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BIT

, Volume 30, Issue 2, pp 332–346 | Cite as

Newton interpolation at Leja points

  • Lothar Reichel
Part II Numerical Mathematics
Received:
Revised:

Abstract

The Newton form is a convenient representation for interpolation polynomials. Its sensitivity to perturbations depends on the distribution and ordering of the interpolation points. The present paper bounds the growth of the condition number of the Newton form when the interpolation points are Leja points for fairly general compact sets K in the complex plane. Because the Leja points are defined recursively, they are attractive to use with the Newton form. If K is an interval, then the Leja points are distributed roughly like Chebyshev points. Our investigation of the Newton form defined by interpolation at Leja points suggests an ordering scheme for arbitrary interpolation points.

AMS(MOS) Subject Classification

65D05 

Key words and phrases

polynomial interpolaton Newton form stability Leja points ordering of interpolation points 
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References

  1. [1]
    W. G. Bickley,Two-dimensional potential problems for the space outside a rectangle, Proc. London Math. Soc. 37 (1934), 82–105.zbMATHCrossRefGoogle Scholar
  2. [2]
    H. Björk,Contribution to the problem of least squares approximation, Report TRITA-NA-7137, Dept. of Comp. Sci., Royal Institute of Technology, Stockholm, 1971.Google Scholar
  3. [3]
    C. de Boor,A practical Guide to Splines, Springer, New York, 1978.zbMATHGoogle Scholar
  4. [4]
    G. Dahlquist and Å. Björck,Numerical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1974.Google Scholar
  5. [5]
    P. J. Davis,Interpolation and Approximation, Dover, New York, 1975.zbMATHGoogle Scholar
  6. [6]
    D. Gaier,Konstruktive Methoden der konformen Abbildung, Springer, Berlin, 1964.zbMATHGoogle Scholar
  7. [7]
    W. Gautschi,Questions of numerical condition related to polynomials, inStudies in Numerical Analysis, ed. G. H. Golub, Math. Assoc. Amer., 1984.Google Scholar
  8. [8]
    D. M. Hough and N. Papamichael,An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains, Numer. Math. 41 (1983), 287–307.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    F. Leja,Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme, Ann. Polon. Math. 4 (1957), 8–13.zbMATHMathSciNetGoogle Scholar
  10. [10]
    B. Fischer and L. Reichel,Newton interpolation in Fejér and Chebyshev points, Math. Comp. 53 (1989) 265–278.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    L. Reichel,On polynomial approximation in the uniform norm by the discrete least squares method, BIT 26 (1986), 349–365.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    L. Reichel and G. Opfer,Chebyshev-Vandermonde systems, Math. Comput., to appear.Google Scholar
  13. [13]
    H. Tal-Ezer,Higher degree interpolation polynomial in Newton form, ICASE Report No. 88–39, ICASE, Hampton, VA, 1988.Google Scholar
  14. [14]
    J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, 5th ed., Amer. Math. Soc., Providence, RI, 1969.Google Scholar

Copyright information

© BIT Foundations 1990

Authors and Affiliations

  • Lothar Reichel
    • 1
  1. 1.Bergen Scientific CentreBergenNorway

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