Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation
Abstract: The Fraser-Hart variant of the Remez algorithm is used to determine the best rational Chebyshev approximation to a continuous function on an interval. A necessary and sufficient condition for the matrix of the associated linear system to be nonsingular at the solution to the approximation problem is given. It is shown that the Fraser-Hart method may fail even if started arbitrarily close to the solution of the approximation problem. Use of the secant method in place of the Fraser-Hart iteration is also considered.
-  N. I. Ahiezer, Lekcii po Teorii Approksimacii, OGIZ, Moscow-Leningrad,], 1947 (Russian). MR 0025598
-  Richard B. Barrar and Henry L. Loeb, On the Remez algorithm for non-linear families, Numer. Math. 15 (1970), 382–391. MR 0267724, https://doi.org/10.1007/BF02165509
-  W. J. Cody, Handbook Series Methods of Approximation: Rational Chebyshev approximation using linear equations, Numer. Math. 12 (1968), no. 4, 242–251. MR 1553964, https://doi.org/10.1007/BF02162506
-  W. FRASER & J. F. HART, "On the computation of rational approximations to continuous functions," Comm. ACM, v. 5, 1962, pp. 401-403.
-  Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900
-  Anthony Ralston, Rational Chebyshev approximation by Remes’ algorithms, Numer. Math. 7 (1965), 322–330. MR 0186983, https://doi.org/10.1007/BF01436526
-  Mathematical methods for digital computers. Vol. II, Edited by Anthony Ralston and Herbert S. Wilf, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0211638
- N. I. ACHIESER (AHIEZER), Lectures On the Theory of Approximation, OGIZ, Moscow, 1947; English transl., Ungar, New York, 1956. MR 10, 33; 20 #1872. MR 0025598 (10:33b)
- R. BARRAR & H. LOEB, "On the Remez algorithm for nonlinear families," Numer Math., v. 15, 1970, pp. 382-391. MR 0267724 (42:2626)
- W. J. CODY, W. FRASER & J. F. HART, "Rational Chebyshev approximation using linear equations," Numer. Math., v. 12, 1968, pp. 242-251. MR 1553964
- W. FRASER & J. F. HART, "On the computation of rational approximations to continuous functions," Comm. ACM, v. 5, 1962, pp. 401-403.
- P. HENRICI, Elements of Numerical Analysis, Wiley, New York, 1964. MR 29 #4173. MR 0166900 (29:4173)
- A. RALSTON, "Rational Chebyshev approximation by Remes' algorithms," Numer. Math., v. 7, 1965, pp. 322-330. MR 32 #4438. MR 0186983 (32:4438)
- A. RALSTON, "Rational Chebyshev approximation," A. RALSTON & H. WILF, (Editors), Mathematical Methods for Digital Computers. Vol. 2, Wiley, New York, 1967, pp. 264-284. MR 35 #2516. MR 0211638 (35:2516)
Retrieve articles in Mathematics of Computation with MSC: 65D15
Retrieve articles in all journals with MSC: 65D15
Keywords: Chebyshev approximation, rational approximation, Fraser-Hart iteration
Article copyright: © Copyright 1975 American Mathematical Society