f08yx:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

Description

nag_lapack_ztgevc (f08yx) computes some or all of the right and/or left generalized eigenvectors of the matrix pair

(A, B)

which is assumed to be in upper triangular form. If the matrix pair

(A, B)

is not upper triangular then the function nag_lapack_zhgeqz (f08xs) should be called before invoking nag_lapack_ztgevc (f08yx).

The right generalized eigenvector

x

and the left generalized eigenvector

y

(A, B)

corresponding to a generalized eigenvalue

λ

are defined by

(A - λ B) x = 0

and

y^{H} (A - λ B) = 0 .

If a generalized eigenvalue is determined as

0 / 0

, which is due to zero diagonal elements at the same locations in both

A

and

B

, a unit vector is returned as the corresponding eigenvector.

Note that the generalized eigenvalues are computed using nag_lapack_zhgeqz (f08xs) but nag_lapack_ztgevc (f08yx) does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by nag_lapack_ztgevc (f08yx).

If all eigenvectors are requested, the function may either return the matrices

X

and/or

Y

of right or left eigenvectors of

(A, B)

, or the products

Z X

and/or

Q Y

, where

Z

and

Q

are two matrices supplied by you. Usually,

Q

and

Z

are chosen as the unitary matrices returned by nag_lapack_zhgeqz (f08xs). Equivalently,

Q

and

Z

are the left and right Schur vectors of the matrix pair supplied to nag_lapack_zhgeqz (f08xs). In that case,

Q Y

and

Z X

are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to nag_lapack_zhgeqz (f08xs).

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 and the corresponding left and right eigenvectors, 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}) .

To compute generalized eigenvalues, it is required to call 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 via the

Q Z

algorithm.

function f08yx_example


fprintf('f08yx 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);

% Normalize eigenvectors to have Euclidean norm 1 and largest component real.
incv = int64(1);
n = int64(4);
for j = 1:n
  [k, r] = f16js(n, VL(:,j), incv);
  scal = conj(VL(k,j))/abs(VL(k,j))/norm(VL(:,j),2);
  VL(:,j) = VL(:,j)*scal;
end
for j = 1:n
  [k, r] = f16js(n, VR(:,j), incv);
  scal = conj(VR(k,j))/abs(VR(k,j))/norm(VR(:,j),2);
  VR(:,j) = VR(:,j)*scal;
end

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

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

f08yx example results

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

Right Eigenvectors
   0.7186 + 0.0000i  -0.4649 + 0.0156i  -0.3946 + 0.0246i  -0.6788 - 0.1233i
  -0.6208 - 0.2009i   0.7652 + 0.0000i   0.7921 + 0.0000i   0.7184 + 0.0000i
   0.2251 + 0.0762i  -0.4275 - 0.0912i  -0.4554 + 0.0334i  -0.0886 - 0.0067i
  -0.0372 - 0.0088i   0.0707 + 0.0442i   0.0824 - 0.0322i  -0.0048 + 0.0006i

Left Eigenvectors
   0.7397 + 0.0000i  -0.3722 - 0.0016i  -0.3240 - 0.1559i  -0.4118 - 0.2276i
  -0.5812 + 0.2589i   0.8003 + 0.0000i   0.8063 - 0.0000i   0.8681 - 0.0000i
   0.1875 - 0.1097i  -0.4606 - 0.0279i  -0.4523 + 0.0903i  -0.1564 + 0.0136i
  -0.0219 + 0.0195i   0.0839 + 0.0311i   0.0755 - 0.0453i   0.0206 - 0.0038i

NAG Toolbox: nag_lapack_ztgevc (f08yx)

▸▿ Contents

Purpose

Syntax