Linear Rational Interpolation and its Application in Approximation and in Solving Boundary Value Problems
Two recent results will be presented. After recalling some pitfalls of polynomial interpolation (e.g., slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant with all nodes prescribed, written in its barycentric form, and optimizing the placement of the poles in such a way as to minimize the sup-norm of the error. In the same spirit we suggest an algorithm which improves on the polynomial pseudospectral method for solving two-point boundary value problems. The substantial improvements obtained for both problems are shown with numerical examples.