Rational functions for guaranteed and experimentally well-conditioned global interpolation

Abstract

Polynomial interpolation is known to be ill-conditioned if the interpolating points are not chosen in special ways; classical rational interpolation can give better results, but does not work in all cases and the corresponding functions can show poles in the interval of interpolation. We present here rational functions which guarantee well-conditioned interpolation on a real interval or a circle and cannot have any poles there. They can be evaluated at least as efficiently as the corresponding interpolation polynomials and the accuracy of their approximation to a given function often compares favorably with that of spline interpolants.