Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zggev (f08wn)

## Purpose

nag_lapack_zggev (f08wn) computes for a pair of $n$ by $n$ complex nonsymmetric matrices $\left(A,B\right)$ the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the $QZ$ algorithm.

## Syntax

[a, b, alpha, beta, vl, vr, info] = f08wn(jobvl, jobvr, a, b, 'n', n)
[a, b, alpha, beta, vl, vr, info] = nag_lapack_zggev(jobvl, jobvr, a, b, 'n', n)

## Description

A generalized eigenvalue for a pair of matrices $\left(A,B\right)$ is a scalar $\lambda$ or a ratio $\alpha /\beta =\lambda$, such that $A-\lambda B$ is singular. It is usually represented as the pair $\left(\alpha ,\beta \right)$, as there is a reasonable interpretation for $\beta =0$, and even for both being zero.
The right generalized eigenvector ${v}_{j}$ corresponding to the generalized eigenvalue ${\lambda }_{j}$ of $\left(A,B\right)$ satisfies
 $A vj = λj B vj .$
The left generalized eigenvector ${u}_{j}$ corresponding to the generalized eigenvalue ${\lambda }_{j}$ of $\left(A,B\right)$ satisfies
 $ujH A = λj ujH B ,$
where ${u}_{j}^{\mathrm{H}}$ is the conjugate-transpose of ${u}_{j}$.
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem $Ax=\lambda Bx$, where $A$ and $B$ are complex, square matrices, are determined using the $QZ$ algorithm. The complex $QZ$ algorithm consists of three stages:
1. $A$ is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time $B$ is reduced to upper triangular form.
2. $A$ is further reduced to triangular form while the triangular form of $B$ is maintained and the diagonal elements of $B$ are made real and non-negative. This is the generalized Schur form of the pair $\left(A,B\right)$.
This function does not actually produce the eigenvalues ${\lambda }_{j}$, but instead returns ${\alpha }_{j}$ and ${\beta }_{j}$ such that
 $λj=αj/βj, j=1,2,…,n.$
The division by ${\beta }_{j}$ becomes your responsibility, since ${\beta }_{j}$ may be zero, indicating an infinite eigenvalue.
3. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the $QZ$ algorithm Linear Algebra Appl. 28 285–303

## 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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, $\mathrm{max}\left|{\alpha }_{j}\right|$ will always be less than and usually comparable with ${‖{\mathbf{a}}‖}_{2}$ in magnitude, and $\mathrm{max}\left|{\beta }_{j}\right|$ 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 ${\mathbf{jobvl}}=\text{'V'}$, $\mathit{ldvl}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldvl}=1$.
The second dimension of the array vl will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ 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 ${\mathbf{jobvr}}=\text{'V'}$, $\mathit{ldvr}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldvr}=1$.
The second dimension of the array vr will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ 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  ${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
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

The computed eigenvalues and eigenvectors are exact for a nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E,F F = Oε A,B F ,$
and $\epsilon$ is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
Note:  interpretation of results obtained with the $QZ$ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of ${\alpha }_{j}$ and ${\beta }_{j}$. It should be noted that if ${\alpha }_{j}$ and ${\beta }_{j}$ are both small for any $j$, it may be that no reliance can be placed on any of the computed eigenvalues ${\lambda }_{i}={\alpha }_{i}/{\beta }_{i}$. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_lapack_dggev (f08wa).

## Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair $\left(A,B\right)$, where
 $A = -21.10-22.50i 53.50-50.50i -34.50+127.50i 7.50+00.50i -0.46-07.78i -3.50-37.50i -15.50+058.50i -10.50-01.50i 4.30-05.50i 39.70-17.10i -68.50+012.50i -7.50-03.50i 5.50+04.40i 14.40+43.30i -32.50-046.00i -19.00-32.50i$
and
 $B = 1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i 0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i 1.00+0.00i 2.40+1.80i -4.00-5.00i 0.00-3.00i 0.00+1.00i -1.80+2.40i 0.00-4.00i 4.00-5.00i .$
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```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

```