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NAG Toolbox: nag_sparseig_real_proc (f12ac)
Purpose
nag_sparseig_real_proc (f12ac) is a postprocessing function that must be called following a final exit from
nag_sparseig_real_iter (f12ab). These are part of a suite of functions for the solution of real sparse eigensystems. The suite also includes
nag_sparseig_real_init (f12aa),
nag_sparseig_real_option (f12ad) and
nag_sparseig_real_monit (f12ae).
Syntax
[
nconv,
dr,
di,
z,
v,
comm,
icomm,
ifail] = f12ac(
sigmar,
sigmai,
resid,
v,
comm,
icomm)
[
nconv,
dr,
di,
z,
v,
comm,
icomm,
ifail] = nag_sparseig_real_proc(
sigmar,
sigmai,
resid,
v,
comm,
icomm)
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
nag_sparseig_real_iter (f12ab),
nag_sparseig_real_proc (f12ac) 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_sparseig_real_proc (f12ac) 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_sparseig_real_proc (f12ac), is a postprocessing function that must be called following a successful final exit from
nag_sparseig_real_iter (f12ab).
nag_sparseig_real_proc (f12ac) uses data returned from
nag_sparseig_real_iter (f12ab) and options, set either by default or explicitly by calling
nag_sparseig_real_option (f12ad), to return the converged approximations to selected eigenvalues and (optionally):
– 
the corresponding approximate eigenvectors; 
– 
an orthonormal basis for the associated approximate invariant subspace; 
– 
both. 
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
Parameters
Compulsory Input Parameters
 1:
$\mathrm{sigmar}$ – double scalar

If one of the
Shifted Inverse Real modes have been selected then
sigmar contains the real part of the shift used; otherwise
sigmar is not referenced.
 2:
$\mathrm{sigmai}$ – double scalar

If one of the
Shifted Inverse Real modes have been selected then
sigmai contains the imaginary part of the shift used; otherwise
sigmai is not referenced.
 3:
$\mathrm{resid}\left(:\right)$ – double array

The dimension of the array
resid
must be at least
${\mathbf{n}}$ (see
nag_sparseig_real_init (f12aa))
Must not be modified following a call to
nag_sparseig_real_iter (f12ab) since it contains data required by
nag_sparseig_real_proc (f12ac).
 4:
$\mathrm{v}\left(\mathit{ldv},:\right)$ – double array

The first dimension of the array
v must be at least
${\mathbf{n}}$.
The second dimension of the array
v must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncv}}\right)$.
The
ncv columns of
v contain the Arnoldi basis vectors for
$\mathrm{OP}$ as constructed by
nag_sparseig_real_iter (f12ab).
 5:
$\mathrm{comm}\left(:\right)$ – double array

The dimension of the array
comm
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see
nag_sparseig_real_init (f12aa))
On initial entry: must remain unchanged from the prior call to
nag_sparseig_real_iter (f12ab).
 6:
$\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

The dimension of the array
icomm
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see
nag_sparseig_real_init (f12aa))
On initial entry: must remain unchanged from the prior call to
nag_sparseig_real_iter (f12ab).
Optional Input Parameters
None.
Output Parameters
 1:
$\mathrm{nconv}$ – int64int32nag_int scalar

The number of converged eigenvalues as found by
nag_sparseig_real_iter (f12ab).
 2:
$\mathrm{dr}\left(:\right)$ – double array

The dimension of the array
dr will be
${\mathbf{nev}}$ (see
nag_sparseig_real_init (f12aa))
The first
nconv locations of the array
dr contain the real parts of the converged approximate eigenvalues.
 3:
$\mathrm{di}\left(:\right)$ – double array

The dimension of the array
di will be
${\mathbf{nev}}$ (see
nag_sparseig_real_init (f12aa))
The first
nconv locations of the array
di contain the imaginary parts of the converged approximate eigenvalues.
 4:
$\mathrm{z}\left({\mathbf{n}}\times \left({\mathbf{nev}}+1\right)\right)$ – double array

If the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
nag_sparseig_real_option (f12ad)) 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 columns. The first column holds the real part of the eigenvector and the second column 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.
 5:
$\mathrm{v}\left(\mathit{ldv},:\right)$ – double array

The first dimension of the array
v will be
${\mathbf{n}}$.
The second dimension of the array
v will be
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncv}}\right)$.
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 columns of
v will contain approximate Schur vectors that span the desired invariant subspace.
 6:
$\mathrm{comm}\left(:\right)$ – double array

The dimension of the array
comm will be
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see
nag_sparseig_real_init (f12aa))
Contains data on the current state of the solution.
 7:
$\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

The dimension of the array
icomm will be
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see
nag_sparseig_real_init (f12aa))
Contains data on the current state of the solution.
 8:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
 ${\mathbf{ifail}}=1$

On entry, $\mathit{ldz}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ or $\mathit{ldz}<1$ when no vectors are required.
 W ${\mathbf{ifail}}=2$

On entry, the option ${\mathbf{Vectors}}=\text{Select}$ was selected, but this is not yet implemented.
 ${\mathbf{ifail}}=3$

The number of eigenvalues found to sufficient accuracy prior to calling
nag_sparseig_real_proc (f12ac), as communicated through the argument
icomm, is zero.
 ${\mathbf{ifail}}=4$

The number of converged eigenvalues as calculated by
nag_sparseig_real_iter (f12ab) differ from the value passed to it through the argument
icomm.
 ${\mathbf{ifail}}=5$

Unexpected error during calculation of a real Schur form: there was a failure to compute all the converged eigenvalues. Please contact
NAG.
 ${\mathbf{ifail}}=6$

Unexpected error: the computed Schur form could not be reordered by an internal call. Please contact
NAG.
 ${\mathbf{ifail}}=7$

Unexpected error in internal call while calculating eigenvectors. Please contact
NAG.
 ${\mathbf{ifail}}=8$

Either the solver function
nag_sparseig_real_iter (f12ab) has not been called prior to the call of this function or a communication array has become corrupted.
 ${\mathbf{ifail}}=9$

The function was unable to dynamically allocate sufficient internal workspace. Please contact
NAG.
 ${\mathbf{ifail}}=10$

An unexpected error has occurred. Please contact
NAG.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
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
Tolerance used is the
machine precision given by
nag_machine_precision (x02aj).
Further Comments
None.
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.
Open in the MATLAB editor:
f12ac_example
function f12ac_example
fprintf('f12ac example results\n\n');
n = int64(100);
nev = int64(4);
ncv = int64(20);
h = 1/(double(n)+1);
rho = 10;
md = repmat(4*h, double(n), 1);
me = repmat(h, double(n1), 1);
irevcm = int64(0);
resid = zeros(n,1);
v = zeros(n, ncv);
x = zeros(n, 1);
mx = zeros(n);
dd = 2/h;
dl = 1/h  rho/2;
du = 1/h + rho/2;
y = zeros(n,1);
[icomm, comm, ifail] = f12aa( ...
n, nev, ncv);
[icomm, comm, ifail] = f12ad( ...
'Regular Inverse', icomm, comm);
[icomm, comm, ifail] = f12ad( ...
'Generalized', icomm, comm);
[md, me, info] = f07jd(md, me);
while (irevcm ~= 5)
[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
f12ab( ...
irevcm, resid, v, x, mx, comm, icomm);
if (irevcm == 1  irevcm == 1)
y(1) = dd*x(1) + du*x(2);
for i = 2:n1
y(i) = dl*x(i1) + dd*x(i) + du*x(i+1);
end
y(n) = dl*x(n1) + dd*x(n);
[x, info] = f07je(md, me, y);
elseif (irevcm == 2)
y(1) = 4*x(1) + x(2);
for i=2:n1
y(i) = x(i1) + 4*x(i) + x(i+1);
end
y(n) = x(n1) + 4*x(n);
x = h*y;
elseif (irevcm == 4)
[niter, nconv, ritzr, ritzi, rzest] = ...
f12ae(icomm, comm);
if (niter == 1)
fprintf('\n');
end
fprintf('Iteration %2d No. converged = %d ', niter, nconv);
fprintf('Norm of estimates = %10.2e\n', norm(rzest));
end
end
[nconv, dr, di, z, v, comm, icomm, ifail] = ...
f12ac(0, 0, resid, v, comm, icomm);
fprintf('\nThe %4d Ritz values of largest magnitude are:\n\n',nconv);
fprintf('%9.4f %+9.4fi\n', [dr di]');
f12ac example results
Iteration 1 No. converged = 0 Norm of estimates = 5.56e+03
Iteration 2 No. converged = 0 Norm of estimates = 5.45e+03
Iteration 3 No. converged = 0 Norm of estimates = 5.30e+03
Iteration 4 No. converged = 0 Norm of estimates = 6.24e+03
Iteration 5 No. converged = 0 Norm of estimates = 7.16e+03
Iteration 6 No. converged = 0 Norm of estimates = 5.45e+03
Iteration 7 No. converged = 0 Norm of estimates = 6.43e+03
Iteration 8 No. converged = 0 Norm of estimates = 5.11e+03
Iteration 9 No. converged = 0 Norm of estimates = 7.19e+03
Iteration 10 No. converged = 1 Norm of estimates = 5.78e+03
Iteration 11 No. converged = 2 Norm of estimates = 4.73e+03
Iteration 12 No. converged = 3 Norm of estimates = 5.00e+03
The 4 Ritz values of largest magnitude are:
20383.0384 +0.0000i
20338.7563 +0.0000i
20265.2844 +0.0000i
20163.1142 +0.0000i
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