Approximating leading singular triplets of a matrix function

TL;DR

This paper develops an inexact Golub-Kahan-Lanczos method to approximate the leading singular triplets of a matrix function, enabling efficient computation of matrix norms for large matrices.

Contribution

It introduces a novel inexact bidiagonalization algorithm tailored for matrix functions, with specific stopping criteria and analysis for approximating leading singular triplets.

Findings

Algorithm effectively approximates leading singular values and vectors.

Numerical experiments demonstrate practical efficiency and accuracy.

Method handles large matrices where direct computation is infeasible.

Abstract

Given a large square matrix $A$ and a sufficiently regular function $f$ so that $f(A)$ is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of $f(A)$, and in particular of $\|f(A)\|$, where $\|\cdot \|$ is the matrix norm induced by the Euclidean vector norm. Since neither $f(A)$ nor $f(A)v$ can be computed exactly, we introduce and analyze an inexact Golub-Kahan-Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations $f(A)v$, $f(A)^*v$. Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.