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A '''minimax approximation algorithm''' (or '''L<sup>∞</sup> approximation'''<ref>{{cite book | title = Handbook of Floating-Point Arithmetic | page = 376 | publisher = Springer | year = 2009 | isbn = 081764704X | first1=Jean-Michel | last1=Muller }}</ref> or '''uniform approximation'''<ref name="phillips">{{cite book | page = 87 | title = Interpolation and Approximation by Polynomials | first = George M. | last = Phillips | publisher = Springer | year = 2003 | isbn = 0387002154}}</ref>) is a method which aims to find an approximation such that the maximum error is minimized. Suppose we seek to approximate the function f(''x'') by a function p(''x'') on the interval [''a'',''b'']. Then a minimax approximation algorithm will aim to minimize<ref name="powell">{{cite book | chapter = 7: The theory of minimax approximation | first = M. J. D. | last= Powell | authorlink=Michael J. D. Powell | year = 1981 | publisher= Cambridge University Press | title = Approximation Theory and Methods | isbn = 0521295149}}</ref>
::<math>\max_{a \leq x \leq b}|f(x)-p(x)|.</math>
==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 by [[Karl Weierstrass]] states that for any function f ∈ [[continuous function|C]][-1,1] and any ''ε'' > 0 then there exists a polynomial p such that<ref name="phillips" />
::<math>|f(x)-p(x)| < \epsilon,\quad -1\leq x \leq 1.</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.
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