183 research outputs found
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 [{\em 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 such basis they project the QEP onto the subspace and
compute the Ritz pairs and the refined Ritz pairs, respectively. We develop
implicitly restarted GSOAR and RGSOAR algorithms, in which we propose certain
exact and refined shifts for respective use within the two algorithms.
Numerical experiments on real-world problems illustrate the efficiency of the
restarted algorithms and the superiority of the restarted RGSOAR to the
restarted GSOAR. The experiments also demonstrate that both IGSOAR and IRGSOAR
generally perform much better than the implicitly restarted Arnoldi method
applied to the corresponding linearization problems, in terms of the accuracy
and the computational efficiency.Comment: 30 pages, 6 figure
Augmented Block Householder Arnoldi Method
AbstractComputing the eigenvalues and eigenvectors of a large sparse nonsymmetric matrix arises in many applications and can be a very computationally challenging problem. In this paper we propose the Augmented Block Householder Arnoldi (ABHA) method that combines the advantages of a block routine with an augmented Krylov routine. A public domain MATLAB code ahbeigs has been developed and numerical experiments indicate that the code is competitive with other publicly available codes
On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method
Using a new analysis approach, we establish a general convergence theory of
the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple
eigenvalue nearest to a given target and the associated eigenvector.
In SIRA, a subspace expansion vector at each step is obtained by solving a
certain inner linear system. We prove that the inexact SIRA method mimics the
exact SIRA well, that is, the former uses almost the same outer iterations to
achieve the convergence as the latter does if all the inner linear systems are
iteratively solved with {\em low} or {\em modest} accuracy during outer
iterations. Based on the theory, we design practical stopping criteria for
inner solves. Our analysis is on one step expansion of subspace and the
approach applies to the Jacobi--Davidson (JD) method with the fixed target
as well, and a similar general convergence theory is obtained for it.
Numerical experiments confirm our theory and demonstrate that the inexact SIRA
and JD are similarly effective and are considerably superior to the inexact
SIA.Comment: 20 pages, 8 figure
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