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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dggev (f08wa)

## Purpose

nag_lapack_dggev (f08wa) computes for a pair of n$n$ by n$n$ real 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, alphar, alphai, beta, vl, vr, info] = f08wa(jobvl, jobvr, a, b, 'n', n)
[a, b, alphar, alphai, beta, vl, vr, info] = nag_lapack_dggev(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 eigenvector vj${v}_{j}$ corresponding to the eigenvalue λj${\lambda }_{j}$ of (A,B)$\left(A,B\right)$ satisfies
 A vj = λj B vj . $A vj = λj B vj .$
The left eigenvector uj${u}_{j}$ corresponding to the 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 generalized eigenproblem Ax = λBx$Ax=\lambda Bx$, where A$A$ and B$B$ are real, square matrices, are determined using the QZ$QZ$ algorithm. The QZ$QZ$ algorithm consists of four stages:
1. A$A$ is reduced to upper Hessenberg form and at the same time B$B$ is reduced to upper triangular form.
2. A$A$ is further reduced to quasi-triangular form while the triangular form of B$B$ is maintained. This is the real generalized Schur form of the pair (A,B) $\left(A,B\right)$.
3. The quasi-triangular form of A$A$ is reduced to triangular form and the eigenvalues extracted. 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. Pairs of complex eigenvalues occur with αj / βj${\alpha }_{j}/{\beta }_{j}$ and αj + 1 / βj + 1${\alpha }_{j+1}/{\beta }_{j+1}$ complex conjugates, even though αj${\alpha }_{j}$ and αj + 1${\alpha }_{j+1}$ are not conjugate.
4. 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, : $:$) – double 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, : $:$) – double 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

### Output Parameters

1:     a(lda, : $:$) – double 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, : $:$) – double 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:     alphar(n) – double array
The element alphar(j)${\mathbf{alphar}}\left(j\right)$ contains the real part of αj${\alpha }_{j}$.
4:     alphai(n) – double array
The element alphai(j)${\mathbf{alphai}}\left(j\right)$ contains the imaginary part of αj${\alpha }_{j}$.
5:     beta(n) – double array
(alphar(j) + alphai(j) × i) / beta(j)$\left({\mathbf{alphar}}\left(\mathit{j}\right)+{\mathbf{alphai}}\left(\mathit{j}\right)×i\right)/{\mathbf{beta}}\left(\mathit{j}\right)$, for j = 1,2,,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues.
If alphai(j)${\mathbf{alphai}}\left(j\right)$ is zero, then the j$j$th eigenvalue is real; if positive, then the j$j$th and (j + 1)$\left(j+1\right)$st eigenvalues are a complex conjugate pair, with alphai(j + 1)${\mathbf{alphai}}\left(j+1\right)$ negative.
Note:  the quotients alphar(j) / beta(j)${\mathbf{alphar}}\left(j\right)/{\mathbf{beta}}\left(j\right)$ and alphai(j) / beta(j)${\mathbf{alphai}}\left(j\right)/{\mathbf{beta}}\left(j\right)$ 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${‖{\mathbf{b}}‖}_{2}$.
6:     vl(ldvl, : $:$) – double 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 eigenvectors uj${u}_{j}$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.
If the j$j$th eigenvalue is real, then uj = vl( : ,j)${u}_{j}={\mathbf{vl}}\left(:,j\right)$, the j$j$th column of vl${\mathbf{vl}}$.
If the j$j$th and (j + 1)$\left(j+1\right)$th eigenvalues form a complex conjugate pair, then uj = vl( : ,j) + i × vl( : ,j + 1)${u}_{j}={\mathbf{vl}}\left(:,j\right)+i×{\mathbf{vl}}\left(:,j+1\right)$ and u(j + 1) = vl( : ,j)i × vl( : ,j + 1)$u\left(j+1\right)={\mathbf{vl}}\left(:,j\right)-i×{\mathbf{vl}}\left(:,j+1\right)$. Each eigenvector will be scaled so the largest component has |real part| + |imag. part| = 1$|\text{real part}|+|\text{imag. part}|=1$.
If jobvl = 'N'${\mathbf{jobvl}}=\text{'N'}$, vl is not referenced.
7:     vr(ldvr, : $:$) – double 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 eigenvectors vj${v}_{j}$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues.
If the j$j$th eigenvalue is real, then vj = vr( : ,j)${v}_{j}={\mathbf{vr}}\left(:,j\right)$, the j$j$th column of VR$\mathrm{VR}$.
If the j$j$th and (j + 1)$\left(j+1\right)$th eigenvalues form a complex conjugate pair, then vj = vr( : ,j) + i × vr( : ,j + 1)${v}_{j}={\mathbf{vr}}\left(:,j\right)+i×{\mathbf{vr}}\left(:,j+1\right)$ and vj + 1 = vr( : ,j)i × vr( : ,j + 1)${v}_{j+1}={\mathbf{vr}}\left(:,j\right)-i×{\mathbf{vr}}\left(:,j+1\right)$. Each eigenvector will be scaled so the largest component has |real part| + |imag. part| = 1$|\text{real part}|+|\text{imag. part}|=1$.
If jobvr = 'N'${\mathbf{jobvr}}=\text{'N'}$, vr is not referenced.
8:     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: 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 INFO = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
The QZ$QZ$ iteration failed. No eigenvectors have been calculated, but alphar(j)${\mathbf{alphar}}\left(j\right)$, alphai(j)${\mathbf{alphai}}\left(j\right)$, 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_dhgeqz (f08xe).
INFO = N + 2${\mathbf{INFO}}={\mathbf{N}}+2$
Error returned from nag_lapack_dtgevc (f08yk).

## 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 complex analogue of this function is nag_lapack_zggev (f08wn).

## Example

```function nag_lapack_dggev_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
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];
[aOut, bOut, alphar, alphai, beta, vl, vr, info] = nag_lapack_dggev(jobvl, jobvr, a, b);

small = nag_machine_real_safe;

for j=1:4
if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
fprintf('\nEigenvalue(%d) is numerically infinite or undetermined\n');
fprintf('alphar(%d) = %11.4e, alphai(%d) = %11.4e, beta(%d) = %11.4e\n', ...
j, alphar(j), j, alphai(j), j, beta(j));
else
fprintf('\nEigenvalue(%d) = %s\n\n', j, num2str(complex(alphar(j), alphai(j))/beta(j)));
end

fprintf('\nEigenvector(%d)\n', j);
if alphai(j) == 0
disp(vr(:, j));
elseif alphai(j) > 0
disp(complex(vr(:, j), vr(:, j+1)));
else
disp(complex(vr(:, j-1), vr(:, j)));
end
end
```
```

Eigenvalue(1) = 2

Eigenvector(1)
1.0000
0.0057
0.0629
0.0629

Eigenvalue(2) = 3+4i

Eigenvector(2)
-0.4398 - 0.5602i
-0.0880 - 0.1120i
-0.1424 + 0.0031i
-0.1424 + 0.0031i

Eigenvalue(3) = 3-4i

Eigenvector(3)
-0.4398 - 0.5602i
-0.0880 - 0.1120i
-0.1424 + 0.0031i
-0.1424 + 0.0031i

Eigenvalue(4) = 4

Eigenvector(4)
-1.0000
-0.0111
0.0333
-0.1556

```
```function f08wa_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
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];
[aOut, bOut, alphar, alphai, beta, vl, vr, info] = f08wa(jobvl, jobvr, a, b);

small = x02am;

for j=1:4
if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
fprintf('\nEigenvalue(%d) is numerically infinite or undetermined\n');
fprintf('alphar(%d) = %11.4e, alphai(%d) = %11.4e, beta(%d) = %11.4e\n', ...
j, alphar(j), j, alphai(j), j, beta(j));
else
fprintf('\nEigenvalue(%d) = %s\n\n', j, num2str(complex(alphar(j), alphai(j))/beta(j)));
end

fprintf('\nEigenvector(%d)\n', j);
if alphai(j) == 0
disp(vr(:, j));
elseif alphai(j) > 0
disp(complex(vr(:, j), vr(:, j+1)));
else
disp(complex(vr(:, j-1), vr(:, j)));
end
end
```
```

Eigenvalue(1) = 2

Eigenvector(1)
1.0000
0.0057
0.0629
0.0629

Eigenvalue(2) = 3+4i

Eigenvector(2)
-0.4398 - 0.5602i
-0.0880 - 0.1120i
-0.1424 + 0.0031i
-0.1424 + 0.0031i

Eigenvalue(3) = 3-4i

Eigenvector(3)
-0.4398 - 0.5602i
-0.0880 - 0.1120i
-0.1424 + 0.0031i
-0.1424 + 0.0031i

Eigenvalue(4) = 4

Eigenvector(4)
-1.0000
-0.0111
0.0333
-0.1556

```