NAG CL Interface
f12acc (real_proc)
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 f12adc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adc for a detailed description of the specification of the optional parameters.
1
Purpose
f12acc is a postprocessing function that must be called following a final exit from
f12abc. These are part of a suite of functions for the solution of real sparse eigensystems. The suite also includes
f12aac,
f12adc and
f12aec.
2
Specification
void 
f12acc (Integer *nconv,
double dr[],
double di[],
double z[],
double sigmar,
double sigmai,
const double resid[],
double v[],
double comm[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: f12acc, nag_sparseig_real_proc or nag_real_sparse_eigensystem_sol.
3
Description
The suite of functions is designed to calculate some of the eigenvalues, $\lambda $, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ 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.
Following a call to
f12abc,
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.
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.
f12acc, is a postprocessing function that must be called following a successful final exit from
f12abc.
f12acc uses data returned from
f12abc and options, set either by default or explicitly by calling
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 MCSP5471195 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 Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5
Arguments

1:
$\mathbf{nconv}$ – Integer *
Output

On exit: the number of converged eigenvalues as found by
f12abc.

2:
$\mathbf{dr}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
dr
must be at least
${\mathbf{nev}}+1$ (see
f12aac).
On exit: the first
nconv locations of the array
dr contain the real parts of the converged approximate eigenvalues.

3:
$\mathbf{di}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
di
must be at least
${\mathbf{nev}}+1$ (see
f12aac).
On exit: the first
nconv locations of the array
di contain the imaginary parts of the converged approximate eigenvalues.

4:
$\mathbf{z}\left[{\mathbf{n}}\times \left({\mathbf{nev}}+1\right)\right]$ – double
Output

On exit: if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
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
${\mathbf{z}}\left[\mathit{i}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and imaginary parts stored in
${\mathbf{z}}\left[\mathit{i}1\right]$, for $\mathit{i}={\mathbf{n}}+1,2{\mathbf{n}}$.

5:
$\mathbf{sigmar}$ – double
Input

On entry: if one of the
${\mathbf{Shifted\; Inverse\; Real}}$ modes have been selected then
sigmar contains the real part of the shift used; otherwise
sigmar is not referenced.

6:
$\mathbf{sigmai}$ – double
Input

On entry: if one of the
${\mathbf{Shifted\; Inverse\; Real}}$ modes have been selected then
sigmai contains the imaginary part of the shift used; otherwise
sigmai is not referenced.

7:
$\mathbf{resid}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
resid
must be at least
${\mathbf{n}}$ (see
f12aac).
On entry: must not be modified following a call to
f12abc since it contains data required by
f12acc.

8:
$\mathbf{v}\left[{\mathbf{n}}\times {\mathbf{ncv}}\right]$ – double
Input/Output

The $\mathit{i}$th element of the $\mathit{j}$th basis vector is stored in location ${\mathbf{v}}\left[{\mathbf{n}}\times \left(\mathit{j}1\right)+\mathit{i}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{ncv}}$.
On entry: the
ncv sections of
v, of length
$n$, contain the Arnoldi basis vectors for
$\mathrm{OP}$ as constructed by
f12abc.
On exit: if the option
${\mathbf{Vectors}}=\mathrm{SCHUR}$ has been set, or the option
${\mathbf{Vectors}}=\mathrm{RITZ}$ 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
$n$, will contain approximate Schur vectors that span the desired invariant subspace.

9:
$\mathbf{comm}\left[\mathit{dim}\right]$ – double
Communication Array

Note: the actual argument supplied
must be the array
comm supplied to the initialization routine
f12aac.
On initial entry: must remain unchanged from the prior call to
f12abc.
On exit: contains data on the current state of the solution.

10:
$\mathbf{icomm}\left[\mathit{dim}\right]$ – Integer
Communication Array

Note: the actual argument supplied
must be the array
icomm supplied to the initialization routine
f12aac.
On initial entry: must remain unchanged from the prior call to
f12abc.
On exit: contains data on the current state of the solution.

11:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_OPTION

On entry, ${\mathbf{Vectors}}=\mathrm{SELECT}$, but this is not yet implemented.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface 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
$=\u2329\mathit{\text{value}}\u232a$, number expected
$=\u2329\mathit{\text{value}}\u232a$. This usually indicates that a communication array has been altered or has become corrupted between calls to
f12abc and
f12acc.
 NE_SCHUR_EIG_FAIL

During calculation of a real Schur form, there was a failure to compute $\u2329\mathit{\text{value}}\u232a$ eigenvalues in a total of $\u2329\mathit{\text{value}}\u232a$ iterations.
 NE_SCHUR_REORDER

The computed Schur form could not be reordered by an internal call. This function returned with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\u2329\mathit{\text{value}}\u232a$. 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
f12abc.
7
Accuracy
The relative accuracy of a Ritz value,
$\lambda $, is considered acceptable if its Ritz estimate
$\le {\mathbf{Tolerance}}\times \left\lambda \right$. The default
${\mathbf{Tolerance}}$ used is the
machine precision given by
X02AJC.
8
Parallelism and Performance
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 implementationspecific information.
None.
10
Example
This example solves $Ax=\lambda Bx$ in regularinvert mode, where $A$ and $B$ are obtained from the standard central difference discretization of the onedimensional convectiondiffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$
on $\left[0,1\right]$, with zero Dirichlet boundary conditions.
10.1
Program Text
10.2
Program Data
10.3
Program Results