Remez algorithm

The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm L_∞ sense.

A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function. In this case, the form of the solution is precised by the equioscillation theorem.

The Remez algorithm starts with the function f to be approximated and a set X of ${\displaystyle n+2}$ sample points ${\displaystyle x_{1},x_{2},...,x_{n+2}}$ in the approximation interval, usually the Chebyshev nodes linearly mapped to the interval. The steps are:

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