Error estimates for quadrature rules based on the Arnoldi process

Abstract

The Arnoldi process can be applied to inexpensively approximate matrix functionals of the form IA(f)=uTf(A)v, where A∈RN×N is a large nonsymmetric matrix, u,v∈RN, and the superscript T denotes transposition. Here f is a function such that f(A) is a well defined matrix function. The computed approximations may be regarded as quadrature rules. This paper presents a new inexpensive approach, based on an extension of quadrature rules introduced by Spalević (2007), to compute estimates of the error in quadrature rules for the approximation of IA(f). Knowledge of the quadrature error is helpful for determining the number of nodes of the quadrature rule to be used. Numerical experiments illustrate the accuracy of the computed error estimates.