if $x$ is a right eigenvector of $A^{''}$ , $P^{T} D^{- 1} x$ is a right eigenvector of $A$ ;
if $y$ is a left eigenvector of $A^{''}$ , $P^{T} D y$ is a left eigenvector of $A$ .

References

None.

Parameters

Compulsory Input Parameters

1: $job$ – string (length ≥ 1)

This must be the same argument job as supplied to nag_lapack_dgebal (f08nh).

Constraint:

job ='N'

'P'

'S'

'B'

2: $side$ – string (length ≥ 1)

Indicates whether left or right eigenvectors are to be transformed.

$side ='L'$: The left eigenvectors are transformed.
$side ='R'$: The right eigenvectors are transformed.

Constraint:

side ='L'

'R'

3: $ilo$ – int64int32nag_int scalar

4: $ihi$ – int64int32nag_int scalar

The values

i_{lo}

and

i_{hi}

, as returned by nag_lapack_dgebal (f08nh).

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

5: $scale (:)$ – double array

The dimension of the array scale must be at least

\max (1, n)

Details of the permutations and/or the scaling factors used to balance the original real nonsymmetric matrix, as returned by nag_lapack_dgebal (f08nh).

6: $v (ldv :)$ – double array

The first dimension of the array v must be at least

\max (1, n)

The second dimension of the array v must be at least

\max (1, m)

The matrix of left or right eigenvectors to be transformed.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array v.
$n$ , the number of rows of the matrix of eigenvectors.

Constraint: $n \geq 0$ .
2: $m$ – int64int32nag_int scalar: Default: the second dimension of the array v.
$m$ , the number of columns of the matrix of eigenvectors.

Constraint: $m \geq 0$ .

Output Parameters

1: $v (ldv :)$ – double array: The first dimension of the array v will be $\max (1, n)$ .
The second dimension of the array v will be $\max (1, m)$ .

The transformed eigenvectors.
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , 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

The errors are negligible.

Further Comments

The total number of floating-point operations is approximately proportional to

n m

The complex analogue of this function is nag_lapack_zgebak (f08nw).

Example

See Example in nag_lapack_dgebal (f08nh).

Open in the MATLAB editor: f08nj_example

function f08nj_example


fprintf('f08nj example results\n\n');

n = int64(4);
a = [ 5.14, 0.91,  0.00, -32.80;
      0.91, 0.20,  0.00,  34.50;
      1.90, 0.80, -0.40,  -3.00;
     -0.33, 0.35,  0.00,   0.66];

% Balance a
[a, ilo, ihi, scale, info] = f08nh( ...
                                    'Both', a);

% Reduce a to upper Hessenberg form
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

% Form Q
[Q, info] = f08nf( ...
                   ilo, ihi, H, tau);

% Calculate the eigenvalues and Schur factorisation of A
[H, wr, wi, Z, info] = f08pe( ...
                              'Schur Form', 'Vectors', ilo, ihi, H, Q);

w = wr + i*wi;
disp('Eigenvalues:');
disp(w);

% Calculate the eigenvectors of A
[select, ~, VR, m, info] = ...
f08qk( ...
       'Right', 'Backtransform', false, H, zeros(1), Z, n);

% Back scale to get eigenvectors of A
[VR, info] = f08nj( ...
                    'Both', 'Right', ilo, ihi, scale, VR);

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(VR(:,j)));
  VR(:,j) =VR(:,j)/norm(VR(:,j));
  if VR(k,j) < 0;
    VR(:,j) = -VR(:,j);
  end
end                            

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

f08nj example results

Eigenvalues:
   -0.4000
   -4.0208
    3.0136
    7.0072

Eigenvectors:
         0   -0.4381    0.4654    0.9513
         0    0.8923    0.7888   -0.1714
    1.0000   -0.0481    0.3981    0.2494
         0   -0.0976    0.0521   -0.0589

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox