f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_dggevx (f08wbc)

1  Purpose

nag_dggevx (f08wbc) 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.

2  Specification

 #include #include
 void nag_dggevx (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double vl[], Integer pdvl, double vr[], Integer pdvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], NagError *fail)

3  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 Section 3 in nag_dggbal (f08whc).

4  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 (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

5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 2.3.1.3 in How to Use the NAG Library and its Documentation 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_BalanceTypeInput
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_LeftVecsTypeInput
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_RightVecsTypeInput
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_RCondTypeInput
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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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 real Schur form of the ‘balanced’ versions of the input $A$ and $B$.
8:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{b}\left[\mathit{dim}\right]$doubleInput/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:  $\mathbf{pdb}$IntegerInput
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:  $\mathbf{alphar}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the element ${\mathbf{alphar}}\left[j-1\right]$ contains the real part of ${\alpha }_{j}$.
12:  $\mathbf{alphai}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the element ${\mathbf{alphai}}\left[j-1\right]$ contains the imaginary part of ${\alpha }_{j}$.
13:  $\mathbf{beta}\left[{\mathbf{n}}\right]$doubleOutput
On exit: $\left({\mathbf{alphar}}\left[\mathit{j}-1\right]+{\mathbf{alphai}}\left[\mathit{j}-1\right]×i\right)/{\mathbf{beta}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues.
If ${\mathbf{alphai}}\left[j-1\right]$ 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 ${\mathbf{alphai}}\left[j\right]$ negative.
Note:  the quotients ${\mathbf{alphar}}\left[j-1\right]/{\mathbf{beta}}\left[j-1\right]$ and ${\mathbf{alphai}}\left[j-1\right]/{\mathbf{beta}}\left[j-1\right]$ 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, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left|{\alpha }_{j}\right|\right)$ will always be less than and usually comparable with ${‖A‖}_{2}$ in magnitude, and $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left|{\beta }_{j}\right|\right)$ will always be less than and usually comparable with ${‖B‖}_{2}$.
14:  $\mathbf{vl}\left[\mathit{dim}\right]$doubleOutput
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 $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.
If ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, vl is not referenced.
15:  $\mathbf{pdvl}$IntegerInput
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 ${\mathbf{pdvl}}\ge 1$.
16:  $\mathbf{vr}\left[\mathit{dim}\right]$doubleOutput
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 $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.
If ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, vr is not referenced.
17:  $\mathbf{pdvr}$IntegerInput
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 ${\mathbf{pdvr}}\ge 1$.
18:  $\mathbf{ilo}$Integer *Output
19:  $\mathbf{ihi}$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 ${\mathbf{ihi}}={\mathbf{n}}$.
20:  $\mathbf{lscale}\left[{\mathbf{n}}\right]$doubleOutput
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 ${\mathbf{ilo}}-1$.
21:  $\mathbf{rscale}\left[{\mathbf{n}}\right]$doubleOutput
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 ${\mathbf{ilo}}-1$.
22:  $\mathbf{abnrm}$double *Output
On exit: the $1$-norm of the balanced matrix $A$.
23:  $\mathbf{bbnrm}$double *Output
On exit: the $1$-norm of the balanced matrix $B$.
24:  $\mathbf{rconde}\left[\mathit{dim}\right]$doubleOutput
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. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus ${\mathbf{rconde}}\left[j-1\right]$, ${\mathbf{rcondv}}\left[j-1\right]$, and the $j$th columns of vl and vr all correspond to the $j$th eigenpair.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$, rconde is not referenced.
25:  $\mathbf{rcondv}\left[\mathit{dim}\right]$doubleOutput
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 eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$, rcondv is not referenced.
26:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENVECTORS
A failure occurred in nag_dtgevc (f08ykc) 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_ITERATION_QZ
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 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for 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
 $ε × abnrm,bbnrm2 / rconde[i-1] .$
An approximate error bound for the angle between the $i$th computed eigenvector ${u}_{j}$ or ${v}_{j}$ is given by
 $ε × abnrm,bbnrm2 / rcondv[i-1] .$
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.

8  Parallelism and Performance

nag_dggevx (f08wbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggevx (f08wbc) 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 complex analogue of this function is nag_zggevx (f08wpc).

10  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.

10.1  Program Text

Program Text (f08wbce.c)

10.2  Program Data

Program Data (f08wbce.d)

10.3  Program Results

Program Results (f08wbce.r)