Matrices for the direct determination of the barycentric weights of rational interpolation

Jean Paul Berrut, Hans Mittelmann

Research output: Contribution to journalArticle

24 Scopus citations

Abstract

Let x0,...,xN be N + 1 interpolation points (nodes) and f0,...,fN be N + 1 interpolation data. Then every rational function r with numerator and denominator degrees ≤N interpolating these values can be written in its barycentric form r(x) = ∑Nk=0uk/x - xkfk/∑Nk=0uk/x - xk, which is completely determined by a vector u of N + 1 barycentric weights uk. Finding u is therefore an alternative to the determination of the coefficients in the canonical form of r; it is advantageous inasmuch as u contains information about unattainable points and poles. In classical rational interpolation the numerator and the denominator of r are made unique (up to a constant factor) by restricting their respective degrees. We determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u's. The method is of complexity script O sign(n3) in terms of the denominator degree n; it seems on the other hand to be among the most stable ones.

Original languageEnglish (US)
Pages (from-to)355-370
Number of pages16
JournalJournal of Computational and Applied Mathematics
Volume78
Issue number2
DOIs
StatePublished - Feb 17 1997

Keywords

  • Barycentric representation
  • Barycentric weights
  • Interpolation
  • Rational interpolation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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