NAG Toolbox: nag_lapack_zgebak (f08nw)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{job}$ – string (length ≥ 1)

This must be the same argument job as supplied to nag_lapack_zgebal (f08nv).

Constraint: $\mathbf{job}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.

2:     $\mathrm{side}$ – string (length ≥ 1)

Indicates whether left or right eigenvectors are to be transformed.

$\mathbf{side}=\text{'L'}$

The left eigenvectors are transformed.

$\mathbf{side}=\text{'R'}$

The right eigenvectors are transformed.

Constraint: $\mathbf{side}=\text{'L'}$ or $\text{'R'}$.

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

4:     $\mathrm{ihi}$ – int64int32nag_int scalar

The values $i_{\mathrm{lo}}$ and $i_{\mathrm{hi}}$, as returned by nag_lapack_zgebal (f08nv).

Constraints:

• if $\mathbf{n}>0$, $1\le \mathbf{ilo}\le \mathbf{ihi}\le \mathbf{n}$;
• if $\mathbf{n}=0$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=0$.

5:     $\mathrm{scale}\left(:\right)$ – double array

The dimension of the array scale must be at least $\max \left(1,\mathbf{n}\right)$

Details of the permutations and/or the scaling factors used to balance the original complex general matrix, as returned by nag_lapack_zgebal (f08nv).

6:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex array

The first dimension of the array v must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array v must be at least $\max \left(1,\mathbf{m}\right)$.

The matrix of left or right eigenvectors to be transformed.

Optional Input Parameters

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

Default: the first dimension of the array v.

$n$, the number of rows of the matrix of eigenvectors.

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

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

Default: the second dimension of the array v.

$m$, the number of columns of the matrix of eigenvectors.

Constraint: $\mathbf{m}\ge 0$.

Output Parameters

1:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex array

The first dimension of the array v will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array v will be $\max \left(1,\mathbf{m}\right)$.

The transformed eigenvectors.

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

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: job, 2: side, 3: n, 4: ilo, 5: ihi, 6: scale, 7: m, 8: v, 9: ldv, 10: info.

It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08nw_example

function f08nw_example


fprintf('f08nw example results\n\n');

n = int64(4);
a = [  1.50 - 2.75i,  0.00 + 0.00i,  0.00 + 0.00i,  0.00 + 0.00i;
      -8.06 - 1.24i, -2.50 - 0.50i,  0.00 + 0.00i, -0.75 + 0.50i;
      -2.09 + 7.56i,  1.39 + 3.97i, -1.25 + 0.75i, -4.82 - 5.67i;
       6.18 + 9.79i, -0.92 - 0.62i,  0.00 + 0.00i, -2.50 - 0.50i];

% Balance A
job = 'Both';
[a, ilo, ihi, scale, info] = f08nv(job, a);

% Reduce a to upper Hessenberg form
[H, tau, info] = f08ns(ilo, ihi, a);

% Form Q
[Q, info] = f08nt(ilo, ihi, H, tau);

% Calculate the eigenvalues and Schur factorisation of A
[H, w, Z, info] = f08ps( ...
                         'Schur Form', 'Vectors', ilo, ihi, H, Q);

disp('Eigenvalues:');
disp(w);

% Calculate the eigenvectors of A
[select, ~, VR, m, info] = ...
f08qx( ...
       'Right', 'Backtransform', false, H, complex(zeros(1)), Z, n);
% Rescale
[VR, info] = f08nw( ...
                    'Both', 'Right', ilo, ihi, scale, VR);

% Normalize: largest elements are real
for i = 1:n
  [~,k] = max(abs(real(VR(:,i)))+abs(imag(VR(:,i))));
  VR(:,i) = VR(:,i)*conj(VR(k,i))/abs(VR(k,i))/norm(VR(:,i));
end

disp('Eigenvectors:');
disp(VR);

f08nw example results

Eigenvalues:
  -1.2500 + 0.7500i
  -1.5000 - 0.4975i
  -3.5000 - 0.5025i
   1.5000 - 2.7500i

Eigenvectors:
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1418 - 0.0407i
   0.0000 + 0.0000i  -0.1015 + 0.0009i   0.1756 - 0.4131i  -0.2711 - 0.1812i
   1.0000 + 0.0000i   0.9884 + 0.0000i   0.7420 + 0.0000i   0.8213 + 0.0000i
   0.0000 + 0.0000i   0.0941 + 0.0619i   0.4170 - 0.2722i   0.1110 + 0.4303i