|Title||Jacobi correction equation, line search, and conjugate gradients in Hermitian eigenvalue computation II: computing several extreme eigenvalues|
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.
|Keywords||Block conjugate gradients, convergence estimates, eigenvalue computation|
|Journal||SIAM Journal on Numerical Analysis|
|Journal citation||46 (5), pp. 2593-2619|
|Publisher||Society for Industrial and Applied Mathematics|
|Digital Object Identifier (DOI)||doi:10.1137/070688754|