An application of gaussian elimination to interpolation by generalized rational functions

Abstract

We consider the calculation of r(ξ), where ξ is a given number, and where {r(x)=p(x)/q(x); xε IR} is a generalized rational function whose coefficients should satisfy some interpolation conditions. We study a procedure that obtains r(ξ)=p(ξ)/q(ξ) by applying Gaussian elimination to remove the unknown coefficients from a system of linear equations. It is shown that the procedure breaks down only if r(ξ) or the coefficients of the rational function are not properly defined. It is proved that the intermediate equations of Gaussian elimination are related to rational interpolating functions that depend on subsets of the coefficients and data. A numerical example demonstrates the procedure.