Polynomial algebra for Birkhoff interpolants

Authors:
John C. Butcher;Robert M. Corless;Laureano Gonzalez-Vega;Azar Shakoori
Affiliations:
Department of Mathematics, University of Auckland, Auckland, New Zealand;Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B7;Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria, Santander, Spain 39071;Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria, Santander, Spain 39071
Venue:
Numerical Algorithms
Year:
2011

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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.