Category of elements: Difference between revisions
en>Cesme'es mNo edit summary |
en>Mark viking Added wl |
||
| Line 1: | Line 1: | ||
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|last2=Brisebarre | first2=Nicolas | last3=de Dinechin | first3=Florent | last4=Jeannerod | first4=Claude-Pierre | last5=Lefèvre | first5=Vincent | last6=Melquiond | first6=Guillaume | last7=Revol | first7=Nathalie | last8=Stehlé | first8=Damien | last9=Torres | first9=Serge | display-authors=1 }}</ref> or '''uniform approximation'''<ref name="phillips">{{cite doi | 10.1007/0-387-21682-0_2}}</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 find a function p(''x'') 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 [[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" /> | |||
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. | |||
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> | |||
==External links== | |||
*[http://mathworld.wolfram.com/MinimaxApproximation.html Minimax approximation algorithm at MathWorld] | |||
==References== | |||
{{Reflist}} | |||
[[Category:Numerical analysis]] | |||
{{algorithm-stub}} | |||
Revision as of 01:38, 12 January 2014
A minimax approximation algorithm (or L∞ approximation[1] or uniform approximation[2]) 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 find a function p(x) to minimize[3]
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.[2]
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.
One popular minimax approximation algorithm is the Remez algorithm. Chebyshev polynomials of the first kind closely approximate the minimax polynomial.[4]
External links
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 Template:Cite doi
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite web