A Rational Arnoldi Approach for Sobolev Rational Least Squares Problems

Abstract

ABSTRACTIn this article, we propose a rational Arnoldi‐based algorithm for solving Sobolev rational least squares problems. The classical approach to tackle least squares problems is to solve a system with a Vandermonde‐type coefficient matrix. Unfortunately, even for modest sizes, the ill‐conditioning of the coefficient matrix does not typically allow for accurate numerical computations. To overcome this drawback we exploit the link between the coefficient matrix and (rational) Krylov subspaces. We extend the theory by allowing the Krylov space to be generated not only by a diagonal matrix, but also by Jordan matrices, which links back to least squares problems involving approximations of the derivatives as well. This allows us to tackle Sobolev polynomial and Sobolev rational least squares problems. Examples are provided to illustrate the performance of the proposed approach.