Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems

TL;DR

This paper introduces a preconditioning technique that enables accurate solutions of ill-conditioned linear systems and eigenvalue problems by ensuring the inverse of the preconditioner can be applied accurately, supported by theoretical analysis and numerical examples.

Contribution

It develops a new preconditioning approach that achieves higher accuracy for ill-conditioned problems by focusing on the accurate application of the inverse preconditioner.

Findings

Preconditioning can improve solution accuracy for ill-conditioned systems.

The inverse-equivalent accuracy concept enhances understanding of stability.

Numerical results demonstrate improved eigenvalue computations.

Abstract

This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an ill-conditioned linear system $Ax=b$, we show that, if the inverse of the preconditioner $M^{-1}$ can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.