NAG Library Function Document
nag_real_banded_sparse_eigensystem_sol (f12agc)
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_banded_sparse_eigensystem_sol (f12agc) is the main solver function in a suite of functions consisting of
nag_real_sparse_eigensystem_option (f12adc),
nag_real_banded_sparse_eigensystem_init (f12afc) and
nag_real_banded_sparse_eigensystem_sol (f12agc). It must be called following an initial call to
nag_real_banded_sparse_eigensystem_init (f12afc) and following any calls to
nag_real_sparse_eigensystem_option (f12adc).
nag_real_banded_sparse_eigensystem_sol (f12agc) returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded nonsymmetric matrices. The banded matrix must be stored using the LAPACK
column ordered
storage format for real banded nonsymmetric
(see
Section 3.3.4 in the f07 Chapter Introduction).
2
Specification
#include <nag.h> |
#include <nagf12.h> |
void |
nag_real_banded_sparse_eigensystem_sol (Integer kl,
Integer ku,
const double ab[],
const double mb[],
double sigmar,
double sigmai,
Integer *nconv,
double dr[],
double di[],
double z[],
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 banded, real and nonsymmetric.
Following a call to the initialization function
nag_real_banded_sparse_eigensystem_init (f12afc),
nag_real_banded_sparse_eigensystem_sol (f12agc) 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 banded nonsymmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices
and
must be stored using the LAPACK column ordered storage format for banded nonsymmetric matrices; please refer to
Section 3.3.2 in the f07 Chapter Introduction for details on this storage format.
nag_real_banded_sparse_eigensystem_sol (f12agc) is based on the banded driver functions
dnbdr1 to
dnbdr6 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 banded driver software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose function.
nag_real_banded_sparse_eigensystem_sol (f12agc), is a general purpose function that must be called following initialization by
nag_real_banded_sparse_eigensystem_init (f12afc).
nag_real_banded_sparse_eigensystem_sol (f12agc) uses 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. |
Please note that for problems, the and inverse modes are only appropriate if either or is symmetric semidefinite. Otherwise, if or is non-singular, the problem can be solved using the matrix (say).
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:
– IntegerInput
-
On entry: the number of subdiagonals of the matrices and .
Constraint:
.
- 2:
– IntegerInput
-
On entry: the number of superdiagonals of the matrices and .
Constraint:
.
- 3:
– const doubleInput
-
Note: the dimension,
dim, of the array
ab
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On entry: must contain the matrix
in LAPACK column-ordered banded storage format for nonsymmetric matrices (see
Section 3.3.4 in the f07 Chapter Introduction).
- 4:
– const doubleInput
-
Note: the dimension,
dim, of the array
mb
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On entry: must contain the matrix
in LAPACK column-ordered banded storage format for nonsymmetric matrices (see
Section 3.3.4 in the f07 Chapter Introduction).
- 5:
– doubleInput
-
On entry: if one of the
modes (see
nag_real_sparse_eigensystem_option (f12adc)) have been selected then
sigmar must contain the real part of the shift used; otherwise
sigmar is not referenced.
Section 4.3.4 in the f12 Chapter Introduction describes the use of shift and inverse transformations.
- 6:
– doubleInput
-
On entry: if one of the
modes (see
nag_real_sparse_eigensystem_option (f12adc)) have been selected then
sigmai must contain the imaginary part of the shift used; otherwise
sigmai is not referenced.
Section 4.3.4 in the f12 Chapter Introduction describes the use of shift and inverse transformations.
- 7:
– Integer *Output
-
On exit: the number of converged eigenvalues.
- 8:
– doubleOutput
-
Note: the dimension,
dim, of the array
dr
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On exit: the first
nconv locations of the array
dr contain the real parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
nev since complex values occur as conjugate pairs and the second in the pair can be returned in position
of the array.
- 9:
– doubleOutput
-
Note: the dimension,
dim, of the array
di
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On exit: the first
nconv locations of the array
di contain the imaginary parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
nev since complex values occur as conjugate pairs and the second in the pair can be returned in position
of the array.
- 10:
– doubleOutput
-
On exit: if the default option
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, if is nonzero, the first eigenvector has real parts stored in locations
, for and imaginary parts stored in
, for .
- 11:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
resid
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On entry: need not be set unless the option
has been set in a prior call to
nag_real_sparse_eigensystem_option (f12adc) in which case
resid must contain an initial residual vector.
On exit: contains the final residual vector.
- 12:
– doubleOutput
-
On exit: if the option
(see
nag_real_sparse_eigensystem_option (f12adc)) has been set to Schur or Ritz then the first
nconv sections of
v, of length
, will contain approximate Schur vectors that span the desired invariant subspace.
The th Schur vector is stored in locations
, for and .
- 13:
– doubleCommunication Array
-
Note: the dimension,
dim, of the array
comm
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On exit: contains no useful information.
- 14:
– IntegerCommunication Array
-
Note: the dimension,
dim, of the array
icomm
must be at least
(see
nag_real_banded_sparse_eigensystem_init (f12afc)).
On exit: contains no useful information.
- 15:
– 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_COMP_BAND_FAC
-
Failure during internal factorization of complex banded matrix. Please contact
NAG.
- NE_COMP_BAND_SOL
-
Failure during internal solution of complex banded matrix. Please contact
NAG.
- NE_INITIALIZATION
-
Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
The maximum number of iterations , the option has been set to .
- NE_INTERNAL_EIGVAL_FAIL
-
Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact
NAG.
- NE_INTERNAL_EIGVEC_FAIL
-
Error in internal call to compute eigenvectors. 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_MAX_ITER
-
The maximum number of iterations has been reached. The maximum number of . The number of converged eigenvalues .
- NE_NO_ARNOLDI_FAC
-
Could not build an Arnoldi factorization. The size of the current Arnoldi factorization .
- 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_NO_SHIFTS_APPLIED
-
No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
- NE_OPT_INCOMPAT
-
The options and are incompatible.
- NE_REAL_BAND_FAC
-
Failure during internal factorization of real banded matrix. Please contact
NAG.
- NE_REAL_BAND_SOL
-
Failure during internal solution of real banded matrix. Please contact
NAG.
- NE_SCHUR_EIG_FAIL
-
During calculation of a real Schur form, there was a failure to compute a number of eigenvalues. Please contact
NAG.
- NE_SCHUR_REORDER
-
The computed Schur form could not be reordered by an internal call. Please contact
NAG.
- NE_TRANSFORM_OVFL
-
Overflow occurred during transformation of Ritz values to those of the original problem.
- NE_ZERO_EIGS_FOUND
-
The number of eigenvalues found to sufficient accuracy is zero.
- NE_ZERO_INIT_RESID
-
The option
was selected but the starting vector held in
resid is zero.
- NE_ZERO_SHIFT
-
The option
has been selected and
zero on entry;
sigmai must be nonzero for this mode of operation.
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_banded_sparse_eigensystem_sol (f12agc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_banded_sparse_eigensystem_sol (f12agc) 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 constructs the matrices and using LAPACK band storage format and solves in shifted imaginary mode using the complex shift .
10.1
Program Text
Program Text (f12agce.c)
10.2
Program Data
Program Data (f12agce.d)
10.3
Program Results
Program Results (f12agce.r)