Estimating the error in matrix function approximations

Abstract

The need to compute matrix functions of the form f(A)v, where $A\in {\mathbb {R}}^{N\times N}$ is a large symmetric matrix, f is a function such that f(A) is well defined, and v≠ 0 is a vector, arises in many applications. This paper is concerned with the situation when A is so large that the evaluation of f(A) is prohibitively expensive. Then, an approximation of f(A)v often is computed by applying a few, say 1 ≤ n ≪ N, steps of the symmetric Lanczos process to A with initial vector v to determine a symmetric tridiagonal matrix $T_{n}\in {\mathbb {R}}^{n\times n}$ and a matrix $V_{n}\in {\mathbb {R}}^{N\times n}$, whose orthonormal columns span a Krylov subspace. The expression V_nf(T_n)e₁∥v∥ furnishes an approximation of f(A)v. The evaluation of f(T_n) is inexpensive, because the matrix T_n is small. It is important to be able to estimate the error in the computed approximation. This paper describes a novel approach that is based on a technique proposed by Spalević for estimating the error in Gauss quadrature rules.