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The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934,〔E. Ya. Remez, "Sur la détermination des polynômes d'approximation de degré donnée", Comm. Soc. Math. Kharkov 10, 41 (1934); "Sur un procédé convergent d'approximations successives pour déterminer les polynômes d'approximation, Compt. Rend. Acad. Sc. 198, 2063 (1934); "Sur le calcul effectiv des polynômes d'approximation des Tschebyscheff", Compt. Rend. Acade. Sc. 199, 337 (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''(''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. ==Procedure== The Remez algorithm starts with the function ''f'' to be approximated and a set ''X'' of sample points in the approximation interval, usually the Chebyshev nodes linearly mapped to the interval. The steps are: # Solve the linear system of equations : (where ), :for the unknowns and ''E''. # Use the as coefficients to form a polynomial . # Find the set ''M'' of points of local maximum error . # If the errors at every are of equal magnitude and alternate in sign, then is the minimax approximation polynomial. If not, replace E with ''M'' and repeat the steps above. The result is called the polynomial of best approximation, the Chebyshev approximation, or the minimax approximation. A review of technicalities in implementing the Remez algorithm is given by W. Fraser. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Remez algorithm」の詳細全文を読む スポンサード リンク
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