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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgebak (f08nw)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zgebak (f08nw) transforms eigenvectors of a balanced matrix to those of the original complex general matrix.


[v, info] = f08nw(job, side, ilo, ihi, scale, v, 'n', n, 'm', m)
[v, info] = nag_lapack_zgebak(job, side, ilo, ihi, scale, v, 'n', n, 'm', m)


nag_lapack_zgebak (f08nw) is intended to be used after a complex general matrix A has been balanced by nag_lapack_zgebal (f08nv), and eigenvectors of the balanced matrix A22 have subsequently been computed.
For a description of balancing, see the document for nag_lapack_zgebal (f08nv). The balanced matrix A is obtained as A=DPAPTD-1, where P is a permutation matrix and D is a diagonal scaling matrix. This function transforms left or right eigenvectors as follows:




Compulsory Input Parameters

1:     job – string (length ≥ 1)
This must be the same argument job as supplied to nag_lapack_zgebal (f08nv).
Constraint: job='N', 'P', 'S' or 'B'.
2:     side – string (length ≥ 1)
Indicates whether left or right eigenvectors are to be transformed.
The left eigenvectors are transformed.
The right eigenvectors are transformed.
Constraint: side='L' or 'R'.
3:     ilo int64int32nag_int scalar
4:     ihi int64int32nag_int scalar
The values ilo and ihi, as returned by nag_lapack_zgebal (f08nv).
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
5:     scale: – double array
The dimension of the array scale must be at least max1,n
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:     vldv: – complex array
The first dimension of the array v must be at least max1,n.
The second dimension of the array v must be at least max1,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: n0.
2:     m int64int32nag_int scalar
Default: the second dimension of the array v.
m, the number of columns of the matrix of eigenvectors.
Constraint: m0.

Output Parameters

1:     vldv: – complex array
The first dimension of the array v will be max1,n.
The second dimension of the array v will be max1,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

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.


The errors are negligible.

Further Comments

The total number of real floating-point operations is approximately proportional to nm.
The real analogue of this function is nag_lapack_dgebak (f08nj).


See Example in nag_lapack_zgebal (f08nv).
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);


% 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));


f08nw example results

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

   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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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