Wolberg 67]{Wol67} Wolberg J R. +Prediction Analysis'' +Van Nostrand. (1967) + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Wolfram 09]{Wo09} Wolfram Research +\verb|mathworld.wolfram.com/Quaternion.html| + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Wu 87]{WU87} Wu, W.T. +A Zero Structure Theorem for polynomial equations solving'' +MM Research Preprints, 1987 + +\end{chunk} + +\begin{chunk}{ignore} +\bibitem[Wynn 56]{Wynn56} Wynn P. +On a Device for Computing the $e_m(S_n )$ Transformation'' +Math. Tables Aids Comput. 10 91--96. (1956) + +\end{chunk} + +\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{chunk}{ignore} +\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena +Applications of Hermite transform in computer algebra'' +\verb|www.imfm.si/preprinti/PDF/00835.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Zak02.pdf| abstract = " - We report on the contents and pedagogy of a course in abstract algebra - that was taught with the aid of educational software developed within - the Mathematica system. We describe the topics covered and the - didactical use of the corresponding Mathematica packages, as well as - draw conclusions for future such courses from the students' comments - and our own experience." -} + let $L$ be a linear differential operator with polynomial + coefficients. We show that there is an isomorphism of differential + operators ${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$ + (called the Hermite transform) on functions for which $({\bf + D_\alpha}{\bf L})f(x)=0$ implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We + present an algorithm that computes the Hermite transform of a rational + function and use it to find $n+1$ linearly independent solutions of + ${\bf L}y=0$ when $({\bf D_\alpha}{\bf L})f(x)=0$ has a rational + solution with $n$ distinct finite poles." \end{chunk} \begin{chunk}{axiom.bib} -@InProceedings{Kalt97a, - author = "Kaltofen, E. and Shoup, V.", - title = "Fast polynomial factorization over high algebraic extensions of - finite fields", - booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.", - crossref = "ISSAC97", - pages = "184--188", - url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf", - paper = "Kalt97a.pdf" +@misc{Zdan14, + author = "Zdancewic, Steve and Martin, Milo M.K.", + title = "Vellvm: Verifying the LLVM", + url = "http://www.cis.upenn.edu/~stevez/vellvm" } \end{chunk} \begin{chunk}{ignore} +\bibitem[Zhi 97]{Zhi97} Zhi, Lihong +`Optimal Algorithm for Algebraic Factoring'' +\verb|www.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf| +%\verb|axiom-developer.org/axiom-website/papers/Zhi97.pdf| + abstract = " + This paper presents an optimized method for factoring multivariate + polynomials over algebraic extension fields which defined by an + irreducible ascending set. The basic idea is to convert multivariate + polynomials to univariate polynomials and algebraic extensions fields + to algebraic number fields by suitable integer substitutions, then + factorize the univariate polynomials over the algebraic number fields. + Finally, construct multivariate factors of the original polynomial by + Hensel lemma and TRUEFACTOR test. Some examples with timing are + included." + \end{chunk} + \eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Bibliography} diff --git a/changelog b/changelog index 14f3319..0dac261 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20140920 tpd src/axiom-website/patches.html 20140920.02.tpd.patch +20140920 tpd books/bookvolbib add abstracts, rearrange, add new sections 20140920 tpd src/axiom-website/patches.html 20140920.01.tpd.patch 20140920 tpd books/bookvolbib add abstracts, rearrange, add new entries 20140919 tpd src/axiom-website/patches.html 20140919.01.tpd.patch diff --git a/patch b/patch index a644f10..73fef81 100644 --- a/patch +++ b/patch @@ -1,3 +1,3 @@ -books/bookvolbib add abstracts, rearrange, add new entries +books/bookvolbib add abstracts, rearrange, add new sections Expand and cleanup bibliography diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 5e0a8d3..6c4d579 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4648,6 +4648,8 @@ books/bookvolbib add references
books/axiom.bst use axiom specific bib style