NAG Toolbox: nag_sparseig_real_proc (f12ac)

Purpose

Syntax

Description

References

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 $\max \left(1,\mathbf{ncv}\right)$.

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 $\max \left(1,\mathbf{lcomm}\right)$ (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 $\max \left(1,\mathbf{licomm}\right)$ (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 $\max \left(1,\mathbf{ncv}\right)$.

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 $\max \left(1,\mathbf{lcomm}\right)$ (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 $\max \left(1,\mathbf{licomm}\right)$ (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

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, $\mathit{ldz}<\max \left(1,\mathbf{n}\right)$ 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

Further Comments

Example

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(n-1), 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);

% Construct m and factorise
[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:n-1
      y(i) = dl*x(i-1) + dd*x(i) + du*x(i+1);
    end
    y(n) = dl*x(n-1) + dd*x(n);
    [x, info] = f07je(md, me, y);
  elseif (irevcm == 2)
    y(1) = 4*x(1) + x(2);
    for i=2:n-1
      y(i) = x(i-1) + 4*x(i) + x(i+1);
    end
    y(n) = x(n-1) + 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