Efficiently preconditioned Inexact Newton methods for large symmetric eigenvalue problems
Abstract
In this paper we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the Preconditioned Conjugate Gradient solution of the linearized Newton system to solve $A \mathbf{u} = q(\mathbf{u}) \mathbf{u}$, $q(\mathbf{u})$ being the Rayleigh Quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the JacobiDavidson method on all the test problems.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 DOI:
 10.48550/arXiv.1312.1553
 arXiv:
 arXiv:1312.1553
 Bibcode:
 2013arXiv1312.1553B
 Keywords:

 Mathematics  Numerical Analysis;
 65F05;
 65F15;
 65H1
 EPrint:
 20 pages, 6 figures. Submitted to Optimization Methods and Software