An Arnoldi-based approach to polynomial and rational least squares problems

TL;DR

This paper introduces an Arnoldi-based method leveraging Krylov subspaces to improve the accuracy of polynomial and rational least squares problems, especially at higher degrees where ill-conditioning is problematic.

Contribution

The work establishes a novel connection between orthogonal polynomials, rational functions, and Krylov subspaces, enabling more stable least squares solutions using Arnoldi orthogonalization.

Findings

Krylov subspace methods improve approximation accuracy

The approach effectively handles high-degree least squares problems

Numerical examples demonstrate the method's performance

Abstract

In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.