NAG Toolbox: nag_sparseig_complex_band_init (f12at)

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

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 nag_sparseig_complex_band_solve (f12au).

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

Contains data to be communicated to nag_sparseig_complex_band_solve (f12au).

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

   $\mathbf{ifail}=1$

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

   $\mathbf{ifail}=2$

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

   $\mathbf{ifail}=3$

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

   $\mathbf{ifail}=4$

The length of the integer array icomm is too small $\mathit{licomm}=\_$, but must be at least $\_$.

   $\mathbf{ifail}=5$

The length of the complex array comm is too small $\mathit{lcomm}=\_$, but must be at least $\_$.

   $\mathbf{ifail}=-99$

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: f12at_example

function f12at_example


fprintf('f12at example results\n\n');

nx = int64(10);
n = nx*nx;
nev = int64(4);
ncv = int64(10);
sigma = 0.4 + 0.6i;

% Construct the matrix A in banded form and store in AB.
% KU, KL are number of superdiagonals and subdiagonals within
% the band of matrices A and M.
kl = nx;
ku = nx;

% Construct ab and mb
ab = zeros(2*kl+ku+1,n);
mb = zeros(2*kl+ku+1,n);

% Main diagonal of A.
idiag = kl + ku + 1;
ab(idiag,1:n) = 4;
mb(idiag,1:n) = ab(idiag,1);

% First subdiagonal and superdiagonal of A.
isup = kl + ku;
isub = kl + ku + 2;
rho = 100;
h = 1/double(nx+1);
for i=1:nx
  lo = (i-1)*nx;
  for j=lo+1:lo+nx-1
    ab(isub,j+1) = -1 + 0.5*h*rho;
    ab(isup,j) = -1 - 0.5*h*rho;
  end
end
mb(isub,2:n)   = 1;
mb(isup,1:n-1) = 1;

% kl-th subdiagonal and ku-th super-diagonal.
isup = kl + 1;
isub = 2*kl + ku + 1;
for i = 1:nx - 1
  lo = (i-1)*nx;
  for j = lo + 1:lo + nx
    ab(isup,nx+j) = -1;
    ab(isub,j) = -1;
  end
end

resid = complex(zeros(100,1));

[icomm, comm, ifail] = f12at( ...
                              n, nev, ncv);
[icomm, comm, ifail] = f12ar( ...
                              'SHIFTED INVERSE', icomm, comm);
[icomm, comm, ifail] = f12ar( ...
                              'GENERALIZED', icomm, comm);

% Find eigenvalues closest in value to SIGMA and corresponding eigenvectors
[nconv, d, z, resid, v, comm, icomm, ifail] = ...
  f12au( ...
         kl, ku, complex(ab), complex(mb), sigma, resid, comm, icomm);

fprintf('\nRitz values closest to sigma:\n');
disp(d);

f12at example results


Ritz values closest to sigma:
   0.3610 + 0.7223i
   0.4598 + 0.7199i
   0.2868 + 0.7241i
   0.2410 + 0.7257i