*S**OFTWARE PAGE of QIANG YE*

**EIGIFP.m****: - **A matlab
program that computes a few (algebraically) smallest or largest eigenvalues of a large symmetric matrix A or the
generalized eigenvalue problem for a pencil (A, B):

*A
x = lambda x *or

where A and B are symmetric and B is positive definite.

- It is a black-box implementation of the inverse free preconditioned Krylov subspace method of

** G. Golub** and

Symmetric Generalized Eigenvalue Problems,

**Features**:

1. A
two level iteration with a projection on Krylov
subspaces generated by a shifted matrix *A-lambda_k B* in the inner iteration; Either the Lanczos algorithm or the Arnoldi algorithm is employed for the
projection; Adaptive choice of inner iterations;

2. A preconditioning technique based on a congruence transformation to accelerate convergence;

3. A
built-in preconditioner using threshold ILU
factorization; *Particularly suitable*
for problems where any of the following applies:

a) factorization of B (i.e. inverting B) is difficult;

b) factorization of a shifted matrix A-lambda_k B (i.e. inverting it) is difficult;

c) no *a priori* information is available on the location
of the spectrum sought.

**Usage**:

[lambda, x] = eigifp(A):
computes the (algebraically) smallest eigenvalue/eigenvector
of A;

[Lambda, X] = eigifp(A, k): computes the k (algebraicaly)
smallest eigenvalue/eigenvector of A;

[lambda, x] = eigifp(A,
B): computes the (algebraically)
smallest eigenvalue/eigenvector of (A, B);

[Lambda, X] = eigifp(A, B, k): computes the
(algebraically) smallest eigenvalue/eigenvector
of (A, B);

To compute the largest eigenvalues, one simply computes the smallest eigenvalue of –A (or (-A, B)) and then negates the eigenvalue. Example:

[Lambda, X] = eigifp(-A,B,k); Lambda=-Lambda; => k largest eigenvalues/eigenvectors of (A,B);

To use any a priori information or
to control computational cost and optimize performance,

eigifp(A,
opt), eigifp(A, k, opt), eigifp(A,
B, opt) and eigifp(A, B, k, opt) take optional
inputs

defined by opt. Use

*help eigifp*

to find out the details. The following is a
documentation of the program.

We are also working to extend program's capabilities such as dealing with severely clustered eigenvalues

or interior eigenvalues.
So, check back for the future versions.

**Download:**

To download version 2.1.1, please enter your e-mail address in the form below
and then click Download. (Please note
that the e-mail will be added to a distribution list for related
announcements.) EIGIFP is develped by Qiang Ye. This
work is supported in part by NSF grants CCR - 0098133
and DMS-0411502.

Please note that this program is provided for research and educational
use. Neither redistribution nor commercial use is permitted without
consent of the author.

Your suggestions, comments, bug reports, and reports of
failure as well as success would all be very important to this project.
Please send them to:** qye3 “at” uky . edu**.

**BLEIGIFP: - **is based on a block generalization of the inverse
free preconditioned Krylov subspace method that can
compute multiple or severely clustered eigenvalues.
The present version implements either a preconditioned algorithm or
non-preconditioned algorithm, but it has a scheme to choose block size
adaptively. See the BLEIGIFP page for details and
download information.

Other programs for preconditioned eigenvalue iterations can be found in Andrew Knyazev's eigensolvers page .

Return
to Ye's
Homepage.