NAG Library Function Document

nag_complex_sparse_eigensystem_init (f12anc)

+− 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_complex_sparse_eigensystem_init (f12anc) is a setup function in a suite of functions consisting of nag_complex_sparse_eigensystem_init (f12anc), nag_complex_sparse_eigensystem_iter (f12apc), nag_complex_sparse_eigensystem_sol (f12aqc), nag_complex_sparse_eigensystem_option (f12arc) and nag_complex_sparse_eigensystem_monit (f12asc). It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by complex nonsymmetric matrices.

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

2 Specification

#include <nag.h>

#include <nagf12.h>

void	nag_complex_sparse_eigensystem_init (Integer n, Integer nev, Integer ncv, Integer icomm[], Integer licomm, Complex comm[], Integer lcomm, NagError *fail)

3 Description

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

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard complex eigenvalue problem

A x = λ x

, or of a generalized complex eigenvalue problem

A x = λ B x

of order

n

, where

n

is large and the coefficient matrices

A

and

B

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

nag_complex_sparse_eigensystem_init (f12anc) is a setup function which must be called before nag_complex_sparse_eigensystem_iter (f12apc), the reverse communication iterative solver, and before nag_complex_sparse_eigensystem_option (f12arc), the options setting function. nag_complex_sparse_eigensystem_sol (f12aqc) is a post-processing function that must be called following a successful final exit from nag_complex_sparse_eigensystem_iter (f12apc), while nag_complex_sparse_eigensystem_monit (f12asc) can be used to return additional monitoring information during the computation.

This setup function initializes the communication arrays, sets (to their default values) all options that can be set by you via the option setting function nag_complex_sparse_eigensystem_option (f12arc), 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 11.1 in nag_complex_sparse_eigensystem_option (f12arc).

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: 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[ $\max (1, licomm)$ ] – IntegerCommunication Array: On exit: contains data to be communicated to the other functions in the suite.
5: licomm – IntegerInput: On entry: the dimension of the array icomm.
If $licomm = - 1$ , a workspace query is assumed and the function only calculates the required dimensions of icomm and comm, which it returns in $icomm [0]$ and $comm [0]$ respectively.

Constraint: $licomm \geq 140 or licomm = - 1$ .
6: comm[ $\max (1, lcomm)$ ] – ComplexCommunication Array: On exit: contains data to be communicated to the other functions in the suite.
7: lcomm – IntegerInput: On entry: the dimension of the array comm.
If $lcomm = - 1$ , a workspace query is assumed and the function only calculates the dimensions of icomm and comm required by nag_complex_sparse_eigensystem_iter (f12apc), which it returns in $icomm [0]$ and $comm [0]$ respectively.

Constraint: $lcomm \geq 3 \times n + 3 \times ncv \times ncv + 5 \times ncv + 60 or lcomm = - 1$ .
8: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
On entry, $nev = ⟨value⟩$ .
Constraint: $nev > 0$ .
NE_INT_2: The length of the integer array icomm is too small $licomm = ⟨value⟩$ , but must be at least $⟨value⟩$ .
NE_INT_3: On entry, $lcomm = ⟨value⟩$ , $n = ⟨value⟩$ and $ncv = ⟨value⟩$ .
Constraint: $lcomm \geq 3 \times n + 3 \times ncv \times ncv + 5 \times ncv + 60$ .
On entry, $ncv = ⟨value⟩$ , $nev = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $ncv \geq nev + 1$ and $ncv \leq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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. The eigenvalues of largest magnitude are found.

NAG Library Function Documentnag_complex_sparse_eigensystem_init (f12anc)