A Restarted Lanczos Approximation to Functions of a Symmetric Matrix

Abstract

In this paper, we investigate a method for restarting the Lanczos method for approximating the matrix-vector product $f(A)b$, where $A in mathbb{R}^{n imes n }$ is a symmetric matrix. For analytic $f$ we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an error bound for the proposed restart scheme. We also present an error bound for the restarted Lanczos approximation of $f(A)b$ for symmetric positive definite $A$ when $f$ is in a particular class of completely monotone functions. We illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved.