Numerical Algorithms

, Volume 56, Issue 3, pp 319–347 | Cite as

Polynomial algebra for Birkhoff interpolants

  • John C. Butcher
  • Robert M. Corless
  • Laureano Gonzalez-Vega
  • Azar Shakoori
Original   Paper

Abstract

We introduce a unifying formulation of a number of related problems which can all be solved using a contour integral formula. Each of these problems requires finding a non-trivial linear combination of possibly some of the values of a function f, and possibly some of its derivatives, at a number of data points. This linear combination is required to have zero value when f is a polynomial of up to a specific degree p. Examples of this type of problem include Lagrange, Hermite and Hermite–Birkhoff interpolation; fixed-denominator rational interpolation; and various numerical quadrature and differentiation formulae. Other applications include the estimation of missing data and root-finding.

Keywords

Lagrange, Hermite, and Hermite–Birkhoff interpolation Contour integrals Barycentric form Fixed-denominator rational interpolation Root-finding 

Mathematics Subject Classifications (2010)

41A05 65D05 65D25 65D30 

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Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • John C. Butcher
    • 1
  • Robert M. Corless
    • 2
  • Laureano Gonzalez-Vega
    • 3
  • Azar Shakoori
    • 3
  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Departamento de Matematicas, Estadistica y ComputacionUniversidad de CantabriaSantanderSpain

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