Approximating leading singular triplets of a matrix function
This work addresses the problem of efficiently computing singular triplets of matrix functions for practitioners in numerical linear algebra and related fields, offering a practical algorithm with theoretical guarantees.
The paper introduces an inexact Golub-Kahan-Lanczos bidiagonalization method to approximate leading singular triplets of a matrix function f(A) without computing f(A) exactly, and provides stopping criteria to handle the lack of a true residual. Numerical experiments demonstrate the algorithm's effectiveness on typical applications.
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.