The Remez algorithm is a numerical method used to find the best uniform approximation of a continuous function by a polynomial. It is particularly useful in the context of Chebyshev approximations and is a technique for minimizing the maximum deviation (error) between a function and its polynomial approximation. The algorithm is named after the Russian mathematician Evgeny Remez.
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The Remez algorithm searches for a best uniform approximation by solving alternation equations on a reference set, finding extrema of the error, and exchanging reference points. For real degree-at-most- polynomials, the equioscillation theorem characterizes success by ordered alternating maximal errors. The alternating-reference error bound supplies a certified lower bound at each step, while the largest absolute error supplies an upper bound. Their gap measures how close the current polynomial is to optimal, provided the upper bound is established globally rather than only on a finite grid.