Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

Implicitly Restarted Arnoldi/lanczos Methods for Large Scale Eigenvalue Calculations

This report provides an introductory overview of the numerical solution of large scale algebraic eigenvalue problems. The main focus is on a class of methods called Krylov subspace projection methods. The Lanczos method is the premier member of this class and the Arnoldi method is a generalization to the nonsymmetric case. A recently developed and very promising variant of the Arnoldi/Lanczos s...

Implicitly Restarted Generalized Second-order Arnoldi Type Algorithms for the Quadratic Eigenvalue Problem

We investigate the generalized second-order Arnoldi (GSOAR) method, a generalization of the SOAR method proposed by Bai and Su [SIAM J. Matrix Anal. Appl., 26 (2005): 640–659.], and the Refined GSOAR (RGSOAR) method for the quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedure to generate an orthonormal basis of a given generalized second-order Krylov subspace, and with su...

ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods

A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem: M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi’s method applie...

A Refined Second-order Arnoldi (RSOAR) Method for the Quadratic Eigenvalue Problem and Implicitly Restarted Algorithms

To implicitly restart the second-order Arnoldi (SOAR) method proposed by Bai and Su for the quadratic eigenvalue problem (QEP), it appears that the SOAR procedure must be replaced by a modified SOAR (MSOAR) one. However, implicit restarts fails to work provided that deflation takes place in the MSOAR procedure. In this paper, we first propose a Refined MSOAR (abbreviated as RSOAR) method that i...