NAG Toolbox: nag_lapack_dggbak (f08wj)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Specifies the backward transformation step required.

$\mathbf{job}=\text{'N'}$

No transformations are done.

$\mathbf{job}=\text{'P'}$

Only do backward transformations based on permutations.

$\mathbf{job}=\text{'S'}$

Only do backward transformations based on scaling.

$\mathbf{job}=\text{'B'}$

Do backward transformations for both permutations and scaling.

Note:  this must be the same argument job as supplied to nag_lapack_dggbal (f08wh).

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

$i_{\mathrm{lo}}$ and $i_{\mathrm{hi}}$ as determined by a previous call to nag_lapack_dggbal (f08wh).

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

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

Details of the permutations and scaling factors applied to the left side of the matrices $A$ and $B$, as returned by a previous call to nag_lapack_dggbal (f08wh).

6:     $\mathrm{rscale}\left(:\right)$ – double array

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

Details of the permutations and scaling factors applied to the right side of the matrices $A$ and $B$, as returned by a previous call to nag_lapack_dggbal (f08wh).

7:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double 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 right or left eigenvectors, as returned by nag_lapack_dggbal (f08wh).

Optional Input Parameters

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

Default: the first dimension of the array v.

$n$, the order of the matrices $A$ and $B$ of the generalized eigenvalue problem.

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

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

Default: the second dimension of the array v.

$m$, the required number of left or right eigenvectors.

Constraint: $0\le \mathbf{m}\le \mathbf{n}$.

Output Parameters

1:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double 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 right or left 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: lscale, 7: rscale, 8: m, 9: v, 10: ldv, 11: 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: f08wj_example

function f08wj_example


fprintf('f08wj example results\n\n');

n = 5;
a = [ 1.0   1.0    1.0    1.0     1.0;
      2.0   4.0    8.0   16.0    32.0;
      3.0   9.0   27.0   81.0   243.0;
      4.0  16.0   64.0  256.0  1024.0;
      5.0  25.0  125.0  625.0  3125.0];
b = a';

%' Balance A and B
job = 'B';
[a, b, ilo, ihi, lscale, rscale, info] = ...
  f08wh(job, a, b);

bbal = b(ilo:ihi,ilo:ihi);
abal = a(ilo:ihi,ilo:ihi);

% QR factorize balanced B
[QR, tau, info] = f08ae(bbal);

% Perform C = Q^T*A
side = 'Left';
trans = 'Transpose';
[c, info] = f08ag( ...
		   side, trans, QR, tau, abal);

% Generalized Hessenberg form (C,R) -> (H,T)
% Form Q explicitly and let Z = I.
[q, info] = f08af(QR, tau);

z         = eye(n);
jlo       = int64(1);
jhi       = int64(ihi-ilo+1);
compq     = 'Vectors Q';
compz     = 'Vectors Z';
[H, T, q, z, info] = ...
  f08we( ...
         compq, compz, jlo, jhi, c, QR, q, z);

% Find eigenvalues of generalized Hessenberg form
%    = eigenvalues of (A,B).
% and return Schur form for computing eigenvectors
job = 'Schur form';
[HS, TS, alphar, alphai, beta, q, z, info] = ...
  f08xe( ...
	 job, compq, compz, jlo, jhi, H, T, q, z);

disp('Generalized eigenvalues of (A,B):');
w = complex(alphar+i*alphai);
disp(w./beta);

% Obtain scaled eigenvectors from Schur form
side = 'Both sides';
howmny = 'Backtransformed using Q and Z';
select = [false];
[q, z, m, info] = f08yk( ...
			 side, howmny, select, HS, TS, q, z, jhi);

% rescale to obtain left and right eigenvectors of (A,B)
job  = 'Back scale';
side = 'Left';
[VL, info] = f08wj( ...
		    job, side, jlo, jhi, lscale, rscale, q);
side = 'Right';
[VR, info] = f08wj( ...
		    job, side, jlo, jhi, lscale, rscale, z);

disp('Left Eigenvectors');
disp(VL);

disp('Right Eigenvectors');
disp(VR);

f08wj example results

Generalized eigenvalues of (A,B):
  -2.4367 + 0.0000i
   0.6069 + 0.7948i
   0.6069 - 0.7948i
   1.0000 + 0.0000i
  -0.4104 + 0.0000i

Left Eigenvectors
   -0.0695   -0.2092   -0.0053   -0.0741    0.0494
    0.1361    0.1635    0.1137    0.1354   -0.1061
   -0.1000   -0.0463   -0.0537   -0.1000    0.1000
    0.0319    0.0059    0.0148    0.0265   -0.0438
   -0.0036   -0.0002   -0.0021   -0.0037    0.0070

Right Eigenvectors
   -0.0494   -0.2077    0.0257   -0.0741   -0.0695
    0.1061    0.1785    0.0883    0.1354    0.1361
   -0.1000   -0.0537   -0.0463   -0.1000   -0.1000
    0.0438    0.0080    0.0138    0.0265    0.0319
   -0.0070   -0.0006   -0.0021   -0.0037   -0.0036