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Newton interpolation at Leja points

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.

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References

  1. W. G. Bickley,Two-dimensional potential problems for the space outside a rectangle, Proc. London Math. Soc. 37 (1934), 82–105.

    MATH  Article  Google Scholar 

  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. C. de Boor,A practical Guide to Splines, Springer, New York, 1978.

    MATH  Google Scholar 

  4. G. Dahlquist and Å. Björck,Numerical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1974.

    Google Scholar 

  5. P. J. Davis,Interpolation and Approximation, Dover, New York, 1975.

    MATH  Google Scholar 

  6. D. Gaier,Konstruktive Methoden der konformen Abbildung, Springer, Berlin, 1964.

    MATH  Google Scholar 

  7. W. Gautschi,Questions of numerical condition related to polynomials, inStudies in Numerical Analysis, ed. G. H. Golub, Math. Assoc. Amer., 1984.

  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.

    MATH  Article  MathSciNet  Google Scholar 

  9. F. Leja,Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme, Ann. Polon. Math. 4 (1957), 8–13.

    MATH  MathSciNet  Google Scholar 

  10. B. Fischer and L. Reichel,Newton interpolation in Fejér and Chebyshev points, Math. Comp. 53 (1989) 265–278.

    MATH  Article  MathSciNet  Google Scholar 

  11. L. Reichel,On polynomial approximation in the uniform norm by the discrete least squares method, BIT 26 (1986), 349–365.

    MATH  Article  MathSciNet  Google Scholar 

  12. L. Reichel and G. Opfer,Chebyshev-Vandermonde systems, Math. Comput., to appear.

  13. H. Tal-Ezer,Higher degree interpolation polynomial in Newton form, ICASE Report No. 88–39, ICASE, Hampton, VA, 1988.

    Google Scholar 

  14. J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, 5th ed., Amer. Math. Soc., Providence, RI, 1969.

    Google Scholar 

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Additional information

Research supported in part by NSF under Grant DMS-8704196 and by U.S. Air Force Grant AFSOR-87-0102.

On leave from University of Kentucky, Department of Mathematics, Lexington, KY 40506, U.S.A.

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Reichel, L. Newton interpolation at Leja points. BIT 30, 332–346 (1990). https://doi.org/10.1007/BF02017352

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  • DOI: https://doi.org/10.1007/BF02017352

AMS(MOS) Subject Classification

  • 65D05

Key words and phrases

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