Computer algebra system

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A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.

Contents

Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.

General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as:

The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions.

This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. Significant systems include Axiom, GAP, Maxima, Magma, Maple, Mathematica, and SageMath.

History

A Texas Instruments TI-Nspire calculator that contains a computer algebra system Computer algebra system.jpg
A Texas Instruments TI-Nspire calculator that contains a computer algebra system

Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence.

A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martinus Veltman, who designed a program for symbolic mathematics, especially high-energy physics, called Schoonschip (Dutch for "clean ship") in 1963. Another early system was FORMAC.

Using Lisp as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial-intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory"), which is a system for numerical computation built 15 years later at the University of New Mexico.

In 1987, Hewlett-Packard introduced the first hand-held calculator CAS with the HP-28 series. [1] Other early handheld calculators with symbolic algebra capabilities included the Texas Instruments TI-89 series and TI-92 calculator, and the Casio CFX-9970G. [2]

The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a copyleft version of Macsyma is called Maxima. Reduce became free software in 2008. [3] Commercial systems include Mathematica [4] and Maple, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include SageMath (which can act as a front-end to several other free and nonfree CAS). Other significant systems include Axiom, GAP, Maxima and Magma.

The movement to web-based applications in the early 2000s saw the release of WolframAlpha, an online search engine and CAS which includes the capabilities of Mathematica. [5]

More recently, computer algebra systems have been implemented using artificial neural networks, though as of 2020 they are not commercially available. [6]

Symbolic manipulations

The symbolic manipulations supported typically include:

In the above, the word some indicates that the operation cannot always be performed.

Additional capabilities

Many also include:

Some include:

Some computer algebra systems focus on specialized disciplines; these are typically developed in academia and are free. They can be inefficient for numeric operations as compared to numeric systems.

Types of expressions

The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include floating-point representation of real numbers, integers (of unbounded size), complex (floating-point representation), interval representation of reals, rational number (exact representation) and algebraic numbers.

Use in education

There have been many advocates for increasing the use of computer algebra systems in primary and secondary-school classrooms. The primary reason for such advocacy is that computer algebra systems represent real-world math more than do paper-and-pencil or hand calculator based mathematics. [12] This push for increasing computer usage in mathematics classrooms has been supported by some boards of education. It has even been mandated in the curriculum of some regions. [13]

Computer algebra systems have been extensively used in higher education. [14] [15] Many universities offer either specific courses on developing their use, or they implicitly expect students to use them for their course work. The companies that develop computer algebra systems have pushed to increase their prevalence among university and college programs. [16] [17]

CAS-equipped calculators are not permitted on the ACT, the PLAN, and in some classrooms [18] though it may be permitted on all of College Board's calculator-permitted tests, including the SAT, some SAT Subject Tests and the AP Calculus, Chemistry, Physics, and Statistics exams. [19]

Mathematics used in computer algebra systems

See also

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References

  1. ↑ Nelson, Richard. "Hewlett-Packard Calculator Firsts". Hewlett-Packard. Archived from the original on 2010-07-03.
  2. ↑ Coons, Albert (October 1999), "Getting started with symbolic mathematics systems: a productivity tool", Technology Tips, The Mathematics Teacher, 92 (7): 620–622, doi:10.5951/mt.92.7.0620, JSTOR   27971125
  3. ↑ "REDUCE Computer Algebra System at SourceForge". reduce-algebra.sourceforge.net. Retrieved 2015-09-28.
  4. ↑ Interview with Gaston Gonnet, co-creator of Maple Archived 2007-12-29 at the Wayback Machine , SIAM History of Numerical Analysis and Computing, March 16, 2005.
  5. ↑ Bhattacharya, Jyotirmoy (2022-05-12). "Wolfram|Alpha: a free online computer algebra system". The Hindu. ISSN   0971-751X . Retrieved 2023-04-26.
  6. ↑ Ornes, Stephen. "Symbolic Mathematics Finally Yields to Neural Networks". Quanta Magazine. Retrieved 2020-11-04.
  7. ↑ Dana-Picard, Thierry Noah. "Computer Assisted Proofs and Automated Methods in Mathematics Education". arxiv.org. Retrieved 2024-06-23.
  8. ↑ "dsolve - Maple Programming Help". www.maplesoft.com. Retrieved 2020-05-09.
  9. ↑ "DSolve - Wolfram Language Documentation". www.wolfram.com. Retrieved 2020-06-28.
  10. ↑ "Basic Algebra and Calculus — Sage Tutorial v9.0". doc.sagemath.org. Retrieved 2020-05-09.
  11. ↑ "Symbolic algebra and Mathematics with Xcas" (PDF).
  12. ↑ "Teaching kids real math with computers". Ted.com. Retrieved 2017-08-12.
  13. ↑ "Mathematics - Manitoba Education". Edu.gov.mb.ca. Retrieved 2017-08-12.
  14. ↑ "Mathematica for Faculty, Staff, and Students : Information Technology - Northwestern University". It.northwestern.edu. Retrieved 2017-08-12.
  15. ↑ "Mathematica for Students - Columbia University Information Technology". cuit.columbia.edu. Retrieved 2017-08-12.
  16. ↑ "Mathematica for Higher Education: Uses for University & College Courses". Wolfram.com. Retrieved 2017-08-12.
  17. ↑ "MathWorks - Academia - MATLAB & Simulink". Mathworks.com. Retrieved 2017-08-12.
  18. ↑ ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test Archived August 31, 2009, at the Wayback Machine
  19. ↑ "AP Exams Calculator Policy". AP Students. College Board. Retrieved 2024-05-24.
  20. 1 2 B. Buchberger; G.E. Collins; R. Loos (2013-06-29). Computer Algebra: Symbolic and Algebraic Computation. Springer Science & Business Media. ISBN   978-3-7091-3406-1.
  21. ↑ Joachim von zur Gathen; Jürgen Gerhard (2013-04-25). Modern Computer Algebra. Cambridge University Press. ISBN   978-1-107-03903-2.
  22. ↑ Keith O. Geddes; Stephen R. Czapor; George Labahn (2007-06-30). Algorithms for Computer Algebra. Springer Science & Business Media. ISBN   978-0-585-33247-5.
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