Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation

Author: Charles B. Dunham
Journal: Math. Comp. 29 (1975), 1078-1082
MSC: Primary 65D15
MathSciNet review: 0388732
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] 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)
  • [2] R. BARRAR & H. LOEB, "On the Remez algorithm for nonlinear families," Numer Math., v. 15, 1970, pp. 382-391. MR 0267724 (42:2626)
  • [3] W. J. CODY, W. FRASER & J. F. HART, "Rational Chebyshev approximation using linear equations," Numer. Math., v. 12, 1968, pp. 242-251. MR 1553964
  • [4] W. FRASER & J. F. HART, "On the computation of rational approximations to continuous functions," Comm. ACM, v. 5, 1962, pp. 401-403.
  • [5] P. HENRICI, Elements of Numerical Analysis, Wiley, New York, 1964. MR 29 #4173. MR 0166900 (29:4173)
  • [6] A. RALSTON, "Rational Chebyshev approximation by Remes' algorithms," Numer. Math., v. 7, 1965, pp. 322-330. MR 32 #4438. MR 0186983 (32:4438)
  • [7] 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)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D15

Retrieve articles in all journals with MSC: 65D15

Additional Information

Keywords: Chebyshev approximation, rational approximation, Fraser-Hart iteration
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society