f08xs:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

nag_lapack_zhgeqz (f08xs) implements a single-shift version of the

Q Z

method for finding the generalized eigenvalues of the complex matrix pair

(A, B)

which is in the generalized upper Hessenberg form. If the matrix pair

(A, B)

is not in the generalized upper Hessenberg form, then the function nag_lapack_zgghrd (f08ws) should be called before invoking nag_lapack_zhgeqz (f08xs).

The argument job specifies two options. If

job ='S'

then the matrix pair

(A, B)

is simultaneously reduced to Schur form by applying one unitary transformation (usually called

Q

) on the left and another (usually called

Z

) on the right. That is,

\begin{matrix} A \leftarrow Q^{H} A Z \\ B \leftarrow Q^{H} B Z \end{matrix}

job ='E'

, then at each iteration the same transformations are computed but they are only applied to those parts of

A

and

B

which are needed to compute

α

and

β

. This option could be used if generalized eigenvalues are required but not generalized eigenvectors.

job ='S'

and

compq ='V'

'I'

, and

compz ='V'

'I'

, then the unitary transformations used to reduce the pair

(A, B)

are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to

job ='S'

and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to

compq ='V'

'I'

rather than

compq ='N'

compq ='I'

, then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix

Q

. However, if

compq ='V'

then the transformations are accumulated in the user-supplied matrix

Q_{0}

in array q on entry and thus on exit q contains the matrix product

Q Q_{0}

. A similar convention is used for compz.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

This example computes the

α

and

β

arguments, which defines the generalized eigenvalues, of the matrix pair

(A, B)

given by

A = (\begin{array}{r} 1.0 + 3.0 i & 1.0 + 4.0 i & 1.0 + 5.0 i & 1.0 + 6.0 i \\ 2.0 + 2.0 i & 4.0 + 3.0 i & 8.0 + 4.0 i & 16.0 + 5.0 i \\ 3.0 + 1.0 i & 9.0 + 2.0 i & 27.0 + 3.0 i & 81.0 + 4.0 i \\ 4.0 + 0.0 i & 16.0 + 1.0 i & 64.0 + 2.0 i & 256.0 + 3.0 i \end{array})

and

B = (\begin{array}{r} 1.0 + 0.0 i & 2.0 + 1.0 i & 3.0 + 2.0 i & 4.0 + 3.0 i \\ 1.0 + 1.0 i & 4.0 + 2.0 i & 9.0 + 3.0 i & 16.0 + 4.0 i \\ 1.0 + 2.0 i & 8.0 + 3.0 i & 27.0 + 4.0 i & 64.0 + 5.0 i \\ 1.0 + 3.0 i & 16.0 + 4.0 i & 81.0 + 5.0 i & 256.0 + 6.0 i \end{array}) .

This requires calls to five functions: nag_lapack_zggbal (f08wv) to balance the matrix, nag_lapack_zgeqrf (f08as) to perform the

Q R

factorization of

B

, nag_lapack_zunmqr (f08au) to apply

Q

A

, nag_lapack_zgghrd (f08ws) to reduce the matrix pair to the generalized Hessenberg form and nag_lapack_zhgeqz (f08xs) to compute the eigenvalues using the

Q Z

algorithm.

function f08xs_example


fprintf('f08xs 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 = 'No Q';
compz = 'No Z';
z = complex(eye(4));
q = complex(eye(4));
jlo = int64(1);
jhi = int64(ihi-ilo+1);
[H, T, ~, ~, info] = ...
  f08ws( ...
         compq, compz, jlo, jhi, c, QR, q, z);

% Find eigenvalues of generalized Hessenberg form
%    = eigenvalues of (A,B).
job = 'Eigenvalues';
ilo = int64(1);
ihi = int64(4);
[~, ~, alpha, beta, ~, ~, info] = ...
  f08xs( ...
         job, compq, compz, jlo, jhi, H, T, q, z);

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

NAG Toolbox: nag_lapack_zhgeqz (f08xs)

▸▿ Contents

Purpose

Syntax

Description