2-norm error bounds and estimates for Lanczos approximations to linear systems and rational matrix functions

TL;DR

This paper introduces a novel method using secondary Lanczos processes to efficiently compute 2-norm error bounds for Lanczos approximations of linear systems and rational matrix functions, improving error estimation accuracy.

Contribution

It presents a new approach leveraging secondary Lanczos processes to obtain computable, efficient error bounds, including upper bounds in the 2-norm, for rational matrix functions and linear system solutions.

Findings

Efficient computation of 2-norm error bounds using secondary Lanczos process.

Ability to obtain upper bounds for errors when the smallest eigenvalue is known.

Numerical experiments demonstrate the effectiveness of the proposed method.

Abstract

The Lanczos process constructs a sequence of orthonormal vectors v_m spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process starting at an arbitrary Lanczos vector v_m. This secondary process is then used to efficiently obtain computable error estimates and error bounds for the Lanczos approximations to the action of a rational matrix function on a vector. This includes, as a special case, the Lanczos approximation to the solution of a linear system Ax = b. Our approach uses the relation between the Lanczos process and quadrature as developed by Golub and Meurant. It is different from methods known so far because of its use of the secondary Lanczos process. With our approach, it is now in particular possible to efficiently obtain {\em upper bounds} for the error in the {\em 2-norm}, provided a lower bound on the smallest eigenvalue of $A$ is known. This holds in particular for a large class of rational matrix functions including best rational approximations to the inverse square root and the sign function. We compare our approach to other existing error estimates and bounds known from the literature and include results of several numerical experiments.