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NAG Toolbox: nag_lapack_zggbak (f08ww)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_lapack_zggbak (f08ww) forms the right or left eigenvectors of the real generalized eigenvalue problem Ax=λBx, by backward transformation on the computed eigenvectors given by nag_lapack_ztgevc (f08yx). It is necessary to call this function only if the optional balancing function nag_lapack_zggbal (f08wv) was previously called to balance the matrix pair A,B.

Syntax

[v, info] = f08ww(job, side, ilo, ihi, lscale, rscale, v, 'n', n, 'm', m)
[v, info] = nag_lapack_zggbak(job, side, ilo, ihi, lscale, rscale, v, 'n', n, 'm', m)

Description

If the matrix pair has been previously balanced using the function nag_lapack_zggbal (f08wv) then nag_lapack_zggbak (f08ww) backtransforms the eigenvector solution given by nag_lapack_ztgevc (f08yx). This is usually the sixth and last step in the solution of the generalized eigenvalue problem.
For a description of balancing, see the document for nag_lapack_zggbal (f08wv).

References

Ward R C (1981) Balancing the generalized eigenvalue problem SIAM J. Sci. Stat. Comp. 2 141–152

Parameters

Compulsory Input Parameters

1:     job – string (length ≥ 1)
Specifies the backtransformation step required.
job='N'
No transformations are done.
job='P'
Only do backward transformations based on permutations.
job='S'
Only do backward transformations based on scaling.
job='B'
Do backward transformations for both permutations and scaling.
Note:  this must be identical to the argument job as supplied to nag_lapack_dggbal (f08wh).
Constraint: job='N', 'P', 'S' or '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' or 'R'.
3:     ilo – int64int32nag_int scalar
4:     ihi – int64int32nag_int scalar
ilo and ihi as determined by a previous call to nag_lapack_zggbal (f08wv).
Constraints:
  • if n>0, 1≤ ilo≤ ihi≤ n ;
  • if n=0, ilo=1 and ihi=0.
5:     lscale: – double array
The dimension of the array lscale must be at least max1,n
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_zggbal (f08wv).
6:     rscale: – double array
The dimension of the array rscale must be at least max1,n
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_zggbal (f08wv).
7:     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 right or left eigenvectors, as returned by nag_lapack_zggbal (f08wv).

Optional Input Parameters

1:     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: n≥0.
2:     m – int64int32nag_int scalar
Default: the second dimension of the array v.
m, the required number of left or right eigenvectors.
Constraint: 0≤m≤n.

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 right or left 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: 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

The errors are negligible.

Further Comments

The number of operations is proportional to n2.
The real analogue of this function is nag_lapack_dggbak (f08wj).

Example

See Example in nag_lapack_zhgeqz (f08xs) and nag_lapack_ztgevc (f08yx).

Open in the MATLAB editor: f08ww_example

function f08ww_example


fprintf('f08ww example results\n\n');

% Generalized eigenvalues of matrix pair (A,B) , where
a = [ 1.0+3.0i  1.0+4.0i  1.0+5.0i   1.0+6.0i;
      2.0+2.0i  4.0+3.0i  8.0+4.0i  16.0+5.0i;
      3.0+1.0i  9.0+2.0i 27.0+3.0i  81.0+4.0i;
      4.0+0.0i 16.0+1.0i 64.0+2.0i 256.0+3.0i];
b = [ 1.0+0.0i  2.0+1.0i  3.0+2.0i   4.0+3.0i;
      1.0+1.0i  4.0+2.0i  9.0+3.0i  16.0+4.0i;
      1.0+2.0i  8.0+3.0i 27.0+4.0i  64.0+5.0i;
      1.0+3.0i 16.0+4.0i 81.0+5.0i 256.0+6.0i];

% Balance matrix pair
job = 'B';
[a, b, ilo, ihi, lscale, rscale, info] = ...
  f08wv(job, a, b);
bbal = b(ilo:ihi,ilo:ihi);
abal = a(ilo:ihi,ilo:ihi);

% QR factorize balanced B
[QR, tau, info] = f08as(bbal);

% Perform C = Q^H*A
side = 'Left';
trans = 'Conjugate transpose';
[c, info] = f08au( ...
		   side, trans, QR, tau, abal);

% Generalized Hessenberg form (C,R) -> (H,T)
compq = 'Vectors Q';
compz = 'Vectors Z';

% Form Q explicitly and let Z = I.
[q, info] = f08at(QR, tau);
z = complex(eye(4));

jlo = int64(1);
jhi = int64(ihi-ilo+1);
[H, T, q, z, info] = ...
  f08ws( ...
	 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';
ilo = int64(1);
ihi = int64(4);
[HS, TS, alpha, beta, q, z, info] = ...
  f08xs( ...
	 job, compq, compz, jlo, jhi, H, T, q, z);

disp('Generalized eigenvalues of (A,B):');
disp(alpha./beta);

% Obtain scaled eigenvectors from Schur form
side = 'Both sides';
howmny = 'Backtransformed using Q and Z';
select = [false];
[q, z, m, info] = f08yx( ...
			 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] = f08ww( ...
		    job, side, jlo, jhi, lscale, rscale, q);
side = 'Right';
[VR, info] = f08ww( ...
		    job, side, jlo, jhi, lscale, rscale, z);

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

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



f08ww example results

Generalized eigenvalues of (A,B):
  -0.6354 + 1.6529i
   0.4580 - 0.8426i
   0.4934 + 0.9102i
   0.6744 - 0.0499i

Left Eigenvectors
  -0.1725 - 0.2037i  -0.0406 - 0.0402i   0.0560 + 0.0022i   0.2371 + 0.1695i
   0.2068 + 0.0997i   0.0877 + 0.0860i  -0.1153 + 0.0499i  -0.5341 - 0.0621i
  -0.0739 - 0.0261i  -0.0475 - 0.0525i   0.0591 - 0.0409i   0.0972 + 0.0028i
   0.0105 + 0.0015i   0.0058 + 0.0124i  -0.0080 + 0.0112i  -0.0129 + 0.0009i

Right Eigenvectors
   0.1437 - 0.1675i  -0.0889 + 0.0018i   0.0693 - 0.0167i   0.7361 + 0.0355i
  -0.1710 + 0.1045i   0.1462 + 0.0020i  -0.1406 + 0.0248i  -0.7608 + 0.1006i
   0.0628 - 0.0372i  -0.0815 - 0.0185i   0.0798 - 0.0202i   0.0947 - 0.0053i
  -0.0095 + 0.0069i   0.0134 + 0.0086i  -0.0136 + 0.0083i   0.0050 - 0.0014i
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
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