Jacobi correction equation, line search, and conjugate gradients in Hermitian eigenvalue computation II: computing several extreme eigenvalues

Ovtchinnikov, E. 2008. Jacobi correction equation, line search, and conjugate gradients in Hermitian eigenvalue computation II: computing several extreme eigenvalues. SIAM Journal on Numerical Analysis. 46 (5), pp. 2593-2619.

TitleJacobi correction equation, line search, and conjugate gradients in Hermitian eigenvalue computation II: computing several extreme eigenvalues
AuthorsOvtchinnikov, E.
Abstract

This paper addresses the question of how to efficiently adapt the conjugate gradient (CG) method to the computation of several leftmost or rightmost eigenvalues and corresponding eigenvectors of Hermitian problems. A generic block CG algorithm instantiated by some available block CG algorithms is considered whereby the new approximate eigenpairs are computed by applying the Rayleigh-Ritz procedure in the trial subspace spanning current approximate eigenvectors and the search direction vectors, each of the latter being a linear combination of the respective gradient of the Rayleigh quotient and all search directions from the previous iteration. An approach related to the so-called Jacobi orthogonal complement correction equation is exploited in the local convergence analysis of this CG algorithm. Based on theoretical considerations, a new block conjugation scheme (a way to compute search directions) is suggested that enjoys a certain kind of optimality and has proved to be competitive in practical eigenvalue computation.

KeywordsBlock conjugate gradients, convergence estimates, eigenvalue computation
JournalSIAM Journal on Numerical Analysis
Journal citation46 (5), pp. 2593-2619
ISSN0036-1429
YearMay 2008
PublisherSociety for Industrial and Applied Mathematics
Digital Object Identifier (DOI)doi:10.1137/070688754
Publication dates
PublishedMay 2008

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