A Critique of the Mathematical Abilities of CA Systems

An earlier review of the same ilk is A Review of CAS Mathematical Capabilities. Also see a Comparison of Polynomial-Oriented Computer Algebra Systems for some timing studies.

A 14-page table of CAS synonyms.

Results:

Boolean Logic and Quantifier Elimination

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Set Theory

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Numbers

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Numerical Analysis

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Statistics

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Combinatorial Theory

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Number Theory

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Algebra

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Trigonometry

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Special Functions

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

The Complex Domain

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Determining Zero Equivalence

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Equations

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Inequalities

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Vector Analysis

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Matrix Theory

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Tensor Analysis

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Sums

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Products

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Limits

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Calculus

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Indefinite Integrals

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Definite Integrals

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Series

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Transforms

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Axiom,

Ordinary Difference and Differential Equations

Axiom, Derive, Derive (ODE), Macsyma, Maple, Mathematica, MuPAD, Reduce

Partial Differential Equations

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce (Heat equation)

Operators

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Programming and Miscellaneous

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce (differentiation)

Mathematics vs Computer Science

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce (local variables)

Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce (matching)

References:

[Asl96]

Helmer Aslaksen, ``Multiple-valued complex functions and computer algebra'', SIGSAM Bulletin, Volume 30, Number 2, June 1996, 12-20. (original)

[Ber96]

Laurent Bernardin, ``A Review of Symbolic Solvers'', SIGSAM Bulletin, Volume 30, Number 1, March 1996, 9-20. (original)

[Fat96]

Richard J. Fateman, ``Symbolic Mathematics System Evaluators'', Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, 1996, 86-94. (original)

[Fau95]

J. C. Faugère, ``GB: State of GB + Tutorial, Proceedings of the POSSO Workshop on Software, draft version, edited by Jean-Charles Faugère, Joel Marchand and Renaud Rioboo, Paris, March 1-4, 1995, 55-60.

[Grä95]

Hans-Gert Gräbe, ``On Factorized Gröbner Bases'', Computer Algebra in Science and Engineering, edited by Fleischer, Grabmeier, Hehl and Küchlin, World Scientific Singapore 1995, 77-89. (original)

[Grä96]

Hans-Gert Gräbe, About the Polynomial System Solve Facility of Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce, Preprint 11/96 des Instituts für Informatik, Universität Leipzig, Germany, December 1996. (original)

[Gru96]

Dominik Gruntz, On Computing Limits in a Symbolic Manipulation System, Ph.D. dissertation, Swiss Federal Institute of Technology, Zürich, Switzerland, Diss. ETH No. 11432, 1996. (original)

[Har91]

David Harper, Chris Wooff and David Hodgkinson, A Guide to COMPUTER ALGEBRA SYSTEMS, John Wiley & Sons, 1991.

[Her94]

W. Hereman, ``Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations'', Euromath Bulletin, Volume 1, Number 2, 1994, 45-79.

[Her95]

Willy Hereman, ``Visual data analysis: maths made easy'', Physics World, Volume 8, Number 4, April 1995, 49-53.

[Her96]

Willy Hereman, ``Computer algebra: lightening the load'', Physics World, Volume 9, Number 3, March 1996, 47-52.

[Koe95]

Wolfram Koepf, ``Efficient Computation of Orthogonal Polynomials in Computer Algebra'', Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Preprint SC 95-42, December 1995. (original)

[Pos96]

Frank Postel and Paul Zimmermann, ``A review of the ODE solvers of Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD and Reduce'', submitted to the 5th Rhine Workshop on Computer Algebra to be held in Saint-Louis, France, April 1-3, 1996. (original)

[Rob93]

Nicolas Robidoux, ``Does Axiom Solve Systems of O.D.E.'s Like Mathematica?'', LA-UR-93-2235, Los Alamos National Laboratory, Los Alamos, New Mexico, 1993.

[Sim92]

Barry Simon, ``Comparative CAS Reviews'', Notices of the American Mathematical Society, Volume 39, Number 7, September 1992, 700-710.

[Sim95]

Barry Simon, ``Symbolic Math Powerhouses'', Desktop Engineering, Volume 1, Issue 1, September/October 1995, 42-51.

[Sim97]

Barry Simon, ``Symbolic Math Powerhouses Revisited'', Desktop Engineering, Volume 2, Issue 7, March 1997, 12-25, 52-53.

[Sto91]

David R. Stoutemyer, ``Crimes and Misdemeanors in the Computer Algebra Trade'', Notices of the American Mathematical Society, Volume 38, Number 7, September 1991, 778-785.

[Wes94]

Michael Wester, ``A Review of CAS Mathematical Capabilities'', Computer Algebra Nederland Nieuwsbrief, Number 13, December 1994, ISSN 1380-1260, 41-48 (newer version of the paper below).

[Wes95]

Michael Wester, ``A Review of CAS Mathematical Capabilities'', Applied Mechanics in the Americas, Volume III, edited by Luis A. Godoy, Sergio R. Idelsohn, Patricio A. A. Laura and Dean T. Mook, American Academy of Mechanics and Asociacion Argentina de Mecanica Computacional, Santa Fe, Argentina, 1995, 450-455.

[Zim95]

Paul Zimmermann, ``Wester's test suite in MuPAD 1.2.2'', Computer Algebra Nederland Nieuwsbrief, Number 14, April 1995, ISSN 1380-1260, 53-64. (original)

[Zim96]

Paul Zimmermann, ``Wester's test suite in MuPAD 1.3'', The SAC Newsletter, Number 1, December 1996, 53-69. (original)

Several of the above papers were presented in the ACA'96 session on Comparative CAS Reviews and Philosophy.

Michael Wester's home page