Category of elements

A minimax approximation algorithm (or L^∞ approximation or uniform approximation) 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

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

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.

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.

External links

• Minimax approximation algorithm at MathWorld

References

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