Minimax approximation algorithm
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 minimize[3]
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.[3] 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
- ^ Muller, Jean-Michel (2009). Handbook of Floating-Point Arithmetic. Springer. p. 376. ISBN 081764704X.
- ^ a b Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi: 10.1007/0-387-21682-0_2, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi= 10.1007/0-387-21682-0_2instead. - ^ a b Powell, M. J. D. (1981). "7: The theory of minimax approximation". Approximation Theory and Methods. Cambridge University Press. ISBN 0521295149.
- ^ "Minimax Polynomial". Wolfram MathWorld. Retrieved 2012-09-03.
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