Computer algebra: Difference between revisions

Revision as of 15:17, 5 July 2020

Revision as of 16:18, 5 July 2020

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Unlike in usual mathematics, "canonical form" and "normal form" are not synonymous in computer algebra.<ref>Davenport, J. H., Siret, Y., & Tournier, É. (1988). Computer algebra. London: Academic.</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 by expressions 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 by [[distributivity]] every product, while it is not necessary with a normal form (see below). Secondly, Itit 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.

==History==