  This paper is concerned with the convergence analysis of an extended
variation of the locally optimal preconditioned conjugate gradient method
(LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which
admits some extended form of Rayleigh quotient. This work is a generalization
of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008).
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.
