EFFICIENT EXPANSION OF SUBSPACES IN THE JACOBI-DAVIDSON METHOD FOR STANDARD AND GENERALIZED EIGENPROBLEMS

We discuss approaches for an efficient handling of the correction equation in the Jacobi-Davidson method. The correction equation is effective in a subspace orthogonal to the current eigenvector approximation. The operator in the correction equation is a dense matrix, but it is composed from three factors that allow for a sparse representation. If the given matrix eigenproblem is sparse then one often aims for the construction of a preconditioner for that matrix. We discuss how to restrict this preconditioner effectively to the subspace orthogonal to the current eigenvector. The correction equation itself is formulated in terms of approximations for an eigenpair. In order to avoid misconvergence one has to make the right selection for the approximations, and this aspect will be discussed as well.

[1]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[2]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[3]  I. Gustafsson.  A class of first order factorization methods , 1978 .

[4]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  J. Olsen,et al.  Passing the one-billion limit in full configuration-interaction (FCI) calculations , 1990 .

[7]  R. Morgan.  Computing Interior Eigenvalues of Large Matrices , 1991 .

[8]  H. V. D. Vorst,et al.  The effect of incomplete decomposition preconditioning on the convergence of conjugate gradients , 1992 .

[9]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[10]  D. R. Fokkema,et al.  BICGSTAB( L ) FOR LINEAR EQUATIONS INVOLVING UNSYMMETRIC MATRICES WITH COMPLEX , 1993 .

[11]  Axel Ruhe.  Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs , 1994 .

[12]  Axel Ruhe.  Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems , 1994 .

[13]  H. V. D. Vorst,et al.  Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems , 1995 .

[14]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..

[15]  H. V. D. Vorst,et al.  Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils , 1996 .

[16]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM J. Matrix Anal. Appl..

[17]  Gerard L. G. Sleijpen,et al.  Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils , 1998, SIAM J. Sci. Comput..

[18]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM Rev..

[19]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .