The stabilization of weights in the Lanczos and Conjugate Gradient method
The Lanczos and the Conjugate Gradient method both compute implicitly a sequence of Gauss quadrature approximations to a certain Riemann-Stieltjes integral. In finite precision computations the corresponding weight function will be slightly perturbed. The purpose of this paper is to solve a conjecture posed by Anne Greenbaum and Zdenek Strakos on the stabilization of weights for the Gauss quadrature approximations, i.e. in particular we prove that for a tight well separated cluster of Ritz values (nodes) an upper bound for the change in the sum of the corresponding weights can be developed that depends mainly on the ratio of the cluster diameter and the gap in the spectrum.
stabilization of weights conjugate gradient method lanczos algorithm tridiagonal matrix gauss-quadrature