Computer algebra: Difference between revisions

Revision as of 16:33, 6 February 2025

Revision as of 09:12, 8 February 2025

Line 60:

Some fundamental mathematical questions arise when one wants to manipulate [[expression (mathematics)|mathematical expressions]] in a computer. We consider mainly the case of the [[Multivariate polynomial|multivariate]] [[rational fraction]]s. This is not a real restriction, because, as soon as the [[irrational function]]s appearing in an expression are simplified, they are usually considered as new indeterminates. For example,

:<math>(\sin(x+y)^2+ \log(z^2-5))^3</math>

is viewed as a polynomial in <math>\sin(x+y)</math> and <math>\log(z^2-5)</math>.

=== Equality ===

There are two notions of equality for [[expression (mathematics)|mathematical expressions]]. '''Syntactic equality''' is the equality of their representation in a computer. This is easy to test in a program. ''Semantic equality'' is when two expressions represent the same mathematical object, as in

:<math> (x+y)^2=x^2+2xy+y^2.</math>

Line 72:

In computer algebra, "canonical form" and "normal form" are not synonymous.<ref>{{cite book |last1=Davenport |first1=J. H. |last2=Siret |first2=Y. |last3=Tournier |first3=É. |title=Computer Algebra: Systems and Algorithms for Algebraic Computation |publisher=Academic |date=1988 |isbn=0-12-204230-1 |oclc=802584470 }}</ref> A ''canonical form'' is such that two expressions in canonical form are semantically equal if and only if they are syntactically equal, while a ''normal form'' is such that an expression in normal form is semantically zero only if it is syntactically zero. In other words, zero has a unique representation as an expression in normal form.

Normal forms are usually preferred in computer algebra for several reasons. Firstly, canonical forms may be more costly to compute than normal forms. For example, to put a polynomial in canonical form, one has to expand every product through the [[distributivity|distributive law]], while it is not necessary with a normal form (see below). Secondly, it may be the case, like for expressions involving radicals, that a canonical form, if it exists, depends on some arbitrary choices and that these choices may be different for two expressions that have been computed independently. This may make impracticable the use of a canonical form impractical.

==History==