Minimax approximation algorithm: Difference between revisions

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==Polynomial approximations==

The conditions for a best appromation are particularly simple if the function p(''x'') is restricted to polynomials less than a stated degree ''n''.<ref name="powell" /> A theorem byThe [[Karl Weierstrass approximation theorem]] states that forevery anycontinuous function f&nbsp;∈&nbsp;[[continuousdefined on a closed interval function|C]][-1a,1b] andcan anybe ''ε''&nbsp;>&nbsp;0uniformly thenapproximated thereas existsclosely aas polynomialdesired pby sucha thatpolynomial function.<ref name="phillips" />

::<math>\max_{-1 \leq x \leq 1}|f(x)-p(x)| < \epsilon.</math>

Polynomial expansions such as the [[Taylor series]] expansion are often convenient for theoretical work but less useful for practical applications. For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of terms in an effort to reduce computational expense of repeated evaluation.