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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zggevx (f08wp)

Purpose

nag_lapack_zggevx (f08wp) 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.
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, alpha, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = f08wp(balanc, jobvl, jobvr, sense, a, b, 'n', n)
[a, b, alpha, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = nag_lapack_zggevx(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 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.
For details of the balancing option, see Description in nag_lapack_zggbal (f08wv).

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\left(A,B\right)$.
6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a has been overwritten. If ${\mathbf{jobvl}}=\text{'V'}$ or ${\mathbf{jobvr}}=\text{'V'}$ or both, then $A$ contains the first part of the Schur form of the ‘balanced’ versions of the input $A$ and $B$.
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:     $\mathrm{ilo}$int64int32nag_int scalar
8:     $\mathrm{ihi}$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 ${\mathbf{ihi}}={\mathbf{n}}$.
9:     $\mathrm{lscale}\left({\mathbf{n}}\right)$ – 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 ${\mathbf{ilo}}-1$.
10:   $\mathrm{rscale}\left({\mathbf{n}}\right)$ – 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 ${\mathbf{ilo}}-1$.
11:   $\mathrm{abnrm}$ – double scalar
The $1$-norm of the balanced matrix $A$.
12:   $\mathrm{bbnrm}$ – double scalar
The $1$-norm of the balanced matrix $B$.
13:   $\mathrm{rconde}\left(:\right)$ – 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.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'V'}$, rconde is not referenced.
14:   $\mathrm{rcondv}\left(:\right)$ – 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 selected eigenvectors, stored in consecutive elements of the array.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'E'}$, rcondv is not referenced.
15:   $\mathrm{info}$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: alpha, 11: beta, 12: vl, 13: ldvl, 14: vr, 15: ldvr, 16: ilo, 17: ihi, 18: lscale, 19: rscale, 20: abnrm, 21: bbnrm, 22: rconde, 23: rcondv, 24: work, 25: lwork, 26: rwork, 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 ${\mathbf{alpha}}\left(j\right)$ 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.
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 real analogue of this function is nag_lapack_dggevx (f08wb).

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
 $B = 1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i 0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i 1.00+0.00i 2.40+1.80i -4.00-5.00i 0.00-3.00i 0.00+1.00i -1.80+2.40i 0.00-4.00i 4.00-5.00i ,$
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 f08wp_example

fprintf('f08wp 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];

balanc = 'Balance';
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
[~, ~, alpha, beta, ~, VR, ilo, ihi, lscale, rscale, abnrm, bbnrm, ...
rconde, rcondv, info] = ...
f08wp( ...
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(alpha(j))*small >= abs(beta(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(alpha(j)/beta(j));
end

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

fprintf('\nEigenvector (%d):\n', j);
disp(VR(:, j));

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

```
```f08wp example results

Eigenvalue  (1)
3.0000 - 1.0000i

Condition number            =  7.5e+00
Error bound                 =  1.2e-14

Eigenvector (1):
-0.7326 - 0.2674i
-0.1493 + 0.0451i
-0.1307 + 0.0851i
-0.0851 - 0.1307i

Condition number            =  7.8e+00
Error bound                 =  1.2e-14

Eigenvalue  (2)
2.0000 - 5.0000i

Condition number            =  2.7e+00
Error bound                 =  4.3e-15

Eigenvector (2):
-0.5202 + 0.4798i
-0.0007 + 0.0040i
-0.0327 + 0.0302i
-0.0302 - 0.0327i

Condition number            =  1.7e+01
Error bound                 =  2.7e-14

Eigenvalue  (3)
3.0000 - 9.0000i

Condition number            =  2.0e+00
Error bound                 =  3.1e-15

Eigenvector (3):
-0.3614 + 0.6386i
0.0188 + 0.1455i
-0.1455 + 0.0188i
-0.0188 - 0.1455i

Condition number            =  1.9e+01
Error bound                 =  3.1e-14

Eigenvalue  (4)
4.0000 - 5.0000i

Condition number            =  1.6e+00
Error bound                 =  2.6e-15

Eigenvector (4):
-0.3660 + 0.6340i
0.0010 + 0.0081i
0.0122 - 0.0211i
-0.0986 - 0.0569i

Condition number            =  1.6e+01
Error bound                 =  2.6e-14

```