Numerical Algorithms

, Volume 13, Issue 1, pp 107–122 | Cite as

The Newton differential correction algorithm for rational Chebyshev approximation with constrained denominators

  • M. Gugat
Article

Abstract

An algorithm for constrained rational Chebyshev approximation is introduced that combines the idea of an algorithm due to Hettich and Zencke, for which superlinear convergence is guaranteed, with the auxiliary problem used in the well-known original differential correction method. Superlinear convergence of the algorithm is proved. Numerical examples illustrate the fast convergence of the method and its advantages compared with the algorithm of Hettich and Zencke.

Keywords

Rational Chebyshev approximation with constrained denominators parametric auxiliary problem parametric optimization Newton's method superlinear convergence 

AMS subject classification

41A20 65D15 90C32 90C34 

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References

  1. [1]
    I. Barrodale, M. J. D. Powell and F. D. K. Roberts, The differential correction algorithm for rationall approximation, SIAM J. Numer. Anal. 9 (1972) 493–504.Google Scholar
  2. [2]
    E. W. Cheney,Introduction to Approximation Theory (McGraw-Hill, 1966).Google Scholar
  3. [3]
    E. W. Cheney and H. L. Loeb, Two new algorithms for rational approximation, Numer. Math. 3 (1961) 72–75.Google Scholar
  4. [4]
    E. W. Cheney and M. J. D. Powell The differential correction algorithm for generalized rational functions, Constr. Approx. 3 (1987) 249–256.Google Scholar
  5. [5]
    C. Dunham, Chebyshev approximation by rationals with constrained denominators, J. Approx. Theory 37 (1983) 5–11.Google Scholar
  6. [6]
    M. Gugat, Fractional semi-infinite programming, Dissertation, Universität Trier (1994).Google Scholar
  7. [7]
    M. Gugat, One-sided derivatives for the value function in convex parametric programming, Optimization 28 (1994) 301–314.Google Scholar
  8. [8]
    M. Gugat, An algorithm for Chebyshev approximation by rationals with constrained denominators, Constr. Approx. 12 (1996) 197–222.Google Scholar
  9. [9]
    R. Hettich and M. Gugat, Optimization under functional constraints and applications, in:Modern Methods of Optimization, Proc. Bayreuth 1990, Lecture Notes in Economics and Mathematical Systems 378 (Springer, 1990) pp. 90–126.Google Scholar
  10. [10]
    R. Hettich and P. Zencke, An algorithm for general restricted rational Chebyshev approximation, SIAM J. Numer. Anal 27 (1990) 1024–1033.Google Scholar
  11. [11]
    E. H. Kaufmann and G. D. Taylor, Uniform approximation by rational functions having restricted denominators, J. Approx. Theory 32 (1981) 9–26.Google Scholar
  12. [12]
    P. Zencke and R. Hettich, Directional derivatives for the value function in semi-infinite programming, Math. Programming 38 (1987) 323–340.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • M. Gugat
    • 1
  1. 1.Department of MathematicsUniversity of TrierTrierGermany

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