# NAG CL Interfacef08wpc (zggevx)

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## 1Purpose

f08wpc computes for a pair of $n×n$ 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.

## 2Specification

 #include
 void f08wpc (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, Complex alpha[], Complex beta[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], NagError *fail)
The function may be called by the names: f08wpc, nag_lapackeig_zggevx or nag_zggevx.

## 3Description

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. 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. 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. 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 Section 3 in f08wvc.

## 4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the $QZ$ algorithm Linear Algebra Appl. 28 285–303

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{balanc}$Nag_BalanceType Input
On entry: specifies the balance option to be performed.
${\mathbf{balanc}}=\mathrm{Nag_NoBalancing}$
Do not diagonally scale or permute.
${\mathbf{balanc}}=\mathrm{Nag_BalancePermute}$
Permute only.
${\mathbf{balanc}}=\mathrm{Nag_BalanceScale}$
Scale only.
${\mathbf{balanc}}=\mathrm{Nag_BalanceBoth}$
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, ${\mathbf{balanc}}=\mathrm{Nag_BalanceBoth}$ is recommended.
Constraint: ${\mathbf{balanc}}=\mathrm{Nag_NoBalancing}$, $\mathrm{Nag_BalancePermute}$, $\mathrm{Nag_BalanceScale}$ or $\mathrm{Nag_BalanceBoth}$.
3: $\mathbf{jobvl}$Nag_LeftVecsType Input
On entry: if ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, do not compute the left generalized eigenvectors.
If ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, compute the left generalized eigenvectors.
Constraint: ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
4: $\mathbf{jobvr}$Nag_RightVecsType Input
On entry: if ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, do not compute the right generalized eigenvectors.
If ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, compute the right generalized eigenvectors.
Constraint: ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
5: $\mathbf{sense}$Nag_RCondType Input
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\mathrm{Nag_NotRCond}$
None are computed.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$
Computed for eigenvalues only.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$
Computed for eigenvectors only.
${\mathbf{sense}}=\mathrm{Nag_RCondBoth}$
Computed for eigenvalues and eigenvectors.
Constraint: ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$, $\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$.
6: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $A$ in the pair $\left(A,B\right)$.
On exit: a has been overwritten. If ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$ or ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$ or both, then $A$ contains the first part of the Schur form of the ‘balanced’ versions of the input $A$ and $B$.
8: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{b}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
where ${\mathbf{B}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $B$ in the pair $\left(A,B\right)$.
On exit: b has been overwritten.
10: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11: $\mathbf{alpha}\left[{\mathbf{n}}\right]$Complex Output
On exit: see the description of beta.
12: $\mathbf{beta}\left[{\mathbf{n}}\right]$Complex Output
On exit: ${\mathbf{alpha}}\left[\mathit{j}-1\right]/{\mathbf{beta}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues.
Note:  the quotients ${\mathbf{alpha}}\left[j-1\right]/{\mathbf{beta}}\left[j-1\right]$ may easily overflow or underflow, and ${\mathbf{beta}}\left[j-1\right]$ may even be zero. Thus, you should avoid naively computing the ratio ${\alpha }_{j}/{\beta }_{j}$. However, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|{\alpha }_{j}|\right)$ will always be less than and usually comparable with ${‖A‖}_{2}$ in magnitude, and $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|{\beta }_{j}|\right)$ will always be less than and usually comparable with ${‖B‖}_{2}$.
13: $\mathbf{vl}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array vl must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvl}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vl}}\left[\left(j-1\right)×{\mathbf{pdvl}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vl}}\left[\left(i-1\right)×{\mathbf{pdvl}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, 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 $|\text{real part}|+|\text{imag. part}|=1$.
If ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, vl is not referenced.
14: $\mathbf{pdvl}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
• if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvl}}\ge 1$.
15: $\mathbf{vr}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array vr must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvr}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vr}}\left[\left(j-1\right)×{\mathbf{pdvr}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vr}}\left[\left(i-1\right)×{\mathbf{pdvr}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, 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 $|\text{real part}|+|\text{imag. part}|=1$.
If ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, vr is not referenced.
16: $\mathbf{pdvr}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
• if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvr}}\ge 1$.
17: $\mathbf{ilo}$Integer * Output
18: $\mathbf{ihi}$Integer * Output
On exit: 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 ${\mathbf{balanc}}=\mathrm{Nag_NoBalancing}$ or $\mathrm{Nag_BalanceScale}$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}={\mathbf{n}}$.
19: $\mathbf{lscale}\left[{\mathbf{n}}\right]$double Output
On exit: 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:
• ${\mathbf{lscale}}\left[\mathit{j}-1\right]={\mathit{pl}}_{\mathit{j}}$, 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 ${\mathbf{ilo}}-1$.
20: $\mathbf{rscale}\left[{\mathbf{n}}\right]$double Output
On exit: 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:
• ${\mathbf{rscale}}\left[\mathit{j}-1\right]={\mathit{pr}}_{\mathit{j}}$, 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 ${\mathbf{ilo}}-1$.
21: $\mathbf{abnrm}$double * Output
On exit: the $1$-norm of the balanced matrix $A$.
22: $\mathbf{bbnrm}$double * Output
On exit: the $1$-norm of the balanced matrix $B$.
23: $\mathbf{rconde}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array rconde must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array.
If ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVecs}$, rconde is not referenced.
24: $\mathbf{rcondv}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array rcondv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array.
If ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVals}$, rcondv is not referenced.
25: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_EIGENVECTORS
A failure occurred in f08yxc while computing generalized eigenvectors.
NE_ENUM_INT_2
On entry, ${\mathbf{jobvl}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvl}}\ge 1$.
On entry, ${\mathbf{jobvr}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvr}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvl}}>0$.
On entry, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvr}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_ITERATION_QZ
The $QZ$ iteration failed. No eigenvectors have been calculated but alpha and beta should be correct from element $⟨\mathit{\text{value}}⟩$.
The $QZ$ iteration failed with an unexpected error, please contact NAG.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The computed eigenvalues and eigenvectors are exact for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $‖(E,F)‖ F = O(ε) ‖(A,B)‖ F ,$
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
 $ε × ‖abnrm,bbnrm‖2 / rconde[i-1] .$
An approximate error bound for the angle between the $i$th computed eigenvector ${u}_{j}$ or ${v}_{j}$ is given by
 $ε × ‖abnrm,bbnrm‖2 / rcondv[i-1] .$
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.

## 8Parallelism and Performance

f08wpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wpc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is f08wbc.

## 10Example

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.

### 10.1Program Text

Program Text (f08wpce.c)

### 10.2Program Data

Program Data (f08wpce.d)

### 10.3Program Results

Program Results (f08wpce.r)