NAG Toolbox: nag_sparseig_real_symm_init (f12fa)

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 Lanczos 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: $\mathbf{nev}<\mathbf{ncv}\le \mathbf{n}$.

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, $\mathbf{ncv}\le \mathbf{nev}$ or $\mathbf{ncv}>\mathbf{n}$.

   $\mathbf{ifail}=4$

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

   $\mathbf{ifail}=5$

On entry, $\mathit{lcomm}<3\times \mathbf{n}+\mathbf{ncv}\times \mathbf{ncv}+8\times \mathbf{ncv}+60$ 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: f12fa_example

function f12fa_example


fprintf('f12fa example results\n\n');

nx  = int64(10);
n   = nx^2;
nev = int64(4);
ncv = int64(10);
imon = 0;

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

% Initialisation Step
[icomm, comm, ifail] = f12fa( ...
                              n, nev, ncv);

% Set Optional Parameters
[icomm, comm, ifail] = f12fd( ...
                              'SMALLEST MAGNITUDE', icomm, comm);

% Solve
while (irevcm ~= 5)
  [irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
    f12fb( ...
           irevcm, resid, v, x, mx, comm, icomm);
  if (irevcm == 1 || irevcm == -1)
    x = f12fa_Ax(nx, x);
  elseif (irevcm == 4 && imon==1)
    [niter, nconv, ritz, rzest] = f12fe(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 = 0;
[nconv, d, z, v, comm, icomm, ifail] = ...
  f12fc( ...
         sigma, resid, v, comm, icomm);

fprintf('Smallest %d Eigenvalues are:\n',nconv);
disp(d(1:nconv));



function [y] = f12fa_Ax(nx, x)

  y = zeros(nx*nx,1);

  h2 = 1/double((nx+1)^2);

  y(1:nx) = f12fa_Bx(nx, x(1:nx));
  y(1:nx) = y(1:nx) - x(nx+1:2*nx);

  for j=2:nx-1
    lo = (j-1)*nx +1;
    hi = j*nx;

    y(lo:hi) = f12fa_Bx(nx, x(lo:hi));
    y(lo:hi) = y(lo:hi) - x(lo-nx:lo-1) - x(hi+1:hi+nx);
  end

  lo = (nx-1)*nx +1;
  hi = nx*nx;
  y(lo:hi) = f12fa_Bx(nx, x(lo:hi));
  y(lo:hi) = y(lo:hi) - x(lo-nx:lo-1);
  y = y/h2;


function [y] = f12fa_Bx(nx,x)

  y = zeros(nx,1);

  dd = 4;
  dl = -1;
  du = -1;

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

f12fa example results

Smallest 4 Eigenvalues are:
   19.6054
   48.2193
   48.2193
   76.8333