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_dggevx (f08wb)

## Purpose

nag_lapack_dggevx (f08wb) 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.
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 (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.
For details of the balancing option, see Section [Description] in (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$QZ$ algorithm Linear Algebra Appl. 28 285–303

## Parameters

### Compulsory Input Parameters

1:     balanc – string (length ≥ 1)
Specifies the balance option to be performed.
balanc = 'N'${\mathbf{balanc}}=\text{'N'}$
Do not diagonally scale or permute.
balanc = 'P'${\mathbf{balanc}}=\text{'P'}$
Permute only.
balanc = 'S'${\mathbf{balanc}}=\text{'S'}$
Scale only.
balanc = 'B'${\mathbf{balanc}}=\text{'B'}$
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, balanc = 'B'${\mathbf{balanc}}=\text{'B'}$ is recommended.
Constraint: balanc = 'N'${\mathbf{balanc}}=\text{'N'}$, 'P'$\text{'P'}$, 'S'$\text{'S'}$ or 'B'$\text{'B'}$.
2:     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'$\text{'V'}$.
3:     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'$\text{'V'}$.
4:     sense – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
sense = 'N'${\mathbf{sense}}=\text{'N'}$
None are computed.
sense = 'E'${\mathbf{sense}}=\text{'E'}$
Computed for eigenvalues only.
sense = 'V'${\mathbf{sense}}=\text{'V'}$
Computed for eigenvectors only.
sense = 'B'${\mathbf{sense}}=\text{'B'}$
Computed for eigenvalues and eigenvectors.
Constraint: sense = 'N'${\mathbf{sense}}=\text{'N'}$, 'E'$\text{'E'}$, 'V'$\text{'V'}$ or 'B'$\text{'B'}$.
5:     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)$\left(A,B\right)$.
6:     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 iwork bwork

### 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a has been overwritten. If jobvl = 'V'${\mathbf{jobvl}}=\text{'V'}$ or jobvr = 'V'${\mathbf{jobvr}}=\text{'V'}$ or both, then A$A$ contains the first part of the real Schur form of the ‘balanced’ versions of the input A$A$ and B$B$.
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 their eigenvalues.
If the j$j$th eigenvalue is real, then v(j) = vr( : ,j)$v\left(j\right)={\mathbf{vr}}\left(:,j\right)$, 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:     ilo – int64int32nag_int scalar
9:     ihi – int64int32nag_int scalar
ilo and ihi are integer values such that a(i,j) = 0${\mathbf{a}}\left(i,j\right)=0$ and b(i,j) = 0${\mathbf{b}}\left(i,j\right)=0$ if i > j$i>j$ and j = 1,2,,ilo1$j=1,2,\dots ,{\mathbf{ilo}}-1$ or i = ihi + 1,,n$i={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
If balanc = 'N'${\mathbf{balanc}}=\text{'N'}$ or 'S'$\text{'S'}$, ilo = 1${\mathbf{ilo}}=1$ and ihi = n${\mathbf{ihi}}={\mathbf{n}}$.
10:   lscale(n) – double array
Details of the permutations and scaling factors applied to the left side of A$A$ and B$B$.
If plj ${\mathit{pl}}_{j}$ is the index of the row interchanged with row j$j$, and dlj ${\mathit{dl}}_{j}$ is the scaling factor applied to row j$j$, then:
• lscale(j) = plj ${\mathbf{lscale}}\left(\mathit{j}\right)={\mathit{pl}}_{\mathit{j}}$, for j = 1,2,,ilo1$\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• lscale = dlj ${\mathbf{lscale}}={\mathit{dl}}_{\mathit{j}}$, for j = ilo,,ihi$\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• lscale = plj ${\mathbf{lscale}}={\mathit{pl}}_{\mathit{j}}$, for j = ihi + 1,,n$\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ihi + 1${\mathbf{ihi}}+1$, then 1$1$ to ilo1${\mathbf{ilo}}-1$.
11:   rscale(n) – double array
Details of the permutations and scaling factors applied to the right side of A$A$ and B$B$.
If prj${\mathit{pr}}_{j}$ is the index of the column interchanged with column j$j$, and drj${\mathit{dr}}_{j}$ is the scaling factor applied to column j$j$, then:
• rscale(j) = prj${\mathbf{rscale}}\left(\mathit{j}\right)={\mathit{pr}}_{\mathit{j}}$, for j = 1,2,,ilo1$\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• if rscale = drj${\mathbf{rscale}}={\mathit{dr}}_{\mathit{j}}$, for j = ilo,,ihi$\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• if rscale = prj${\mathbf{rscale}}={\mathit{pr}}_{\mathit{j}}$, for j = ihi + 1,,n$\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ihi + 1${\mathbf{ihi}}+1$, then 1$1$ to ilo1${\mathbf{ilo}}-1$.
12:   abnrm – double scalar
The 1$1$-norm of the balanced matrix A$A$.
13:   bbnrm – double scalar
The 1$1$-norm of the balanced matrix B$B$.
14:   rconde( : $:$) – double array
Note: the dimension of the array rconde must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If sense = 'E'${\mathbf{sense}}=\text{'E'}$ or 'B'$\text{'B'}$, 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 rconde(j)${\mathbf{rconde}}\left(j\right)$, rcondv(j)${\mathbf{rcondv}}\left(j\right)$, and the j$j$th columns of vl and vr all correspond to the j$j$th eigenpair.
If sense = 'V'${\mathbf{sense}}=\text{'V'}$, rconde is not referenced.
15:   rcondv( : $:$) – double array
Note: the dimension of the array rcondv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If sense = 'V'${\mathbf{sense}}=\text{'V'}$ or 'B'$\text{'B'}$, 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 sense = 'E'${\mathbf{sense}}=\text{'E'}$, rcondv is not referenced.
16:   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: 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 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.
An approximate error bound on the chordal distance between the i$i$th computed generalized eigenvalue w$w$ and the corresponding exact eigenvalue λ $\lambda$ is
 ε × ‖abnrm,bbnrm‖2 / rconde(i) . $ε × ‖abnrm,bbnrm‖2 / rcondei .$
An approximate error bound for the angle between the i$i$th computed eigenvector vl(i) ${\mathbf{vl}}\left(i\right)$ or vr(i) ${\mathbf{vr}}\left(i\right)$ is given by
 ε × ‖abnrm,bbnrm‖2 / rcondv(i) . $ε × ‖abnrm,bbnrm‖2 / rcondvi .$
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$QZ$ 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_zggevx (f08wp).

## Example

```function nag_lapack_dggevx_example
balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
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, ilo, ihi, lscale, rscale, ...
abnrm, bbnrm, rconde, rcondv, info] = ...
nag_lapack_dggevx(balanc, jobvl, jobvr, sense, a, b);

epsilon = nag_machine_precision
small = nag_machine_real_safe;
absnrm = sqrt(abnrm^2+bbnrm^2);
tol = epsilon*absnrm;

for j=1:4

% Print out information on the jth eigenvalue
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('Reciprocal condition number = %8.1e\n', rconde(j));
if rconde(j) > 0
fprintf('Error bound                 = %8.1e\n', tol/rconde(j));
else
fprintf('Error bound is infinite\n');
end

% Print out information on the jth eigenvector
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
fprintf('Reciprocal condition number = %8.1e\n', rcondv(j));
if rcondv(j) > 0
fprintf('Error bound                 = %8.1e\n', tol/rcondv(j));
else
fprintf('Error bound is infinite\n');
end
end

for i=1:4
end
```
```

epsilon =

1.1102e-16

Eigenvalue(1) = 2

Reciprocal condition number =  9.5e-02
Error bound                 =  2.5e-14

Eigenvector(1)
-1.0000
-0.0057
-0.0629
-0.0629

Reciprocal condition number =  1.3e-01
Error bound                 =  1.9e-14

Eigenvalue(2) = 3+4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(2)
-0.4255 - 0.5745i
-0.0851 - 0.1149i
-0.1430 - 0.0009i
-0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(3) = 3-4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(3)
-0.4255 - 0.5745i
-0.0851 - 0.1149i
-0.1430 - 0.0009i
-0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(4) = 4

Reciprocal condition number =  5.1e-01
Error bound                 =  4.6e-15

Eigenvector(4)
-1.0000
-0.0111
0.0333
-0.1556

Reciprocal condition number =  7.1e-02
Error bound                 =  3.3e-14

```
```function f08wb_example
balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
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, ilo, ihi, lscale, rscale, ...
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=1:4

% Print out information on the jth eigenvalue
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('Reciprocal condition number = %8.1e\n', rconde(j));
if rconde(j) > 0
fprintf('Error bound                 = %8.1e\n', tol/rconde(j));
else
fprintf('Error bound is infinite\n');
end

% Print out information on the jth eigenvector
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
fprintf('Reciprocal condition number = %8.1e\n', rcondv(j));
if rcondv(j) > 0
fprintf('Error bound                 = %8.1e\n', tol/rcondv(j));
else
fprintf('Error bound is infinite\n');
end
end

for i=1:4
end
```
```

epsilon =

1.1102e-16

Eigenvalue(1) = 2

Reciprocal condition number =  9.5e-02
Error bound                 =  2.5e-14

Eigenvector(1)
-1.0000
-0.0057
-0.0629
-0.0629

Reciprocal condition number =  1.3e-01
Error bound                 =  1.9e-14

Eigenvalue(2) = 3+4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(2)
-0.4255 - 0.5745i
-0.0851 - 0.1149i
-0.1430 - 0.0009i
-0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(3) = 3-4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(3)
-0.4255 - 0.5745i
-0.0851 - 0.1149i
-0.1430 - 0.0009i
-0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(4) = 4

Reciprocal condition number =  5.1e-01
Error bound                 =  4.6e-15

Eigenvector(4)
-1.0000
-0.0111
0.0333
-0.1556

Reciprocal condition number =  7.1e-02
Error bound                 =  3.3e-14

```