A '''minimax approximation algorithm''' (or '''L<sup>∞</sup> approximation'''<ref>{{citebookor |'''uniform titleapproximation''') =is Handbooka ofmethod Floating-Pointto Arithmeticfind |an pageapproximation =of 376a |[[mathematical publisherfunction]] =that Springer|minimizes yearmaximum error.<ref name="Muller_2010">{{cite 2009book | isbn author-last1= 081764704XMuller |author-first1=Jean-Michel | last1=Muller|author-last2=Brisebarre |author-first2=Nicolas |author-last3=de Dinechin |author-first3=Florent |author-last4=Jeannerod |author-first4=Claude-Pierre |author-last5=Lefèvre |author-first5=Vincent |author-last6=Melquiond |author-first6=Guillaume |author-last7=Revol |author-first7=Nathalie|author7-link=Nathalie Revol |author-last8=Stehlé |author-first8=Damien |author-last9=Torres |author-first9=Serge |title=Handbook displayof Floating-authorsPoint Arithmetic |url=https://archive.org/details/handbookfloating00mull_867 |url-access=limited |year=2010 |publisher=[[Birkhäuser]] |edition=1 |isbn=978-0-8176-4704-9<!-- print --> |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668<!-- |id=ISBN 978-0-8176-4705-6 (online), ISBN 0-8176-4704-X (print) --> |page=[https://archive.org/details/handbookfloating00mull_867/page/n388 376]}}</ref> or '''uniform approximation'''<ref name="phillips">{{citeCite doibook | doi = 10.1007/0-387-21682-0_2}}< | first = George M. | last = Phillips| chapter = Best Approximation | title = Interpolation and Approximation by Polynomials | url = https:/ref>)/archive.org/details/interpolationapp00phil_282 is| aurl-access method= tolimited find| anseries approximation= thatCMS Books minimizesin maximumMathematics error| pages = [https://archive.org/details/interpolationapp00phil_282/page/n63 49]–11 | year = 2003 | publisher = Springer | isbn = 0-387-00215-4 }}</ref>
==Example==
For example, togiven approximate thea function <math>f(''x'')</math> by a function p(''x'')defined on the interval <math>[''a'',''b'']</math> and a degree bound <math>n</math>, a minimax polynomial approximation algorithm will find a functionpolynomial <math>p(''x'')</math> of degree at most <math>n</math> to minimize ::<math>\max_{a \leq x \leq b}|f(x)-p(x)|.</math><ref name="powell">{{cite book | chapter = 7: The theory of minimax approximation | first = M. J. D. | last= Powell | authorlinkauthor-link=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 [[Weierstrass approximation theorem]] states that every continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.<ref name="phillips" />
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.
Polynomial expansions such as the [[Taylor series]] expansion are often convenient for theoretical work but less useful for practical applications. ForTruncated practical[[Chebyshev workseries]], ithowever, isclosely often desirable to minimizeapproximate the maximum absolute or relative error of aminimax polynomial fit for any given number of terms in an effort to reduce computational expense of repeated evaluation.
One popular minimax approximation algorithm is the [[Remez algorithm]]. [[Chebyshev polynomials of the first kind]] closely approximate the minimax polynomial.<ref>{{cite web | url = http://mathworld.wolfram.com/MinimaxPolynomial.html | title = Minimax Polynomial | publisher = Wolfram MathWorld | accessdate= 2012-09-03}}</ref>
