Local convergence behavior of extended local optimal block preconditioned conjugate gradient method for computing eigenvalues of Hermitian matrices
This work provides incremental improvements in convergence analysis for eigenvalue computation methods, benefiting researchers in numerical linear algebra and optimization.
The paper analyzes the local convergence behavior of an extended version of the locally optimal preconditioned conjugate gradient method for computing extreme eigenvalues of Hermitian matrices, deriving new or sharper convergence rates than previous work, including extensions to generalized problems like Hermitian matrix polynomials.
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.