NAG Toolbox: nag_sparseig_complex_init (f12an)

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:     $\mathrm{comm}\left(\max \left(1,\mathit{lcomm}\right)\right)$ – complex 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, $\mathit{lcomm}<3\times \mathbf{n}+3\times \mathbf{ncv}\times \mathbf{ncv}+5\times \mathbf{ncv}+60$ 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: f12an_example

function f12an_example


fprintf('f12an example results\n\n');

load('west0479.mat')
A   = west0479;
m   = size(A,1);
n   = int64(m);
nev = int64(8);
ncv = int64(20);

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

sigma = complex(0);

% Initialisation Step
[icomm, comm, ifail] = f12an(n, nev, ncv);

% 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)
    x = A*x;
  elseif (irevcm == 4)

    [niter, nconv, ritz, rzest] = f12as( ...
                                         icomm, comm);

    fprintf('Iteration %2d, No. converged = %d', niter, nconv);
    fprintf(' norm of estimates = %10.2e\n', norm(rzest(1:nev),2));
  end
end

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

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

fig1 = figure;
plot(d(1:nconv),'b*'); 
title('Largest eigenvalues of sparse matrix west0479');
xlabel('real');
ylabel('imaginary')

f12an example results

Iteration  1, No. converged = 2 norm of estimates =   2.01e-02
Iteration  2, No. converged = 2 norm of estimates =   1.87e-02
Iteration  3, No. converged = 2 norm of estimates =   3.60e-03
Iteration  4, No. converged = 2 norm of estimates =   4.62e-05
Iteration  5, No. converged = 2 norm of estimates =   8.49e-08
Iteration  6, No. converged = 2 norm of estimates =   2.89e-10
Iteration  7, No. converged = 2 norm of estimates =   1.14e-12


Largest 8 Eigenvalues are:
   1.0e+03 *

   0.0000 + 1.7007i
   0.0000 - 1.7007i
  -0.1009 + 0.0666i
  -0.1009 - 0.0666i
   0.1081 + 0.0541i
   0.1081 - 0.0541i
  -0.0072 + 0.1207i
  -0.0072 - 0.1207i