Local convergence behavior of extended local optimal block preconditioned conjugate gradient method for computing eigenvalues of Hermitian matrices

TL;DR

This paper analyzes the local convergence of an extended LOBPCG method for Hermitian eigenvalue problems, providing sharper rates and extending to generalized problems like matrix polynomials.

Contribution

It offers new, sharper convergence rate analyses for an extended LOBPCG method, including for generalized Hermitian problems, improving upon prior work.

Findings

Derived new convergence rates for the extended LOBPCG method.

Extended analysis to Hermitian matrix polynomials.

Provided sharper convergence bounds than previous studies.

Abstract

This paper provides a comprehensive and detailed analysis of the local convergence behavior of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for computing the extreme eigenvalue of a Hermitian matrix. The convergence rates derived in this work are either obtained for the first time or sharper than those previously established, including those in Ovtchinnikov's work ({\em SIAM J. Numer. Anal.}, 46(5):2567--2592, 2008). The study also extends to generalized problems, including Hermitian matrix polynomials that admit an extended form of the Rayleigh quotient. The new approach used to obtain these rates may also serve as a valuable tool for the convergence analysis of other gradient-type optimization methods.