NAG Toolbox: nag_lapack_dggev (f08wa)

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{alphar}\left(\mathbf{n}\right)$ – double array

The element $\mathbf{alphar}\left(j\right)$ contains the real part of ${\alpha }_j$.

4:     $\mathrm{alphai}\left(\mathbf{n}\right)$ – double array

The element $\mathbf{alphai}\left(j\right)$ contains the imaginary part of ${\alpha }_j$.

5:     $\mathrm{beta}\left(\mathbf{n}\right)$ – double array

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

If $\mathbf{alphai}\left(j\right)$ is zero, then the $j$th eigenvalue is real; if positive, then the $j$th and $\left(j+1\right)$st eigenvalues are a complex conjugate pair, with $\mathbf{alphai}\left(j+1\right)$ negative.

Note:  the quotients $\mathbf{alphar}\left(j\right)/\mathbf{beta}\left(j\right)$ and $\mathbf{alphai}\left(j\right)/\mathbf{beta}\left(j\right)$ 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 ${‖\mathbf{b}‖}_2$.

6:     $\mathrm{vl}\left(\mathit{ldvl},:\right)$ – double 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 eigenvectors $u_j$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.

If the $j$th eigenvalue is real, then $u_j=\mathbf{vl}\left(:,j\right)$, the $j$th column of $\mathbf{vl}$.

If the $j$th and $\left(j+1\right)$th eigenvalues form a complex conjugate pair, then $u_j=\mathbf{vl}\left(:,j\right)+i\times \mathbf{vl}\left(:,j+1\right)$ and $u\left(j+1\right)=\mathbf{vl}\left(:,j\right)-i\times \mathbf{vl}\left(:,j+1\right)$. Each eigenvector will be scaled so the largest component has $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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

7:     $\mathrm{vr}\left(\mathit{ldvr},:\right)$ – double 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 eigenvectors $v_j$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues.

If the $j$th eigenvalue is real, then $v_j=\mathbf{vr}\left(:,j\right)$, the $j$th column of $\mathrm{VR}$.

If the $j$th and $\left(j+1\right)$th eigenvalues form a complex conjugate pair, then $v_j=\mathbf{vr}\left(:,j\right)+i\times \mathbf{vr}\left(:,j+1\right)$ and $v_{j+1}=\mathbf{vr}\left(:,j\right)-i\times \mathbf{vr}\left(:,j+1\right)$. Each eigenvector will be scaled so the largest component has $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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

8:     $\mathrm{info}$ – 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: alphar, 9: alphai, 10: beta, 11: vl, 12: ldvl, 13: vr, 14: ldvr, 15: work, 16: lwork, 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 $\mathbf{alphar}\left(j\right)$, $\mathbf{alphai}\left(j\right)$, 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_dhgeqz (f08xe).

   $\mathbf{info}=\mathbf{n}+2$

Error returned from nag_lapack_dtgevc (f08yk).

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08wa_example

function f08wa_example


fprintf('f08wa example results\n\n');

% Generalized eigenvalues and right eigenvectors of (A, B):
n = 4;
a = [3.9, 12.5, -34.5, -0.5;
     4.3, 21.5, -47.5,  7.5;
     4.3, 21.5, -43.5,  3.5;
     4.4, 26,   -46,    6];
b = [1,    2,    -3,    1;
     1,    3,    -5,    4;
     1,    3,    -4,    3;
     1,    3,    -4,    4];

jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
[~, ~, alphar, alphai, beta, ~, VR, info] = ...
  f08wa( ...
	 jobvl, jobvr, a, b);

small = x02am;

for j=1:n
  if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
    fprintf('\n%4d: Eigenvalue is numerically infinite or undetermined\n',j);
    fprintf('%4d: alphar = %11.4e, alphai = %11.4e, beta = %11.4e\n', ...
        j, alphar(j), alphai(j), beta(j));
  else  
    fprintf('\n%4d: Eigenvalue =  ', j);
    if alphai(j)==0
      fprintf('%10.4e\n',alphar(j)/beta(j));
    elseif alphai(j)>=0
      fprintf('%10.4e + %10.4ei\n',alphar(j)/beta(j), alphai(j)/beta(j));
    else
      fprintf('%10.4e - %10.4ei\n',alphar(j)/beta(j), -alphai(j)/beta(j));
    end
  end

  fprintf('      Eigenvector:\n', j);
  if alphai(j) == 0
    fprintf('%30.4e\n',VR(:, j));
  else
    if alphai(j) > 0
      k = j;
    else   
      k = j-1;
    end
    for l = 1:n
      if VR(l,k+1)>=0
        fprintf('%30.4e + %10.4ei\n',VR(l, k), VR(l, k+1));
      else
        fprintf('%30.4e - %10.4ei\n',VR(l, k), -VR(l, k+1));
      end
    end
  end
end

f08wa example results


   1: Eigenvalue =  2.0000e+00
      Eigenvector:
                    1.0000e+00
                    5.7143e-03
                    6.2857e-02
                    6.2857e-02

   2: Eigenvalue =  3.0000e+00 + 4.0000e+00i
      Eigenvector:
                   -4.3979e-01 - 5.6021e-01i
                   -8.7958e-02 - 1.1204e-01i
                   -1.4241e-01 + 3.1418e-03i
                   -1.4241e-01 + 3.1418e-03i

   3: Eigenvalue =  3.0000e+00 - 4.0000e+00i
      Eigenvector:
                   -4.3979e-01 - 5.6021e-01i
                   -8.7958e-02 - 1.1204e-01i
                   -1.4241e-01 + 3.1418e-03i
                   -1.4241e-01 + 3.1418e-03i

   4: Eigenvalue =  4.0000e+00
      Eigenvector:
                   -1.0000e+00
                   -1.1111e-02
                    3.3333e-02
                   -1.5556e-01