NAG Library Function Document

nag_real_sparse_eigensystem_monit (f12aec)

Note: this function uses optional arguments to define choices in the problem specification. If you wish to use default settings for all of the optional arguments, then the option setting function nag_real_sparse_eigensystem_option (f12adc) need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in nag_real_sparse_eigensystem_option (f12adc) for a detailed description of the specification of the optional arguments.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_real_sparse_eigensystem_monit (f12aec) can be used to return additional monitoring information during computation. It is in a suite of functions consisting of nag_real_sparse_eigensystem_init (f12aac), nag_real_sparse_eigensystem_iter (f12abc), nag_real_sparse_eigensystem_sol (f12acc), nag_real_sparse_eigensystem_option (f12adc) and nag_real_sparse_eigensystem_monit (f12aec).

2 Specification

#include <nag.h>

#include <nagf12.h>

void	nag_real_sparse_eigensystem_monit (Integer niter, Integer nconv, double ritzr[], double ritzi[], double rzest[], const Integer icomm[], const double comm[])

3 Description

The suite of functions 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.

On an intermediate exit from nag_real_sparse_eigensystem_iter (f12abc) with

irevcm = 4

, nag_real_sparse_eigensystem_monit (f12aec) may be called to return monitoring information on the progress of the Arnoldi iterative process. The information returned by nag_real_sparse_eigensystem_monit (f12aec) is:

–	the number of the current Arnoldi iteration;
–	the number of converged eigenvalues at this point;
–	the real and imaginary parts of the converged eigenvalues;
–	the error bounds on the converged eigenvalues.

nag_real_sparse_eigensystem_monit (f12aec) does not have an equivalent function from the ARPACK package which prints various levels of detail of monitoring information through an output channel controlled via an argument value (see Lehoucq et al. (1998) for details of ARPACK routines). nag_real_sparse_eigensystem_monit (f12aec) should not be called at any time other than immediately following an

irevcm = 4

return from nag_real_sparse_eigensystem_iter (f12abc).

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 Arguments

1: niter – Integer *Output: On exit: the number of the current Arnoldi iteration.
2: nconv – Integer *Output: On exit: the number of converged eigenvalues so far.
3: ritzr[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array ritzr must be at least $ncv$ (see nag_real_sparse_eigensystem_init (f12aac)).
On exit: the first nconv locations of the array ritzr contain the real parts of the converged approximate eigenvalues.
4: ritzi[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array ritzi must be at least $ncv$ (see nag_real_sparse_eigensystem_init (f12aac)).
On exit: the first nconv locations of the array ritzi contain the imaginary parts of the converged approximate eigenvalues.
5: rzest[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array rzest must be at least $ncv$ (see nag_real_sparse_eigensystem_init (f12aac)).
On exit: the first nconv locations of the array rzest contain the Ritz estimates (error bounds) on the converged approximate eigenvalues.
6: icomm[ $\dim$ ] – const IntegerCommunication Array: Note: the dimension, dim, of the array icomm must be at least $\max (1, licomm)$ , where licomm is passed to the setup function (see nag_real_sparse_eigensystem_init (f12aac)).
On entry: the array icomm output by the preceding call to nag_real_sparse_eigensystem_iter (f12abc).
7: comm[ $\dim$ ] – const doubleCommunication Array: Note: the dimension, dim, of the array comm must be at least $\max (1, lcomm)$ , where lcomm is passed to the setup function (see nag_real_sparse_eigensystem_init (f12aac)).
On entry: the array comm output by the preceding call to nag_real_sparse_eigensystem_iter (f12abc).

6 Error Indicators and Warnings

None.

7 Accuracy

A Ritz value,

λ

, is deemed to have converged if its Ritz estimate

\leq Tolerance \times |λ|

. The default

Tolerance

used is the machine precision given by nag_machine_precision (X02AJC).

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example solves

A x = λ B x

in shifted-real mode, where

A

is the tridiagonal matrix with

2

on the diagonal,

- 2

on the subdiagonal and

3

on the superdiagonal. The matrix

B

is the tridiagonal matrix with

4

on the diagonal and

1

on the off-diagonals. The shift sigma,

σ

, is a complex number, and the operator used in the shifted-real iterative process is

OP = real ({(A - σ B)}_{- 1} B)

NAG Library Function Documentnag_real_sparse_eigensystem_monit (f12aec)