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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zggev (f08wn)

## Purpose

nag_lapack_zggev (f08wn) computes for a pair of n$n$ by n$n$ complex nonsymmetric matrices (A,B)$\left(A,B\right)$ the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ$QZ$ 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 (A,B)$\left(A,B\right)$ is a scalar λ$\lambda$ or a ratio α / β = λ$\alpha /\beta =\lambda$, such that AλB$A-\lambda B$ is singular. It is usually represented as the pair (α,β)$\left(\alpha ,\beta \right)$, as there is a reasonable interpretation for β = 0$\beta =0$, and even for both being zero.
The right generalized eigenvector vj${v}_{j}$ corresponding to the generalized eigenvalue λj${\lambda }_{j}$ of (A,B)$\left(A,B\right)$ satisfies
 A vj = λj B vj . $A vj = λj B vj .$
The left generalized eigenvector uj${u}_{j}$ corresponding to the generalized eigenvalue λj${\lambda }_{j}$ of (A,B)$\left(A,B\right)$ satisfies
 ujH A = λj ujH B , $ujH A = λj ujH B ,$
where ujH${u}_{j}^{\mathrm{H}}$ is the conjugate-transpose of uj${u}_{j}$.
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem Ax = λBx$Ax=\lambda Bx$, where A$A$ and B$B$ are complex, square matrices, are determined using the QZ$QZ$ algorithm. The complex QZ$QZ$ algorithm consists of three stages:
1. A$A$ is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time B$B$ is reduced to upper triangular form.
2. A$A$ is further reduced to triangular form while the triangular form of B$B$ is maintained and the diagonal elements of B$B$ are made real and non-negative. This is the generalized Schur form of the pair (A,B) $\left(A,B\right)$.
This function does not actually produce the eigenvalues λj${\lambda }_{j}$, but instead returns αj${\alpha }_{j}$ and βj${\beta }_{j}$ such that
 λj = αj / βj,  j = 1,2, … ,n. $λj=αj/βj, j=1,2,…,n.$
The division by βj${\beta }_{j}$ becomes your responsibility, since βj${\beta }_{j}$ 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$QZ$ algorithm Linear Algebra Appl. 28 285–303

## Parameters

### Compulsory Input Parameters

1:     jobvl – string (length ≥ 1)
If jobvl = 'N'${\mathbf{jobvl}}=\text{'N'}$, do not compute the left generalized eigenvectors.
If jobvl = 'V'${\mathbf{jobvl}}=\text{'V'}$, compute the left generalized eigenvectors.
Constraint: jobvl = 'N'${\mathbf{jobvl}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     jobvr – string (length ≥ 1)
If jobvr = 'N'${\mathbf{jobvr}}=\text{'N'}$, do not compute the right generalized eigenvectors.
If jobvr = 'V'${\mathbf{jobvr}}=\text{'V'}$, compute the right generalized eigenvectors.
Constraint: jobvr = 'N'${\mathbf{jobvr}}=\text{'N'}$ or 'V'$\text{'V'}$.
3:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The matrix A$A$ in the pair (A,B)$\left(A,B\right)$.
4:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The matrix B$B$ in the pair (A,B)$\left(A,B\right)$.

### Optional Input Parameters

1:     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$n$, the order of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb ldvl ldvr work lwork rwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
2:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
b has been overwritten.
3:     alpha(n) – complex array
See the description of beta.
4:     beta(n) – complex array
alpha(j) / beta(j)${\mathbf{alpha}}\left(\mathit{j}\right)/{\mathbf{beta}}\left(\mathit{j}\right)$, for j = 1,2,,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues.
Note:  the quotients alpha(j) / beta(j)${\mathbf{alpha}}\left(j\right)/{\mathbf{beta}}\left(j\right)$ may easily overflow or underflow, and beta(j)${\mathbf{beta}}\left(j\right)$ may even be zero. Thus, you should avoid naively computing the ratio αj / βj${\alpha }_{j}/{\beta }_{j}$. However, max|αj|$\mathrm{max}|{\alpha }_{j}|$ will always be less than and usually comparable with a2${‖{\mathbf{a}}‖}_{2}$ in magnitude, and max|βj|$\mathrm{max}|{\beta }_{j}|$ will always be less than and usually comparable with b2${‖{\mathbf{b}}‖}_{2}$.
5:     vl(ldvl, : $:$) – complex array
The first dimension, ldvl, of the array vl will be
• if jobvl = 'V'${\mathbf{jobvl}}=\text{'V'}$, ldvl max (1,n) $\mathit{ldvl}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldvl1$\mathit{ldvl}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobvl = 'V'${\mathbf{jobvl}}=\text{'V'}$, and at least 1$1$ otherwise
If jobvl = 'V'${\mathbf{jobvl}}=\text{'V'}$, the left generalized eigenvectors uj${u}_{j}$ 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 |real part| + |imag. part| = 1$|\text{real part}|+|\text{imag. part}|=1$.
If jobvl = 'N'${\mathbf{jobvl}}=\text{'N'}$, vl is not referenced.
6:     vr(ldvr, : $:$) – complex array
The first dimension, ldvr, of the array vr will be
• if jobvr = 'V'${\mathbf{jobvr}}=\text{'V'}$, ldvr max (1,n) $\mathit{ldvr}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldvr1$\mathit{ldvr}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobvr = 'V'${\mathbf{jobvr}}=\text{'V'}$, and at least 1$1$ otherwise
If jobvr = 'V'${\mathbf{jobvr}}=\text{'V'}$, the right generalized eigenvectors vj${v}_{j}$ 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 |real part| + |imag. part| = 1$|\text{real part}|+|\text{imag. part}|=1$.
If jobvr = 'N'${\mathbf{jobvr}}=\text{'N'}$, vr is not referenced.
7:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [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.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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 INFO = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
The QZ$QZ$ iteration failed. No eigenvectors have been calculated, but alpha(j)${\mathbf{alpha}}\left(j\right)$ and beta(j)${\mathbf{beta}}\left(j\right)$ should be correct for j = info + 1,,n$j={\mathbf{info}}+1,\dots ,{\mathbf{n}}$.
INFO = N + 1${\mathbf{INFO}}={\mathbf{N}}+1$
Unexpected error returned from nag_lapack_zhgeqz (f08xs).
INFO = N + 2${\mathbf{INFO}}={\mathbf{N}}+2$
Error returned from nag_lapack_ztgevc (f08yx).

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices (A + E)$\left(A+E\right)$ and (B + F)$\left(B+F\right)$, where
 ‖(E,F)‖F = O(ε) ‖(A,B)‖F , $‖(E,F)‖ F = O(ε) ‖(A,B)‖ F ,$
and ε$\epsilon$ is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
Note:  interpretation of results obtained with the QZ$QZ$ 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 αj${\alpha }_{j}$ and βj${\beta }_{j}$. It should be noted that if αj${\alpha }_{j}$ and βj${\beta }_{j}$ are both small for any j$j$, it may be that no reliance can be placed on any of the computed eigenvalues λi = αi / βi${\lambda }_{i}={\alpha }_{i}/{\beta }_{i}$. 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 n3${n}^{3}$.
The real analogue of this function is nag_lapack_dggev (f08wa).

## Example

function nag_lapack_zggev_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
a = [ -21.1 - 22.5i,  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;
4.3 - 5.5i,  39.7 - 17.1i,  -68.5 + 12.5i,  -7.5 - 3.5i;
5.5 + 4.4i,  14.4 + 43.3i,  -32.5 - 46i,  -19 - 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];
[aOut, bOut, alpha, beta, vl, vr, info] = nag_lapack_zggev(jobvl, jobvr, a, b)

aOut =

1.0e+02 *

0.1903 - 0.5710i   0.5359 - 0.8982i  -0.8131 - 0.6323i   1.0666 - 0.4479i
0.0000 + 0.0000i   0.1188 - 0.2970i   0.0356 + 0.2763i  -0.0067 - 0.1642i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.1096 - 0.0365i  -0.2502 - 0.0820i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.2187 - 0.2734i

bOut =

6.3443 + 0.0000i   3.3986 + 0.7119i  -0.5152 - 2.3820i   6.5818 + 2.4299i
0.0000 + 0.0000i   5.9409 + 0.0000i  -2.4480 - 0.3427i   5.7385 - 0.7017i
0.0000 + 0.0000i   0.0000 + 0.0000i   3.6536 + 0.0000i  -1.4096 - 3.9326i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   5.4681 + 0.0000i

alpha =

19.0329 -57.0986i
11.8818 -29.7045i
10.9609 - 3.6536i
21.8722 -27.3403i

beta =

6.3443 + 0.0000i
5.9409 + 0.0000i
3.6536 + 0.0000i
5.4681 + 0.0000i

vl =

6.9226e-310 + 0.0000e+00i

vr =

-0.8238 - 0.1762i   0.6397 + 0.3603i   0.9775 + 0.0225i  -0.9062 + 0.0938i
-0.1530 + 0.0707i   0.0042 - 0.0005i   0.1591 - 0.1137i  -0.0074 + 0.0069i
-0.0707 - 0.1530i   0.0402 + 0.0226i   0.1209 - 0.1537i   0.0302 - 0.0031i
0.1530 - 0.0707i  -0.0226 + 0.0402i   0.1537 + 0.1209i  -0.0146 - 0.1410i

info =

0

function f08wn_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
a = [ -21.1 - 22.5i,  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;
4.3 - 5.5i,  39.7 - 17.1i,  -68.5 + 12.5i,  -7.5 - 3.5i;
5.5 + 4.4i,  14.4 + 43.3i,  -32.5 - 46i,  -19 - 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];
[aOut, bOut, alpha, beta, vl, vr, info] = f08wn(jobvl, jobvr, a, b)

aOut =

1.0e+02 *

0.1903 - 0.5710i   0.5359 - 0.8982i  -0.8131 - 0.6323i   1.0666 - 0.4479i
0.0000 + 0.0000i   0.1188 - 0.2970i   0.0356 + 0.2763i  -0.0067 - 0.1642i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.1096 - 0.0365i  -0.2502 - 0.0820i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.2187 - 0.2734i

bOut =

6.3443 + 0.0000i   3.3986 + 0.7119i  -0.5152 - 2.3820i   6.5818 + 2.4299i
0.0000 + 0.0000i   5.9409 + 0.0000i  -2.4480 - 0.3427i   5.7385 - 0.7017i
0.0000 + 0.0000i   0.0000 + 0.0000i   3.6536 + 0.0000i  -1.4096 - 3.9326i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   5.4681 + 0.0000i

alpha =

19.0329 -57.0986i
11.8818 -29.7045i
10.9609 - 3.6536i
21.8722 -27.3403i

beta =

6.3443 + 0.0000i
5.9409 + 0.0000i
3.6536 + 0.0000i
5.4681 + 0.0000i

vl =

0.0000 + 0.0000i

vr =

-0.8238 - 0.1762i   0.6397 + 0.3603i   0.9775 + 0.0225i  -0.9062 + 0.0938i
-0.1530 + 0.0707i   0.0042 - 0.0005i   0.1591 - 0.1137i  -0.0074 + 0.0069i
-0.0707 - 0.1530i   0.0402 + 0.0226i   0.1209 - 0.1537i   0.0302 - 0.0031i
0.1530 - 0.0707i  -0.0226 + 0.0402i   0.1537 + 0.1209i  -0.0146 - 0.1410i

info =

0