An iterative block Arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems
type: Item
instance of: scholarly article
title: An iterative block Arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems
zbMATH Open document ID: 1054.65035
DOI: 10.1016/S0096-3003(03)00655-6
author: Gang Wu
published in: Applied Mathematics and Computation
publication date: 2004-08-06T00:00:00Z
review text: In order to handle large unsymmetric eigenvalue problems the author modifies the \(m\)-step Arnoldi method to its block version. The resulting algorithm updates the block Krylov subspaces \(K_{m}\left( A,V_{1}\right) ,\) spanned by \(V_{1},AV_{1},\dots,AV_{1}^{m-1},\) where \(V_{1}\) is an initial \(N\times p\) orthonormal matrix. It basically uses the Ritz pair \(\left( \lambda _{i}^{(m)},\varphi _{i}^{(m)}\right) \;\)of \(A\) in \(K_{m}\left( A,V_{1}\right) \) to approximate the sought eigenpair \(\left( \lambda _{i},\varphi _{i}\right).\) The two main advantages of the method are: its efficiency to compute clustered eigenvalues and also multiple eigenvalues. The algorithm is also suitable for parallel processing. The convergence of the method is examined and numerical experiments are used to illustrate the efficiency of the method.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65Y05
zbMATH DE Number: 2082102
zbMATH Keywords: Block Arnoldi method
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: orthogonal projection
zbMATH Keywords: Krylov subspace methods
zbMATH Keywords: Ritz values
zbMATH Keywords: Ritz vectors
zbMATH Keywords: algorithm
zbMATH Keywords: multiple eigenvalues
zbMATH Keywords: parallel processing
zbMATH Keywords: convergence
zbMATH Keywords: numerical experiments
reviewed by: Amin Boumenir
describes a project that uses: Harwell-Boeing sparse matrix collection
describes a project that uses: eigs
describes a project that uses: IRAM
MaRDI profile type: Publication
cites work: Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices
cites work: Numerical methods for large eigenvalue problems
cites work: The principle of minimized iterations in the solution of the matrix eigenvalue problem
cites work: A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices
cites work: Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems
cites work: An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices
cites work: Generalized block Lanczos methods for large unsymmetric eigenproblems
cites work: A Simultaneous Iteration Method for the Unsymmetric Eigenvalue Problem
cites work: The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices
cites work: The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems
cites work: An analysis of the Rayleigh-Ritz method for approximating eigenspaces
cites work: Q4851796
cites work: Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems
cites work: A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems
cites work: A variation on the block Arnoldi method for large unsymmetric matrix eigenproblems
cites work: Q4503865
cites work: Reorthogonalization and Stable Algorithms for Updating the Gram-Schmidt QR Factorization
cites work: Implicit Application of Polynomial Filters in a k-Step Arnoldi Method
cites work: A refined subspace iteration algorithm for large sparse eigenproblems
cites work: Q5689624
cites work: Absolute and relative perturbation bounds for invariant subspaces of matrices
cites work: Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm
cites work: Sparse matrix test problems
cites work: Chebyshev Acceleration Techniques for Solving Nonsymmetric Eigenvalue Problems
full work available at URL: s0096-3003(03)00655-6
OpenAlex ID: W1987373851
description: scientific article; zbMATH DE number 2082102