Newton interpolation at Leja points
Part II Numerical Mathematics
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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
65D05Key words and phrases
polynomial interpolaton Newton form stability Leja points ordering of interpolation pointsPreview
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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.MATHCrossRefGoogle 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.MATHGoogle 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.MATHGoogle Scholar
- [6]D. Gaier,Konstruktive Methoden der konformen Abbildung, Springer, Berlin, 1964.MATHGoogle Scholar
- [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]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.MATHCrossRefMathSciNetGoogle 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.MATHMathSciNetGoogle Scholar
- [10]B. Fischer and L. Reichel,Newton interpolation in Fejér and Chebyshev points, Math. Comp. 53 (1989) 265–278.MATHCrossRefMathSciNetGoogle Scholar
- [11]L. Reichel,On polynomial approximation in the uniform norm by the discrete least squares method, BIT 26 (1986), 349–365.MATHCrossRefMathSciNetGoogle Scholar
- [12]L. Reichel and G. Opfer,Chebyshev-Vandermonde systems, Math. Comput., to appear.Google Scholar
- [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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