A modified Remes algorithm

The frequently used Remes algorithm, which finds the best approximation to a continuous function in a finite interval, may not always converge. This iterative algorithm requires the error function of the intermediate approximation at every iteration to have equal magnitude with alternating signs at...

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Bibliographic Details
Published in:SIAM journal on scientific and statistical computing 1988-11, Vol.9 (6), p.1058-1072
Main Author: CHIANG, Y.-L. F
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Polynomials
Sciences and techniques of general use
Signs
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Summary:The frequently used Remes algorithm, which finds the best approximation to a continuous function in a finite interval, may not always converge. This iterative algorithm requires the error function of the intermediate approximation at every iteration to have equal magnitude with alternating signs at a specified number of points. When this requirement cannot be fulfilled, the algorithm fails to converge. In this paper, a property (called property $y$) and a modified Remes algorithm are defined such that the convergence of the new algorithm is guaranteed if the initial approximation has property $y$. Numerical examples of best approximations in various forms are given to show the use and convergence of the new algorithm.
ISSN:0196-5204
1064-8275
2168-3417
1095-7197
DOI:10.1137/0909072