NAG Library Function Document
nag_real_sparse_eigensystem_sol (f12acc)
Note: this function uses optional parameters to define choices in the problem specification. If you wish to use default
settings for all of the optional parameters, 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 parameters.
1
Purpose
nag_real_sparse_eigensystem_sol (f12acc) is a post-processing function that must be called following a final exit from
nag_real_sparse_eigensystem_iter (f12abc). These are part of a suite of functions for the solution of real sparse eigensystems. The suite also includes
nag_real_sparse_eigensystem_init (f12aac),
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_sol (Integer *nconv,
double dr[],
double di[],
double z[],
double sigmar,
double sigmai,
const double resid[],
double v[],
double comm[],
Integer icomm[],
NagError *fail) |
|
3
Description
The suite of functions is designed to calculate some of the eigenvalues, , (and optionally the corresponding eigenvectors, ) of a standard eigenvalue problem , or of a generalized eigenvalue problem of order , where is large and the coefficient matrices and are sparse, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.
Following a call to
nag_real_sparse_eigensystem_iter (f12abc),
nag_real_sparse_eigensystem_sol (f12acc) returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real nonsymmetric matrices. There is negligible additional cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
nag_real_sparse_eigensystem_sol (f12acc) is based on the function
dneupd from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in
Lehoucq and Sorensen (1996) and
Lehoucq (2001) while its use within the ARPACK software is described in great detail in
Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in
Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
nag_real_sparse_eigensystem_sol (f12acc), is a post-processing function that must be called following a successful final exit from
nag_real_sparse_eigensystem_iter (f12abc).
nag_real_sparse_eigensystem_sol (f12acc) uses data returned from
nag_real_sparse_eigensystem_iter (f12abc) and options, set either by default or explicitly by calling
nag_real_sparse_eigensystem_option (f12adc), to return the converged approximations to selected eigenvalues and (optionally):
– |
the corresponding approximate eigenvectors; |
– |
an orthonormal basis for the associated approximate invariant subspace; |
– |
both. |
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:
– Integer *Output
-
On exit: the number of converged eigenvalues as found by
nag_real_sparse_eigensystem_iter (f12abc).
- 2:
– doubleOutput
-
Note: the dimension,
dim, of the array
dr
must be at least
(see
nag_real_sparse_eigensystem_init (f12aac)).
On exit: the first
nconv locations of the array
dr contain the real parts of the converged approximate eigenvalues.
- 3:
– doubleOutput
-
Note: the dimension,
dim, of the array
di
must be at least
(see
nag_real_sparse_eigensystem_init (f12aac)).
On exit: the first
nconv locations of the array
di contain the imaginary parts of the converged approximate eigenvalues.
- 4:
– doubleOutput
-
On exit: if the default option
(see
nag_real_sparse_eigensystem_option (f12adc)) has been selected then
z contains the final set of eigenvectors corresponding to the eigenvalues held in
dr and
di. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive array segments. The first segment holds the real part of the eigenvector and the second holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.
For example, the first eigenvector has real parts stored in locations
, for and imaginary parts stored in
, for .
- 5:
– doubleInput
-
On entry: if one of the
modes have been selected then
sigmar contains the real part of the shift used; otherwise
sigmar is not referenced.
- 6:
– doubleInput
-
On entry: if one of the
modes have been selected then
sigmai contains the imaginary part of the shift used; otherwise
sigmai is not referenced.
- 7:
– const doubleInput
-
Note: the dimension,
dim, of the array
resid
must be at least
(see
nag_real_sparse_eigensystem_init (f12aac)).
On entry: must not be modified following a call to
nag_real_sparse_eigensystem_iter (f12abc) since it contains data required by
nag_real_sparse_eigensystem_sol (f12acc).
- 8:
– doubleInput/Output
-
The th element of the th basis vector is stored in location , for and .
On entry: the
ncv sections of
v, of length
, contain the Arnoldi basis vectors for
as constructed by
nag_real_sparse_eigensystem_iter (f12abc).
On exit: if the option
has been set, or the option
has been set and a separate array
z has been passed (i.e.,
z does not equal
v), then the first
nconv sections of
v, of length
, will contain approximate Schur vectors that span the desired invariant subspace.
- 9:
– doubleCommunication Array
-
Note: the dimension,
dim, of the array
comm
must be at least
(see
nag_real_sparse_eigensystem_init (f12aac)).
On initial entry: must remain unchanged from the prior call to
nag_real_sparse_eigensystem_iter (f12abc).
On exit: contains data on the current state of the solution.
- 10:
– IntegerCommunication Array
-
Note: the dimension,
dim, of the array
icomm
must be at least
(see
nag_real_sparse_eigensystem_init (f12aac)).
On initial entry: must remain unchanged from the prior call to
nag_real_sparse_eigensystem_iter (f12abc).
On exit: contains data on the current state of the solution.
- 11:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INITIALIZATION
-
Either the solver function has not been called prior to the call of this function or a communication array has become corrupted.
- NE_INTERNAL_EIGVEC_FAIL
-
In calculating eigenvectors, an internal call returned with an error. Please contact
NAG.
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_INVALID_OPTION
-
On entry, , but this is not yet implemented.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_RITZ_COUNT
-
Got a different count of the number of converged Ritz values than the value passed to it through the argument
icomm: number counted
, number expected
. This usually indicates that a communication array has been altered or has become corrupted between calls to
nag_real_sparse_eigensystem_iter (f12abc) and
nag_real_sparse_eigensystem_sol (f12acc).
- NE_SCHUR_EIG_FAIL
-
During calculation of a real Schur form, there was a failure to compute eigenvalues in a total of iterations.
- NE_SCHUR_REORDER
-
The computed Schur form could not be reordered by an internal call. This function returned with
. Please contact
NAG.
- NE_ZERO_EIGS_FOUND
-
The number of eigenvalues found to sufficient accuracy, as communicated through the argument
icomm, is zero. You should experiment with different values of
nev and
ncv, or select a different computational mode or increase the maximum number of iterations prior to calling
nag_real_sparse_eigensystem_iter (f12abc).
7
Accuracy
The relative accuracy of a Ritz value,
, is considered acceptable if its Ritz estimate
. The default
used is the
machine precision given by
nag_machine_precision (X02AJC).
8
Parallelism and Performance
nag_real_sparse_eigensystem_sol (f12acc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
None.
10
Example
This example solves in regular-invert mode, where and are obtained from the standard central difference discretization of the one-dimensional convection-diffusion operator on , with zero Dirichlet boundary conditions.
10.1
Program Text
Program Text (f12acce.c)
10.2
Program Data
Program Data (f12acce.d)
10.3
Program Results
Program Results (f12acce.r)