Structure Preserving Restarts of the Non-Symmetric Lanczos Algorithm via the Implicitly shifted LR algorithm

TL;DR

This paper introduces a structure-preserving restart method for the non-symmetric Lanczos algorithm using the implicitly shifted LR iteration, enhancing eigenvalue computations by maintaining matrix structure.

Contribution

It adapts the implicitly shifted LR iteration as a restart technique for the non-symmetric Lanczos algorithm, preserving the tridiagonal structure of the reduced matrix.

Findings

The method effectively preserves matrix structure during restarts.

It improves the efficiency of eigenvalue computations for large matrices.

The approach extends polynomial filtering techniques to non-symmetric cases.

Abstract

The implicitly shifted QR iteration is used as a restart procedure for the Arnoldi method for the calculation of a few dominant eigenvalues of a large matrix. We show that the underlying idea of implicit polynomial filtering can be utilized in much the same manner via the implicitly shifted LR iteration to create a restart procedure for the non-symmetric Lanczos algorithm for eigenvalue computations, which preserves the tri-diagonal structure of the reduced matrix.