Symbolic computation

Symbolic computation manipulates mathematical expressions as exact structures rather than approximating them with numerical values. It includes operations such as simplification, differentiation, integration, and equation solving.

Expression tree: A rooted tree representation in which operators are internal nodes and operands are their children. Symbolic systems represent formulas as trees so they can inspect and transform subexpressions.
Symbolic differentiation: The exact computation of derivative expressions from formulas using differentiation rules. Differentiation is a standard symbolic operation used in calculus and mathematical modeling.
Numerical analysis: The study of algorithms for approximate numerical solutions to mathematical problems. Numerical analysis targets reliable approximations where symbolic computation seeks exact expressions.
Algebraic manipulation: The transformation of algebraic expressions according to mathematical rules. Computer symbolic methods automate forms of manipulation long performed by hand.
Term rewriting: A formal method that transforms expressions by repeatedly applying oriented equations called rewrite rules. Simplification and normalization often work by applying rewrite rules to expression trees.
Symbolic integration: The search for exact antiderivatives or integrals expressed in symbolic form. Integration demonstrates both the power and limits of algorithmic formula manipulation.
Floating-point arithmetic: A finite-precision representation and arithmetic system for approximating real numbers on computers. Rounding error distinguishes floating-point calculations from exact symbolic manipulation.
Macsyma: An early general-purpose computer algebra system developed at MIT beginning in the 1960s. Macsyma helped establish the architecture and capabilities of modern computer algebra systems.
Canonical form: A standardized representation chosen so equivalent objects can be compared or manipulated consistently. Canonical forms help symbolic systems recognize when differently written expressions are equivalent.
Polynomial factorization: The decomposition of a polynomial into products of simpler polynomials over a specified coefficient domain. Factorization is a central exact operation in algebra systems.

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