NAG Toolbox: nag_sparseig_real_band_init (f12af)

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_real_band_solve (f12ag).

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

Contains data to be communicated to nag_sparseig_real_band_solve (f12ag).

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, $\mathit{lcomm}<60$ 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: f12af_example

function f12af_example


fprintf('f12af example results\n\n');

nx  = int64(10);
n   = nx*nx;
nev = int64(4);
ncv = nx;
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;
for j=1:n
  ab(idiag,j) = 4;
  mb(idiag,j) = 4;
end

% 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
for j = 1:n - 1
  mb(isub,j+1) = 1;
  mb(isup,j) = 1;
end

% 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

sigmar = 0.4;
sigmai = 0.6;
resid = zeros(100,1);

[icomm, comm, ifail] = f12af( ...
                              n, nev, ncv);
[icomm, comm, ifail] = f12ad( ...
                              'Shifted imaginary', icomm, comm);
[icomm, comm, ifail] = f12ad( ...
                              'Generalized', icomm, comm);
[nconv, dr, di, z, resid, v, comm, icomm, ifail] = ...
  f12ag( ...
         kl, ku, ab, mb, sigmar, sigmai, resid, comm, icomm);

fprintf('The %4d Ritz values closest to %8.2f %+8.2fi are:\n\n', ...
        nconv, sigmar, sigmai);
fprintf('%9.4f %+9.4fi\n', [dr(1:nconv) di(1:nconv)]');

f12af example results

The    4 Ritz values closest to     0.40    +0.60i are:

   0.3610   +0.7223i
   0.3610   -0.7223i
   0.4598   -0.7199i
   0.4598   +0.7199i