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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dggevx (f08wb)

## Purpose

nag_lapack_dggevx (f08wb) computes for a pair of $n$ by $n$ real nonsymmetric matrices $\left(A,B\right)$ the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the $QZ$ algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

## Syntax

[a, b, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = f08wb(balanc, jobvl, jobvr, sense, a, b, 'n', n)
[a, b, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = nag_lapack_dggevx(balanc, jobvl, jobvr, sense, 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 eigenvector ${v}_{j}$ corresponding to the eigenvalue ${\lambda }_{j}$ of $\left(A,B\right)$ satisfies
 $A vj = λj B vj .$
The left eigenvector ${u}_{j}$ corresponding to the 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 generalized eigenproblem $Ax=\lambda Bx$, where $A$ and $B$ are real, square matrices, are determined using the $QZ$ algorithm. The $QZ$ algorithm consists of four stages:
1. $A$ is reduced to upper Hessenberg form and at the same time $B$ is reduced to upper triangular form.
2. $A$ is further reduced to quasi-triangular form while the triangular form of $B$ is maintained. This is the real generalized Schur form of the pair $\left(A,B\right)$.
3. The quasi-triangular form of $A$ is reduced to triangular form and the eigenvalues extracted. 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. Pairs of complex eigenvalues occur with ${\alpha }_{j}/{\beta }_{j}$ and ${\alpha }_{j+1}/{\beta }_{j+1}$ complex conjugates, even though ${\alpha }_{j}$ and ${\alpha }_{j+1}$ 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.
For details of the balancing option, see Description in nag_lapack_dggbal (f08wh).

## 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:     $\mathrm{balanc}$ – string (length ≥ 1)
Specifies the balance option to be performed.
${\mathbf{balanc}}=\text{'N'}$
Do not diagonally scale or permute.
${\mathbf{balanc}}=\text{'P'}$
Permute only.
${\mathbf{balanc}}=\text{'S'}$
Scale only.
${\mathbf{balanc}}=\text{'B'}$
Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, ${\mathbf{balanc}}=\text{'B'}$ is recommended.
Constraint: ${\mathbf{balanc}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2:     $\mathrm{jobvl}$ – 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 $\text{'V'}$.
3:     $\mathrm{jobvr}$ – 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 $\text{'V'}$.
4:     $\mathrm{sense}$ – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\text{'N'}$
None are computed.
${\mathbf{sense}}=\text{'E'}$
Computed for eigenvalues only.
${\mathbf{sense}}=\text{'V'}$
Computed for eigenvectors only.
${\mathbf{sense}}=\text{'B'}$
Computed for eigenvalues and eigenvectors.
Constraint: ${\mathbf{sense}}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
5:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $\left(A,B\right)$.
6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a has been overwritten. If ${\mathbf{jobvl}}=\text{'V'}$ or ${\mathbf{jobvr}}=\text{'V'}$ or both, then $A$ contains the first part of the real Schur form of the ‘balanced’ versions of the input $A$ and $B$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{alphar}\left({\mathbf{n}}\right)$ – 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
$\left({\mathbf{alphar}}\left(\mathit{j}\right)+{\mathbf{alphai}}\left(\mathit{j}\right)×i\right)/{\mathbf{beta}}\left(\mathit{j}\right)$, 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, $\mathrm{max}\left|{\alpha }_{j}\right|$ 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 ${‖{\mathbf{b}}‖}_{2}$.
6:     $\mathrm{vl}\left(\mathit{ldvl},:\right)$ – double array
The first dimension, $\mathit{ldvl}$, of the array vl will be
• if ${\mathbf{jobvl}}=\text{'V'}$, $\mathit{ldvl}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• 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.
7:     $\mathrm{vr}\left(\mathit{ldvr},:\right)$ – double array
The first dimension, $\mathit{ldvr}$, of the array vr will be
• if ${\mathbf{jobvr}}=\text{'V'}$, $\mathit{ldvr}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• 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.
8:     $\mathrm{ilo}$int64int32nag_int scalar
9:     $\mathrm{ihi}$int64int32nag_int scalar
ilo and ihi are integer values such that ${\mathbf{a}}\left(i,j\right)=0$ and ${\mathbf{b}}\left(i,j\right)=0$ if $i>j$ and $j=1,2,\dots ,{\mathbf{ilo}}-1$ or $i={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
If ${\mathbf{balanc}}=\text{'N'}$ or $\text{'S'}$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}={\mathbf{n}}$.
10:   $\mathrm{lscale}\left({\mathbf{n}}\right)$ – double array
Details of the permutations and scaling factors applied to the left side of $A$ and $B$.
If ${\mathit{pl}}_{j}$ is the index of the row interchanged with row $j$, and ${\mathit{dl}}_{j}$ is the scaling factor applied to row $j$, then:
• ${\mathbf{lscale}}\left(\mathit{j}\right)={\mathit{pl}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• ${\mathbf{lscale}}={\mathit{dl}}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• ${\mathbf{lscale}}={\mathit{pl}}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ${\mathbf{ihi}}+1$, then $1$ to ${\mathbf{ilo}}-1$.
11:   $\mathrm{rscale}\left({\mathbf{n}}\right)$ – double array
Details of the permutations and scaling factors applied to the right side of $A$ and $B$.
If ${\mathit{pr}}_{j}$ is the index of the column interchanged with column $j$, and ${\mathit{dr}}_{j}$ is the scaling factor applied to column $j$, then:
• ${\mathbf{rscale}}\left(\mathit{j}\right)={\mathit{pr}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• if ${\mathbf{rscale}}={\mathit{dr}}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• if ${\mathbf{rscale}}={\mathit{pr}}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ${\mathbf{ihi}}+1$, then $1$ to ${\mathbf{ilo}}-1$.
12:   $\mathrm{abnrm}$ – double scalar
The $1$-norm of the balanced matrix $A$.
13:   $\mathrm{bbnrm}$ – double scalar
The $1$-norm of the balanced matrix $B$.
14:   $\mathrm{rconde}\left(:\right)$ – double array
The dimension of the array rconde will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus ${\mathbf{rconde}}\left(j\right)$, ${\mathbf{rcondv}}\left(j\right)$, and the $j$th columns of vl and vr all correspond to the $j$th eigenpair.
If ${\mathbf{sense}}=\text{'V'}$, rconde is not referenced.
15:   $\mathrm{rcondv}\left(:\right)$ – double array
The dimension of the array rcondv will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{sense}}=\text{'V'}$ or $\text{'B'}$, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.
If ${\mathbf{sense}}=\text{'E'}$, rcondv is not referenced.
16:   $\mathrm{info}$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: balanc, 2: jobvl, 3: jobvr, 4: sense, 5: n, 6: a, 7: lda, 8: b, 9: ldb, 10: alphar, 11: alphai, 12: beta, 13: vl, 14: ldvl, 15: vr, 16: ldvr, 17: ilo, 18: ihi, 19: lscale, 20: rscale, 21: abnrm, 22: bbnrm, 23: rconde, 24: rcondv, 25: work, 26: lwork, 27: iwork, 28: bwork, 29: 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 ${\mathbf{alphar}}\left(j\right)$, ${\mathbf{alphai}}\left(j\right)$, 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

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.
An approximate error bound on the chordal distance between the $i$th computed generalized eigenvalue $w$ and the corresponding exact eigenvalue $\lambda$ is
 $ε × abnrm,bbnrm2 / rcondei .$
An approximate error bound for the angle between the $i$th computed eigenvector ${u}_{j}$ or ${v}_{j}$ is given by
 $ε × abnrm,bbnrm2 / rcondvi .$
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.11 of Anderson et al. (1999).
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 complex analogue of this function is nag_lapack_zggevx (f08wp).

## Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair $\left(A,B\right)$, where
 $A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0 and B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ,$
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.
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 f08wb_example

fprintf('f08wb 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];

balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
[~, ~, alphar, alphai, beta, ~, VR, ~, ~, ~, ~, abnrm, bbnrm, ...
rconde, rcondv, info] = ...
f08wb( ...
balanc, jobvl, jobvr, sense, a, b);

epsilon = x02aj;
small   = x02am;
absnrm  = sqrt(abnrm^2+bbnrm^2);
tol     = epsilon*absnrm;

for j=1:n

% display information on the jth eigenvalue
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

if rconde(j) > 0
fprintf('\n      Condition number            = %8.1e\n', 1/rconde(j));
fprintf('      Error bound                 = %8.1e\n', tol/rconde(j));
else
fprintf('\n      Reciprocal condition number = %8.1e\n', rconde(j));
fprintf('      Error bound is infinite\n');
end

fprintf('\n      Eigenvector:\n');
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

if rcondv(j) > 0
fprintf('\n      Condition number            = %8.1e\n', 1/rcondv(j));
fprintf('      Error bound                 = %8.1e\n', tol/rcondv(j));
else
fprintf('\n      Reciprocal condition number = %8.1e\n', rcondv(j));
fprintf('      Error bound is infinite\n');
end
end

```
```f08wb example results

1: Eigenvalue =  2.0000e+00

Condition number            =  1.1e+01
Error bound                 =  2.5e-14

Eigenvector:
-1.0000e+00
-5.7143e-03
-6.2857e-02
-6.2857e-02

Condition number            =  8.0e+00
Error bound                 =  1.9e-14

2: Eigenvalue =  3.0000e+00 + 4.0000e+00i

Condition number            =  6.1e+00
Error bound                 =  1.4e-14

Eigenvector:
-4.2550e-01 - 5.7450e-01i
-8.5099e-02 - 1.1490e-01i
-1.4298e-01 - 8.6125e-04i
-1.4298e-01 - 8.6125e-04i

Condition number            =  2.6e+01
Error bound                 =  6.2e-14

3: Eigenvalue =  3.0000e+00 - 4.0000e+00i

Condition number            =  6.1e+00
Error bound                 =  1.4e-14

Eigenvector:
-4.2550e-01 - 5.7450e-01i
-8.5099e-02 - 1.1490e-01i
-1.4298e-01 - 8.6125e-04i
-1.4298e-01 - 8.6125e-04i

Condition number            =  2.6e+01
Error bound                 =  6.2e-14

4: Eigenvalue =  4.0000e+00

Condition number            =  1.9e+00
Error bound                 =  4.6e-15

Eigenvector:
-1.0000e+00
-1.1111e-02
3.3333e-02
-1.5556e-01

Condition number            =  1.4e+01
Error bound                 =  3.3e-14
```