Minimax approximation algorithm

A minimax approximation algorithm (or L^∞ approximation or uniform approximation) is a method to find an approximation that minimizes maximum error.

Example

For example, to approximate the function f(x) by a function p(x) on the interval [a,b], a minimax approximation algorithm will find a function p(x) to minimize

$\max _{a\leq x\leq b}|f(x)-p(x)|.$

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. 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. Truncated Chebyshev series, however, closely approximate the minimax polynomial.

One popular minimax approximation algorithm is the Remez algorithm.

External links

• Minimax approximation algorithm at MathWorld