Computer algebra: Difference between revisions

Revision as of 18:55, 10 April 2023

Revision as of 18:55, 10 April 2023

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In computer algebra, "canonical form" and "normal form" are not synonymous.<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 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 eghhhgghhauwuuuuuuuevery product through [[distributivity]], 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.

==History==