Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

Abstract

This article is concerned with the convergence analysis of an extension of the locally optimal preconditioned conjugate gradient method for the extreme eigenvalue of a Hermitian matrix polynomial that possesses some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J Numer Anal. 2008;46(5):2567-92.). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence behavior.

Subspace algorithms for detecting a definite Hermitian matrix pair

“…The convergence of θ + i+1;1 and θ − i+1;1 , obtained by algorithms from [5,6], to some eigenvalues is proven in [49] for…”

“…respectively, but convergence to the extremal eigenvalues λ ± 1 is guaranteed only if θ + i+1;1 < λ + 2 for some i and λ − 2 < θ − i ′ +1;1 for some i ′ . An asymptotic estimate for the convergence of the sequences {θ ± i;1 } is given in [49] for a single vector version.…”