NAG Toolbox: nag_sparseig_complex_proc (f12aq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{sigma}$ – complex scalar

If one of the Shifted Inverse (see nag_sparseig_complex_option (f12ar)) modes has been selected then sigma contains the shift used; otherwise sigma is not referenced.

2:     $\mathrm{resid}\left(:\right)$ – complex array

The dimension of the array resid must be at least $\mathbf{n}$ (see nag_sparseig_complex_init (f12an))

Must not be modified following a call to nag_sparseig_complex_iter (f12ap) since it contains data required by nag_sparseig_complex_proc (f12aq).

3:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex 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_complex_iter (f12ap).

4:     $\mathrm{comm}\left(:\right)$ – complex array

The dimension of the array comm must be at least $\max \left(1,\mathbf{lcomm}\right)$ (see nag_sparseig_complex_init (f12an))

On initial entry: must remain unchanged from the prior call to nag_sparseig_complex_init (f12an).

5:     $\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

The dimension of the array icomm must be at least $\max \left(1,\mathbf{licomm}\right)$ (see nag_sparseig_complex_init (f12an))

On initial entry: must remain unchanged from the prior call to nag_sparseig_complex_init (f12an).

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{nconv}$ – int64int32nag_int scalar

The number of converged eigenvalues as found by nag_sparseig_complex_option (f12ar).

2:     $\mathrm{d}\left(:\right)$ – complex array

The dimension of the array d will be $\mathbf{ncv}$ (see nag_sparseig_complex_init (f12an))

The first nconv locations of the array d contain the converged approximate eigenvalues.

3:     $\mathrm{z}\left(\mathbf{n}\times \mathbf{ncv}\right)$ – complex 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 d. The complex eigenvector associated with an eigenvalue is stored in the corresponding column of z.

4:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex 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 $\mathbf{Vectors}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ 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.

5:     $\mathrm{comm}\left(:\right)$ – complex array

The dimension of the array comm will be $\max \left(1,\mathbf{lcomm}\right)$ (see nag_sparseig_complex_init (f12an))

Contains data on the current state of the solution.

6:     $\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

The dimension of the array icomm will be $\max \left(1,\mathbf{licomm}\right)$ (see nag_sparseig_complex_init (f12an))

Contains data on the current state of the solution.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry, $\mathit{ldz}<\max \left(1,\mathbf{n}\right)$ or $\mathit{ldz}<1$ when no vectors are required.

   $\mathbf{ifail}=2$

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_complex_proc (f12aq), as communicated through the argument icomm, is zero.

   $\mathbf{ifail}=4$

The number of converged eigenvalues as calculated by nag_sparseig_complex_iter (f12ap) differ from the value passed to it through the argument icomm.

   $\mathbf{ifail}=5$

Unexpected error during calculation of a 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_complex_iter (f12ap) 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: f12aq_example

function f12aq_example


fprintf('f12aq example results\n\n');

global rho;

n   = int64(100);
nev = int64(4);
ncv = int64(20);
rho = 10;
imon = 0;

irevcm = int64(0);
resid = complex(zeros(n,1));
v  = complex(zeros(n,ncv));
x  = complex(zeros(n,1));
mx = complex(zeros(n,1));
z  = complex(zeros(n,1));

% Initialisation Step
[icomm, comm, ifail] = f12an( ...
                              n, nev, ncv);
[icomm, comm, ifail] = f12ar( ...
                              'Regular Inverse', icomm, comm);
[icomm, comm, ifail] = f12ar( ...
                              'Generalized', icomm, comm);

% Factorize B
h  = 1/double(n+1);
cl = complex(h*ones(n-1,1));
cd = complex(4*h*ones(n,1));
cu = cl;

[cl, cd, cu, cu2, ipiv, info] = f07cr( ...
                                       cl, cd, cu);

% Solve
while (irevcm ~= 5)
  [irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
    f12ap( ...
           irevcm, resid, v, x, mx, comm, icomm);
  if (irevcm == 1 || irevcm == -1)
    % Solve By = Ax;
    ax = f12aq_Ax(n, x);
    [x, info] = f07cs( ...
                       'N', cl, cd, cu, cu2, ipiv, ax);
  elseif (irevcm == 2)
    % y = Bx
    x = f12aq_Ax(n, x);
  elseif (irevcm == 4 && imon==1)
    [niter, nconv, ritz, rzest] = f12as( ...
                                         icomm, comm);
    fprintf(['Iteration %2d, No. converged = %d, ', ...
             'norm of estimates = %10.2e\n'], ...
            niter, nconv, norm(rzest(1:nev),2));
  end
end

% Post-process to compute eigenvalues/vectors
sigma = complex(0);
[nconv, d, z, v, comm, icomm, ifail] = ...
f12aq( ...
       sigma, resid, v, comm, icomm);

fprintf('Largest %d Eigenvalues are:\n',nconv);
fprintf('%12.4f%+12.4fi\n',[real(d(1:nconv)) imag(d(1:nconv))]');



function [y] = f12aq_Ax(n, x)

  global rho;

  y = complex(ones(n,1));

  h = 1/double(n+1);
  s = rho/2;
  dd = complex(2/h);
  dl = complex(-1/h - s);
  du = complex(-1/h + s);

  y(1) = dd*x(1) + du*x(2);
  for j=2:n-1
    y(j) = dl*x(j-1) + dd*x(j) +du*x(j+1);
  end
  y(n) = dl*x(n-1) + dd*x(n);


function [y] = f12aq_Bx(n,x)

  y = complex(ones(n,1));

  h = 1/double(n+1);
  dd = complex(4*h);
  dl = complex(h);
  du = complex(h);
  y(1) = dd*x(1) +du*x(2);
  for j=2:n-1
    y(j) = dl*x(j-1) + dd*x(j) +du*x(j+1);
  end
  y(n) = dl*x(n-1) + dd*x(n);

f12aq example results

Largest 4 Eigenvalues are:
  20383.0384     -0.0000i
  20338.7563     +0.0000i
  20265.2844     -0.0000i
  20163.1142     +0.0000i