JDQR

From this page you can get a Matlab® implementation of the JDQR algorithm.

The JDQR algorithm can be used for computing a few selected eigenvalues with some desirable property together with the associated eigenvectors of a matrix A. The matrix can be real or complex, Hermitian or non-Hermitian, .... The algorithm is effective especially in case A is sparse and of large size.

The Jacobi-Davidson method is used to compute a partial Schur decomposition of A. The decomposition leads to the wanted eigenpairs.

The Jacobi-Davidson method has been introduced in

G.L.G. Sleijpen and H.A. van der Vorst,: A Jacobi-Davidson iteration method for linear eigenvalue problems,; SIAM J. Matrix Anal. Appl. (SIMAX), 17 (1996), pp. 401-425 .

For the JDQR algorithm, see

D.R. Fokkema, G.L.G. Sleijpen , and H.A. van der Vorst,: Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils,; SIAM J. Sc. Comput., 20:1 (1998), pp. 94-125

The Matlab® implementation here is based on the algorithms as discussed in

Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk Van der Vorst (eds.),: Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide ,; Chapter 4.7 (for symmetric matrices A) and; Chapter 7.12 (for general type of matrices A).

The Matlab® code here contains a number of function M-files (as a GMRES, Bi-CGSTAB, BiCGstab(ell), ... implementation for solving linear systems) that can be used the improve the performance of the JDQR algorithm. These functions M-files are bundled in one file jdqr.m. In this form, jdqr.m requires Matlab version 5.1 or higher.

We also provide a few "test" files that can help to see how the jdqr.m can be used. jdqr.tar.gz contains the M-files jdqr.m plus the test files in tarred and zipped form (apply gunzip jdqr.tar.gz and tar xvf jdqr.tar to unpack). jdqr.m and the test files can also be obtained separately: see below, where you can also find a description of their use.

The code is distributed under the terms of the GNU General Public License (version 2 of the License, or any later version) as published by the Free Software Foundation.

File:	`jdqr.m`
Requires:	Matlab Version 5.1.
Function:	`[X,Lambda,Q,S,HISTORY] = JDQR(matrix/filename,K,SIGMA,OPTIONS);`
Description

File:	`testB.m`
Requires:	`jdqr.m` and Matlab Version 5.1.
Function:	`testB`
Description:	Contains some examples for using `jdqr.m`

File:	`testA.m`
Requires:	`jdqr.m`, `ILU.m, Example1.m, Example2.m` and Matlab Version 5.1.
Function:	`testA`
Description:	Contains some examples for using preconditioning in `jdqr.m`

File:	`Example1.m`
Requires:	Matlab Version 5.1.
Function:	`out=Example1(v)`
Description:	Example of a function file for `jdqr` of a linear operator. `out` is the result of the operator applied to the vector `v`.

File:	`Example2.m`
Requires:	Matlab Version 5.1.
Function:	`out=Example2(v,flag)`
Description:	Example of a function file for `jdqr` of a linear operator. `out` is the result of the operator applied to the vector `v` when applied with 1 input argument. If `flag='dimension'` then `out` is the dimension `N`. If `flag='preconditioner'` then `out` is the result of "a preconditioner".

File:	`ILU.m`
Requires:	Matlab Version 5.1.
Function:	`out=ILU(v)`
Description:	Example of a function file for `jdqr` of a preconditioner. `out` is the result of the preconditioner applied to the vector `v`.