NAG Toolbox: nag_sparseig_real_init (f12aa)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

The order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.

Constraint: $\mathbf{n}>0$.

2:     $\mathrm{nev}$ – int64int32nag_int scalar

The number of eigenvalues to be computed.

Constraint: $0<\mathbf{nev}<\mathbf{n}-1$.

3:     $\mathrm{ncv}$ – int64int32nag_int scalar

The number of Arnoldi basis vectors to use during the computation.

At present there is no a priori analysis to guide the selection of ncv relative to nev. However, it is recommended that $\mathbf{ncv}\ge 2\times \mathbf{nev}+1$. If many problems of the same type are to be solved, you should experiment with increasing ncv while keeping nev fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.

Constraint: $\mathbf{nev}+1<\mathbf{ncv}\le \mathbf{n}$.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{icomm}\left(\max \left(1,\mathit{licomm}\right)\right)$ – int64int32nag_int array

Contains data to be communicated to the other functions in the suite.

2:     $\mathrm{comm}\left(\max \left(1,\mathit{lcomm}\right)\right)$ – double array

Contains data to be communicated to the other functions in the suite.

3:     $\mathrm{ifail}$ – 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, $\mathbf{n}\le 0$.

   $\mathbf{ifail}=2$

On entry, $\mathbf{nev}\le 0$.

   $\mathbf{ifail}=3$

On entry, $\mathbf{ncv}<\mathbf{nev}+2$ or $\mathbf{ncv}>\mathbf{n}$.

   $\mathbf{ifail}=4$

On entry, $\mathit{licomm}<140$ and $\mathit{licomm}\ne -1$.

   $\mathbf{ifail}=5$

On entry, $\mathit{lcomm}<3\times \mathbf{n}+3\times \mathbf{ncv}\times \mathbf{ncv}+6\times \mathbf{ncv}+60$ and $\mathit{lcomm}\ne -1$.

   $\mathbf{ifail}=-99$

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: f12aa_example

function f12aa_example


fprintf('f12aa example results\n\n');

% Discretization is on 12x12 grid, so spacing is h = 1/11;
% Dirichlet conditions reduce this to a system of order 10x10.
nx   = int64(10);
n    = nx^2;
nev  = int64(10);
ncv  = int64(30);

h    = 1/(double(nx)+1);
rho  = 100;

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

dd = 4/h^2;
dl = -1/h^2 - rho/(2*h);
du = -1/h^2 + rho/(2*h);
y = zeros(n,1);

[icomm, comm, ifail] = f12aa( ...
                              n, nev, ncv);
[icomm, comm, ifail] = f12ad( ...
                              'Smallest Mag', icomm, comm);

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)
    % -u_xx-rho*u_x 
    y(1:nx:n) = dd*x(1:nx:n) + du*x(2:nx:n);
    for j = 2:nx-1
      y(j:nx:n) = dl*x(j-1:nx:n) + dd*x(j:nx:n) + du*x(j+1:nx:n);
    end
    y(nx:nx:n) = dl*x(nx-1:nx:n) + dd*x(nx:nx:n);
    % -u_yy
    y(1:nx) = y(1:nx) - (1/h^2)*x(nx+1:2*nx);
    for j = 2:nx-1
      lo = (j-1)*nx;
      y(lo+1:lo+nx) =  y(lo+1:lo+nx) - (1/h^2)*x(lo+nx+1:lo+2*nx) ...
                       - (1/h^2)*x(lo-nx+1:lo);
    end
    lo = (nx-1)*nx;
    y(lo+1:lo+nx) =  y(lo+1:lo+nx) - (1/h^2)*x(lo-nx+1:lo);
    x = y;
  end
end

[nconv, dr, di, z, v, comm, icomm, ifail] = ...
  f12ac( ...
         0, 0, resid, v, comm, icomm);

fprintf('The %4d Ritz values of smallest magnitude are:\n\n',nconv);
fprintf('%9.3f %+9.3fi\n', [dr di]');

f12aa example results

The   10 Ritz values of smallest magnitude are:

  251.803  +152.711i
  251.803  -152.711i
  280.417  +152.711i
  280.417  -152.711i
  325.524  +152.711i
  325.524  -152.711i
  383.470  +152.711i
  383.470  -152.711i
  449.560  +152.711i
  449.560  -152.711i