Quadrature Rules with Applications to Error Estimation

Abstract

This dissertation presents new inexpensive approaches to compute estimates of the error in quadrature rules for the approximation of the matrix functionals of the form u^Tf(A)v, where A is a large nonsymmetric matrix, u,v are given vectors, and the superscript T denotes transposition. Here f is a function such that f(A) is a well-defined matrix function. Knowledge of the quadrature error is helpful for determining the number of nodes of the quadrature rule to be used. Among the approaches considered is the use of an extension of the averaged Gauss rules introduced by Spalević. We also use the difference between enhanced averaged rules and the associated Gauss rule to estimate the quadrature error in the latter. The enhanced averaged rules generalize averaged rules introduced by Laurie. Also, enhanced averaged rules associated with Gauss rules determined by measures with support in the complex plane are described. Finally, we explore the application of pairs of Gauss and optimal averaged Gauss rules, or pairs of Gauss and shifted optimal averaged Gauss rules, to estimate the error in iterates determined by the conjugate gradient method applied to the solution of linear systems of equations with a large symmetric positive definite matrix. The development of techniques for estimating the error in iterates determined by this method has received considerable attention in the literature. Available methods for bracketing the error in the iterates evaluate pairs of Gauss and Gauss-Radau quadrature rules to determine lower and upper bounds for the error in the A-norm. These methods require that a user allocates a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally requires further computations to estimate the location of the smallest eigenvalue of the system matrix. Two approaches to avoid these computations are described in the literature: i) carry out extra steps with the conjugate gradient method, or ii) replace the Gauss-Radau quadrature rule by an anti-Gauss rule. Both approaches yield error estimates instead of error bounds. The former approach increases the computational burden by carrying out more matrix-vector product evaluations, while the latter approach may yield inaccurate error estimates when the anti-Gauss rule has a node very close to the origin. We discuss the application of averaged and optimal averaged quadrature rules to the estimation of the A-norm of the error in iterates computed by the conjugate gradient method.