NAG Toolbox: nag_lapack_zggev (f08wn)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{jobvl}$ – string (length ≥ 1)

If $\mathbf{jobvl}=\text{'N'}$, do not compute the left generalized eigenvectors.

If $\mathbf{jobvl}=\text{'V'}$, compute the left generalized eigenvectors.

Constraint: $\mathbf{jobvl}=\text{'N'}$ or $\text{'V'}$.

2:     $\mathrm{jobvr}$ – string (length ≥ 1)

If $\mathbf{jobvr}=\text{'N'}$, do not compute the right generalized eigenvectors.

If $\mathbf{jobvr}=\text{'V'}$, compute the right generalized eigenvectors.

Constraint: $\mathbf{jobvr}=\text{'N'}$ or $\text{'V'}$.

3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The matrix $A$ in the pair $\left(A,B\right)$.

4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

The first dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

The matrix $B$ in the pair $\left(A,B\right)$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrices $A$ and $B$.

Constraint: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

The first dimension of the array b will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array b will be $\max \left(1,\mathbf{n}\right)$.

b has been overwritten.

3:     $\mathrm{alpha}\left(\mathbf{n}\right)$ – complex array

See the description of beta.

4:     $\mathrm{beta}\left(\mathbf{n}\right)$ – complex array

$\mathbf{alpha}\left(\mathit{j}\right)/\mathbf{beta}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$, will be the generalized eigenvalues.

Note:  the quotients $\mathbf{alpha}\left(j\right)/\mathbf{beta}\left(j\right)$ may easily overflow or underflow, and $\mathbf{beta}\left(j\right)$ may even be zero. Thus, you should avoid naively computing the ratio ${\alpha }_j/{\beta }_j$. However, $\max \left|{\alpha }_j\right|$ will always be less than and usually comparable with ${‖\mathbf{a}‖}_2$ in magnitude, and $\max \left|{\beta }_j\right|$ will always be less than and usually comparable with ${‖\mathbf{b}‖}_2$.

5:     $\mathrm{vl}\left(\mathit{ldvl},:\right)$ – complex array

The first dimension, $\mathit{ldvl}$, of the array vl will be

• if $\mathbf{jobvl}=\text{'V'}$, $\mathit{ldvl}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldvl}=1$.

The second dimension of the array vl will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvl}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobvl}=\text{'V'}$, the left generalized eigenvectors $u_j$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

If $\mathbf{jobvl}=\text{'N'}$, vl is not referenced.

6:     $\mathrm{vr}\left(\mathit{ldvr},:\right)$ – complex array

The first dimension, $\mathit{ldvr}$, of the array vr will be

• if $\mathbf{jobvr}=\text{'V'}$, $\mathit{ldvr}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldvr}=1$.

The second dimension of the array vr will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvr}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobvr}=\text{'V'}$, the right generalized eigenvectors $v_j$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

If $\mathbf{jobvr}=\text{'N'}$, vr is not referenced.

7:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: jobvl, 2: jobvr, 3: n, 4: a, 5: lda, 6: b, 7: ldb, 8: alpha, 9: beta, 10: vl, 11: ldvl, 12: vr, 13: ldvr, 14: work, 15: lwork, 16: rwork, 17: 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.

W  $\mathbf{info}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$

The $QZ$ iteration failed. No eigenvectors have been calculated, but $\mathbf{alpha}\left(j\right)$ and $\mathbf{beta}\left(j\right)$ should be correct for $j=\mathbf{info}+1,\dots ,\mathbf{n}$.

   $\mathbf{info}=\mathbf{n}+1$

Unexpected error returned from nag_lapack_zhgeqz (f08xs).

   $\mathbf{info}=\mathbf{n}+2$

Error returned from nag_lapack_ztgevc (f08yx).

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08wn_example

function f08wn_example


fprintf('f08wn example results\n\n');

% Generalized eigenvalues and right eigenvectors of (A, B):
n = 4;
a = [ -21.10 - 22.50i,  53.5 - 50.5i, -34.5 + 127.5i,    7.5 +  0.5i;
       -0.46 -  7.78i,  -3.5 - 37.5i, -15.5 +  58.5i,  -10.5 -  1.5i;
        4.30 -  5.50i,  39.7 - 17.1i, -68.5 +  12.5i,   -7.5 -  3.5i;
        5.50 +  4.40i,  14.4 + 43.3i, -32.5 -  46.00i, -19.0 - 32.5i];
b = [   1    -  5i,      1.6 +  1.2i,  -3   +   0i,      0   -  1i;
        0.8  -  0.6i,    3   -  5i,    -4   +   3i,     -2.4 -  3.2i;
        1    +  0i,      2.4 +  1.8i,  -4   -   5i,      0   -  3i;
        0    +  1i,     -1.8 +  2.4i,   0   -   4i,      4   -  5i];

jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
[~, ~, alpha, beta, ~, VR, info] = ...
  f08wn( ...
	 jobvl, jobvr, a, b);

small = x02am;

[eigs, pos] = sort(alpha./beta);

for j=1:n
  if (abs(alpha(pos(j)))*small >= abs(beta(pos(j))))
    fprintf('\n%4d: Eigenvalue is numerically infinite or undetermined\n',j);
    fprintf('%4d: alpha = (%11.4e,%11.4e), beta = (%11.4e,%11.4e)\n', ...
	    j, real(alpha(j)), imag(alpha(j)), real(beta(j)), imag(beta(j)));
  else
    fprintf('Eigenvalue  (%d):\n', j);
    disp(eigs(j));
  end

  fprintf('Eigenvector (%d):\n', j);
  disp(VR(:, pos(j))/VR(1, pos(j)));
end

f08wn example results

Eigenvalue  (1):
   3.0000 - 1.0000i

Eigenvector (1):
   1.0000 + 0.0000i
   0.1600 - 0.1200i
   0.1200 - 0.1600i
   0.1600 + 0.1200i

Eigenvalue  (2):
   2.0000 - 5.0000i

Eigenvector (2):
   1.0000 - 0.0000i
   0.0046 - 0.0034i
   0.0629 + 0.0000i
  -0.0000 + 0.0629i

Eigenvalue  (3):
   4.0000 - 5.0000i

Eigenvector (3):
   1.0000 + 0.0000i
   0.0089 - 0.0067i
  -0.0333 + 0.0000i
  -0.0000 + 0.1556i

Eigenvalue  (4):
   3.0000 - 9.0000i

Eigenvector (4):
   1.0000 - 0.0000i
   0.1600 - 0.1200i
   0.1200 + 0.1600i
  -0.1600 + 0.1200i