Minimax approximation algorithm
For example, given a function defined on the interval and a degree bound , a minimax polynomial approximation algorithm will find a polynomial of degree at most to minimize
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.
- Muller, Jean-Michel et al. (2009). Handbook of Floating-Point Arithmetic. Springer. p. 376. ISBN 081764704X.
- Phillips, George M. (2003). "Best Approximation". Interpolation and Approximation by Polynomials. CMS Books in Mathematics. Springer. pp. 49–11. doi:10.1007/0-387-21682-0_2. ISBN 0-387-00215-4.
- Powell, M. J. D. (1981). "7: The theory of minimax approximation". Approximation Theory and Methods. Cambridge University Press. ISBN 0521295149.
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