JACOBI CORRECTION EQUATION, LINE SEARCH, ANDCONJUGATE GRADIENTS IN HERMITIAN EIGENVALUECOMPUTATION II: COMPUTING SEVERALEXTREME EIGENVALUES

摘要

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 correspondingeigenvectors of Hermitian problems. A generic block CG algorithm instantiated by some availableblock CG algorithms is considered whereby the new approximate eigenpairs are computed by applyingthe Rayleigh–Ritz procedure in the trial subspace spanning current approximate eigenvectors andthe search direction vectors, each of the latter being a linear combination of the respective gradientof the Rayleigh quotient and all search directions from the previous iteration. An approach related tothe so-called Jacobi orthogonal complement correction equation is exploited in the local convergenceanalysis 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 hasproved to be competitive in practical eigenvalue computation.