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. https://doi.org/10.1137/070688754
| Title | Jacobi correction equation, line search, and conjugate gradients in Hermitian eigenvalue computation II: computing several extreme eigenvalues |
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| Authors | Ovtchinnikov, 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. |
| Keywords | Block conjugate gradients, convergence estimates, eigenvalue computation |
| Journal | SIAM Journal on Numerical Analysis |
| Journal citation | 46 (5), pp. 2593-2619 |
| ISSN | 0036-1429 |
| Year | May 2008 |
| Publisher | Society for Industrial and Applied Mathematics |
| Digital Object Identifier (DOI) | https://doi.org/10.1137/070688754 |
| Publication dates | |
| Published | May 2008 |