NAG Library Routine Document

F12AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F12AAF is a setup routine in a suite of routines consisting of F12AAF, F12ABF, F12ACF, F12ADF and F12AEF. It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real nonsymmetric matrices.

The suite of routines is suitable for the solution of large sparse, standard or generalized, nonsymmetric eigenproblems where only a few eigenvalues from a selected range of the spectrum are required.

2 Specification

SUBROUTINE F12AAF (

N, NEV, NCV, ICOMM, LICOMM, COMM, LCOMM, IFAIL)

INTEGER	N, NEV, NCV, ICOMM(max(1,LICOMM)), LICOMM, LCOMM, IFAIL
REAL (KIND=nag_wp)	COMM(max(1,LCOMM))

3 Description

The suite of routines is designed to calculate some of the eigenvalues,

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard eigenvalue problem

A x = λ x

, or of a generalized eigenvalue problem

A x = λ B x

of order

n

, where

n

is large and the coefficient matrices

A

and

B

are sparse, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.

F12AAF is a setup routine which must be called before F12ABF, the reverse communication iterative solver, and before F12ADF, the options setting routine. F12ACF is a post-processing routine that must be called following a successful final exit from F12ABF, while F12AEF can be used to return additional monitoring information during the computation.

This setup routine initializes the communication arrays, sets (to their default values) all options that can be set by you via the option setting routine F12ADF, and checks that the lengths of the communication arrays as passed by you are of sufficient length. For details of the options available and how to set them see Section 10.1 in F12ADF.

4 References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562

Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory

Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821

Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

5 Parameters

1: N – INTEGERInput: On entry: the order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.
Constraint: $N > 0$ .
2: NEV – INTEGERInput: On entry: the number of eigenvalues to be computed.
Constraint: $0 < NEV < N - 1$ .
3: NCV – INTEGERInput: On entry: the number of Arnoldi basis vectors to use during the computation.
At present there is no a priori analysis to guide the selection of NCV relative to NEV. However, it is recommended that $NCV \geq 2 \times NEV + 1$ . If many problems of the same type are to be solved, you should experiment with increasing NCV while keeping NEV fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.

Constraint: $NEV + 1 < NCV \leq N$ .
4: ICOMM( $\max (1, LICOMM)$ ) – INTEGER arrayCommunication Array: On exit: contains data to be communicated to the other routines in the suite.
5: LICOMM – INTEGERInput: On entry: the dimension of the array ICOMM as declared in the (sub)program from which F12AAF is called.
If $LICOMM = - 1$ , a workspace query is assumed and the routine only calculates the required dimensions of ICOMM and COMM, which it returns in $ICOMM (1)$ and $COMM (1)$ respectively.

Constraint: $LICOMM \geq 140 or LICOMM = - 1$ .
6: COMM( $\max (1, LCOMM)$ ) – REAL (KIND=nag_wp) arrayCommunication Array: On exit: contains data to be communicated to the other routines in the suite.
7: LCOMM – INTEGERInput: On entry: the dimension of the array COMM as declared in the (sub)program from which F12AAF is called.
If $LCOMM = - 1$ , a workspace query is assumed and the routine only calculates the dimensions of ICOMM and COMM required by F12ABF, which it returns in $ICOMM (1)$ and $COMM (1)$ respectively.

Constraint: $LCOMM \geq 3 \times N + 3 \times NCV \times NCV + 6 \times NCV + 60 or LCOMM = - 1$ .
8: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $N \leq 0$ .

$IFAIL = 2$: On entry, $NEV \leq 0$ .

$IFAIL = 3$: On entry, $NCV < NEV + 2$ or $NCV > N$ .

$IFAIL = 4$: On entry, $LICOMM < 140$ and $LICOMM \neq - 1$ .

$IFAIL = 5$: On entry, $LCOMM < 3 \times N + 3 \times NCV \times NCV + 6 \times NCV + 60$ and $LCOMM \neq - 1$ .

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example solves

A x = λ x

in regular mode, where

A

is obtained from the standard central difference discretization of the convection-diffusion operator

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + ρ \frac{\partial u}{\partial x}

on the unit square, with zero Dirichlet boundary conditions, where

ρ = 100

NAG Library Routine DocumentF12AAF