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 Classification65D05
Key words and phrasespolynomial interpolaton Newton form stability Leja points ordering of interpolation points
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- 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
- G. Dahlquist and Å. Björck,Numerical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1974.Google Scholar
- W. Gautschi,Questions of numerical condition related to polynomials, inStudies in Numerical Analysis, ed. G. H. Golub, Math. Assoc. Amer., 1984.Google Scholar
- L. Reichel and G. Opfer,Chebyshev-Vandermonde systems, Math. Comput., to appear.Google Scholar
- H. Tal-Ezer,Higher degree interpolation polynomial in Newton form, ICASE Report No. 88–39, ICASE, Hampton, VA, 1988.Google Scholar
- J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, 5th ed., Amer. Math. Soc., Providence, RI, 1969.Google Scholar