Gaussian Quadrature Applied to Adaptive Chebyshev Iteration

Abstract

Chebyshev iteration has been a popular iterative scheme for the solution of large linear systems of equations with a symmetric positive definite matrix A. With the advent of parallel processors, there has been a resurgence of interest in this method. In Chebyshev iteration one determines iteration parameters so that the residual polynomials axe scaled Chebyshev polynomials for some interval [a, b] on the positive real axis. Chebyshev iteration is often implemented as an adaptive iteration scheme, in which one during the iterations seeks to determine an interval [a, b] that make the iterates converge rapidly to the solution of the linear system. Roughly, the interval should contain most of the spectrum of A and be as small as possible. Recently, Golub and Kent [6] proposed a new adaptive Chebyshev iteration method, in which inner products of residual vectors are interpreted as modified moments. These modified moments and the recursion coefficients for the residual polynomials yield a symmetric tridiagonal matrix, whose eigenvalues are used to determine an interval [a, b]. The eigenvalues are nodes of a Gaussian quadrature rule. We propose a modification of this scheme, in which the determination of a suitable interval is based on the weights of this quadrature rule also. Computed examples illustrate that a significant reduction in the number of iterations can be achieved by this modification.

Research supported in part by the Design and Manufacturing Institute at Stevens Institute of Technology. Research supported in part by NSF grant CCR-8821078. Research supported in part by NSF grants DMS-9002884 and DMS-9205531.