Computer algebra system

Symbolic manipulations

At its inception around 1968, Macsyma (an acronym for Project MAC's Symbolic Manipulator), was written in LISP and designed to utilize multiple data representations. Its general expression representation was modeled after FORMAC, though adapted to use LISP list structures. While this allowed the software to represent any arbitrary mathematical expression, its built-in simplifier had limited capabilities.

To address specific mathematical operations, Macsyma included a rational function representation—similar to ALPAK—which processed ratios of multivariable polynomials with integer coefficients. This representation relied on a greatest common divisor (GCD) algorithm to automatically maintain these functions in simplified forms. As the software evolved, its developers expanded its architecture to include additional representations, notably for power series.

A Computer Algebra System typically includes symbolic manipulations such as:

• simplification to a smaller expression or some standard form, including automatic simplification with assumptions and simplification with constraints
• substitution of symbols or numeric values for certain expressions
• change of form of expressions: expanding products and powers, partial and full factorization, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, transforming logic expressions, etc.
• partial and total differentiation
• some indefinite and definite integration (see symbolic integration), including multidimensional integrals
• symbolic constrained and unconstrained global optimization
• solution of linear and some non-linear equations over various domains
• solution of some differential and difference equations
• taking some limits
• integral transforms
• series operations such as expansion, summation and products
• matrix operations including products, inverses, etc.
• statistical computation
• theorem proving and verification which is very useful in the area of experimental mathematics
• optimized code generation

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

Additional capabilities

Many also include:

• a programming language, allowing users to implement their own algorithms
• arbitrary-precision numeric operations
• exact integer arithmetic and number theory functionality
• Editing of mathematical expressions in two-dimensional form
• plotting graphs and parametric plots of functions in two and three dimensions, and animating them
• drawing charts and diagrams
• APIs for linking it on an external program such as a database, or using in a programming language to use the computer algebra system
• string manipulation such as matching and searching
• add-ons for use in applied mathematics such as physics, bioinformatics, computational chemistry and packages for physical computation
• solvers for differential equations
• pretty-printing output to conform to standard mathematical notation conventions

Some include:

• graphic production and editing such as computer-generated imagery and signal processing as image processing
• sound synthesis

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 a CAS typically include multivariable polynomials; 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. 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.

Computer algebra systems have been extensively used in higher education. 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.

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

Mathematics used in computer algebra systems

• Knuth–Bendix completion algorithm
• Root-finding algorithms
• Symbolic integration via e.g. Risch algorithm or Risch–Norman algorithm
• Hypergeometric summation via e.g. Gosper's algorithm
• Limit computation via e.g. Gruntz's algorithm
• Polynomial factorization via e.g., over finite fields, Berlekamp's algorithm or Cantor–Zassenhaus algorithm.
• Greatest common divisor via e.g. Euclidean algorithm
• Gaussian elimination
• Gröbner basis via e.g. Buchberger's algorithm; generalization of Euclidean algorithm and Gaussian elimination
• Padé approximant
• Schwartz–Zippel lemma and testing polynomial identities
• Chinese remainder theorem
• Diophantine equations
• Landau's algorithm (nested radicals)
• Derivatives of elementary functions and special functions. (e.g. See derivatives of the incomplete gamma function.)
• Cylindrical algebraic decomposition
• Quantifier elimination over real numbers via cylindrical algebraic decomposition