An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems

An Implicitly-restarted Krylov Subspace Method for Real Symmetric/skew-symmetric Eigenproblems

A new implicitly-restarted Krylov subspace method for real symmetric/skew-symme– tric generalized eigenvalue problems is presented. The new method improves and generalizes the SHIRA method of [37] to the case where the skew symmetric matrix is singular. It computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point. Several applications from control theory are...

An Implicitly Restarted Lanczos Method for Large Symmetric Eigenvalue Problems

The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric n×n matrix. However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. In the implicitly restarted Lanczos method studied in the present paper, this problem is addressed by fixing the num...

IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

Convergence Analysis of Restarted Krylov Subspace Eigensolvers

The A-gradient minimization of the Rayleigh quotient allows to construct robust and fastconvergent eigensolvers for the generalized eigenvalue problem for (A,M) with symmetric and positive definite matrices. The A-gradient steepest descent iteration is the simplest case of more general restarted Krylov subspace iterations for the special case that all step-wise generated Krylov subspaces are tw...

Sharp Ritz Value Estimates for Restarted Krylov Subspace Iterations

Gradient iterations for the Rayleigh quotient are elemental methods for computing the smallest eigenvalues of a pair of symmetric and positive definite matrices. A considerable convergence acceleration can be achieved by preconditioning and by computing Rayleigh-Ritz approximations from subspaces of increasing dimensions. An example of the resulting Krylov subspace eigensolvers is the generaliz...