2-Norm Error Bounds and Estimates for Lanczos Approximations to Linear Systems and Rational Matrix Functions

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 possible in particular to efficiently obtain upper bounds for the error in the 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.