Riccati-based preconditioner for computing invariant subspaces of large matrices

Abstract

This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Non-Hermitian Eigenvalue Problems

“…The idea of tuning the preconditioner for eigenvalue problems was introduced in [7,8] for inexact inverse iteration. There is considerable interest in inexact solves for subspace based methods, especially in relation to the Jacobi-Davidson method (JD) [24,2], [16] and the Riccati-based methods as developed in [18], [3], the latter may be viewed as the block analogue of JD and are useful for computing invariant subspaces. Other useful methods which use inexact solves within inner outer iterations include the trace minimization [22] and the inexact Raleigh quotient (IRQ) iteration [23].…”