# NAG Library Routine Document

## 1Purpose

f08xbf (dggesx) computes the generalized eigenvalues, the generalized real Schur form $\left(S,T\right)$ and, optionally, the left and/or right generalized Schur vectors for a pair of $n$ by $n$ real nonsymmetric matrices $\left(A,B\right)$.
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.

## 2Specification

Fortran Interface
 Subroutine f08xbf ( sort, n, a, lda, b, ldb, sdim, beta, vsl, vsr, work, info)
 Integer, Intent (In) :: n, lda, ldb, ldvsl, ldvsr, lwork, liwork Integer, Intent (Out) :: sdim, iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), vsl(ldvsl,*), vsr(ldvsr,*) Real (Kind=nag_wp), Intent (Out) :: alphar(n), alphai(n), beta(n), rconde(2), rcondv(2), work(max(1,lwork)) Logical, External :: selctg Logical, Intent (Inout) :: bwork(*) Character (1), Intent (In) :: jobvsl, jobvsr, sort, sense
#include <nagmk26.h>
 void f08xbf_ (const char *jobvsl, const char *jobvsr, const char *sort, logical (NAG_CALL *selctg)(const double *ar, const double *ai, const double *b),const char *sense, const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, Integer *sdim, double alphar[], double alphai[], double beta[], double vsl[], const Integer *ldvsl, double vsr[], const Integer *ldvsr, double rconde[], double rcondv[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, logical bwork[], Integer *info, const Charlen length_jobvsl, const Charlen length_jobvsr, const Charlen length_sort, const Charlen length_sense)
The routine may be called by its LAPACK name dggesx.

## 3Description

The generalized real Schur factorization of $\left(A,B\right)$ is given by
 $A = QSZT , B = QTZT ,$
where $Q$ and $Z$ are orthogonal, $T$ is upper triangular and $S$ is upper quasi-triangular with $1$ by $1$ and $2$ by $2$ diagonal blocks. The generalized eigenvalues, $\lambda$, of $\left(A,B\right)$ are computed from the diagonals of $T$ and $S$ and satisfy
 $Az = λBz ,$
where $z$ is the corresponding generalized eigenvector. $\lambda$ is actually returned as the pair $\left(\alpha ,\beta \right)$ such that
 $λ = α/β$
since $\beta$, or even both $\alpha$ and $\beta$ can be zero. The columns of $Q$ and $Z$ are the left and right generalized Schur vectors of $\left(A,B\right)$.
Optionally, f08xbf (dggesx) can order the generalized eigenvalues on the diagonals of $\left(S,T\right)$ so that selected eigenvalues are at the top left. The leading columns of $Q$ and $Z$ then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xbf (dggesx) computes $T$ to have non-negative diagonal elements, and the $2$ by $2$ blocks of $S$ correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the $QZ$ algorithm.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in ${\mathbf{rconde}}\left(1\right)$ and ${\mathbf{rconde}}\left(2\right)$ respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in ${\mathbf{rcondv}}\left(1\right)$ and ${\mathbf{rcondv}}\left(2\right)$. See Section 4.11 of Anderson et al. (1999) for further information.

## 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{jobvsl}$ – Character(1)Input
On entry: if ${\mathbf{jobvsl}}=\text{'N'}$, do not compute the left Schur vectors.
If ${\mathbf{jobvsl}}=\text{'V'}$, compute the left Schur vectors.
Constraint: ${\mathbf{jobvsl}}=\text{'N'}$ or $\text{'V'}$.
2:     $\mathbf{jobvsr}$ – Character(1)Input
On entry: if ${\mathbf{jobvsr}}=\text{'N'}$, do not compute the right Schur vectors.
If ${\mathbf{jobvsr}}=\text{'V'}$, compute the right Schur vectors.
Constraint: ${\mathbf{jobvsr}}=\text{'N'}$ or $\text{'V'}$.
3:     $\mathbf{sort}$ – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
${\mathbf{sort}}=\text{'N'}$
Eigenvalues are not ordered.
${\mathbf{sort}}=\text{'S'}$
Eigenvalues are ordered (see selctg).
Constraint: ${\mathbf{sort}}=\text{'N'}$ or $\text{'S'}$.
4:     $\mathbf{selctg}$ – Logical Function, supplied by the user.External Procedure
If ${\mathbf{sort}}=\text{'S'}$, selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.
If ${\mathbf{sort}}=\text{'N'}$, selctg is not referenced by f08xbf (dggesx), and may be called with the dummy function f08xaz.
The specification of selctg is:
Fortran Interface
 Function selctg ( ar, ai, b)
 Logical :: selctg Real (Kind=nag_wp), Intent (In) :: ar, ai, b
#include <nagmk26.h>
 Nag_Boolean selctg (const double *ar, const double *ai, const double *b)
1:     $\mathbf{ar}$ – Real (Kind=nag_wp)Input
2:     $\mathbf{ai}$ – Real (Kind=nag_wp)Input
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: an eigenvalue $\left({\mathbf{ar}}\left(j\right)+\sqrt{-1}×{\mathbf{ai}}\left(j\right)\right)/{\mathbf{b}}\left(j\right)$ is selected if ${\mathbf{selctg}}\left({\mathbf{ar}}\left(j\right),{\mathbf{ai}}\left(j\right),{\mathbf{b}}\left(j\right)\right)$ is .TRUE.. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy ${\mathbf{selctg}}\left({\mathbf{ar}}\left(j\right),{\mathbf{ai}}\left(j\right),{\mathbf{b}}\left(j\right)\right)=\mathrm{.TRUE.}$ after ordering. ${\mathbf{info}}=\mathbf{n}+{\mathbf{2}}$ in this case.
selctg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08xbf (dggesx) is called. Arguments denoted as Input must not be changed by this procedure.
5:     $\mathbf{sense}$ – Character(1)Input
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\text{'N'}$
None are computed.
${\mathbf{sense}}=\text{'E'}$
Computed for average of selected eigenvalues only.
${\mathbf{sense}}=\text{'V'}$
Computed for selected deflating subspaces only.
${\mathbf{sense}}=\text{'B'}$
Computed for both.
If ${\mathbf{sense}}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, ${\mathbf{sort}}=\text{'S'}$.
Constraint: ${\mathbf{sense}}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
6:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the first of the pair of matrices, $A$.
On exit: a has been overwritten by its generalized Schur form $S$.
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08xbf (dggesx) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the second of the pair of matrices, $B$.
On exit: b has been overwritten by its generalized Schur form $T$.
10:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08xbf (dggesx) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   $\mathbf{sdim}$ – IntegerOutput
On exit: if ${\mathbf{sort}}=\text{'N'}$, ${\mathbf{sdim}}=0$.
If ${\mathbf{sort}}=\text{'S'}$, ${\mathbf{sdim}}=\text{}$ number of eigenvalues (after sorting) for which selctg is .TRUE.. (Complex conjugate pairs for which selctg is .TRUE. for either eigenvalue count as $2$.)
12:   $\mathbf{alphar}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: see the description of beta.
13:   $\mathbf{alphai}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: see the description of beta.
14:   $\mathbf{beta}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: $\left({\mathbf{alphar}}\left(\mathit{j}\right)+{\mathbf{alphai}}\left(\mathit{j}\right)×i\right)/{\mathbf{beta}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues. ${\mathbf{alphar}}\left(\mathit{j}\right)+{\mathbf{alphai}}\left(\mathit{j}\right)×i$, and ${\mathbf{beta}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, are the diagonals of the complex Schur form $\left(S,T\right)$ that would result if the $2$ by $2$ diagonal blocks of the real Schur form of $\left(A,B\right)$ were further reduced to triangular form using $2$ by $2$ complex unitary transformations.
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 $\alpha /\beta$. However, alphar and alphai will always be less than and usually comparable with ${‖A‖}_{2}$ in magnitude, and beta will always be less than and usually comparable with ${‖B‖}_{2}$.
15:   $\mathbf{vsl}\left({\mathbf{ldvsl}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vsl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvsl}}=\text{'V'}$.
On exit: if ${\mathbf{jobvsl}}=\text{'V'}$, vsl will contain the left Schur vectors, $Q$.
If ${\mathbf{jobvsl}}=\text{'N'}$, vsl is not referenced.
16:   $\mathbf{ldvsl}$ – IntegerInput
On entry: the first dimension of the array vsl as declared in the (sub)program from which f08xbf (dggesx) is called.
Constraints:
• if ${\mathbf{jobvsl}}=\text{'V'}$, ${\mathbf{ldvsl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvsl}}\ge 1$.
17:   $\mathbf{vsr}\left({\mathbf{ldvsr}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vsr must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvsr}}=\text{'V'}$.
On exit: if ${\mathbf{jobvsr}}=\text{'V'}$, vsr will contain the right Schur vectors, $Z$.
If ${\mathbf{jobvsr}}=\text{'N'}$, vsr is not referenced.
18:   $\mathbf{ldvsr}$ – IntegerInput
On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xbf (dggesx) is called.
Constraints:
• if ${\mathbf{jobvsr}}=\text{'V'}$, ${\mathbf{ldvsr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvsr}}\ge 1$.
19:   $\mathbf{rconde}\left(2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, ${\mathbf{rconde}}\left(1\right)$ and ${\mathbf{rconde}}\left(2\right)$ contain the reciprocal condition numbers for the average of the selected eigenvalues.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'V'}$, rconde is not referenced.
20:   $\mathbf{rcondv}\left(2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{sense}}=\text{'V'}$ or $\text{'B'}$, ${\mathbf{rcondv}}\left(1\right)$ and ${\mathbf{rcondv}}\left(2\right)$ contain the reciprocal condition numbers for the selected deflating subspaces.
if ${\mathbf{sense}}=\text{'N'}$ or $\text{'E'}$, rcondv is not referenced.
21:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ returns the optimal lwork.
22:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08xbf (dggesx) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
• if ${\mathbf{sense}}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(8×\left({\mathbf{n}}+1\right)+16,2×{\mathbf{n}}+2×{\mathbf{sdim}}×\left({\mathbf{n}}-{\mathbf{sdim}}\right)\right)$;
• otherwise ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,8×{\mathbf{n}},6×{\mathbf{n}}+16\right)$.
Note: that $2×{\mathbf{sdim}}×\left({\mathbf{n}}-{\mathbf{sdim}}\right)\le {\mathbf{n}}×{\mathbf{n}}/2$. Note also that an error is only returned if ${\mathbf{lwork}}<8×\left({\mathbf{n}}+1\right)+16$, but if ${\mathbf{sense}}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$ this may not be large enough. Consider increasing lwork by $\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
23:   $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$ – Integer arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ returns the minimum liwork.
24:   $\mathbf{liwork}$ – IntegerInput
On entry: the dimension of the array iwork as declared in the (sub)program from which f08xbf (dggesx) is called.
If ${\mathbf{liwork}}=-1$, a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
• if ${\mathbf{sense}}=\text{'N'}$ or ${\mathbf{n}}=0$, ${\mathbf{liwork}}\ge 1$;
• otherwise ${\mathbf{liwork}}\ge {\mathbf{n}}+6$.
25:   $\mathbf{bwork}\left(*\right)$ – Logical arrayWorkspace
Note: the dimension of the array bwork must be at least $1$ if ${\mathbf{sort}}=\text{'N'}$, and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ otherwise.
If ${\mathbf{sort}}=\text{'N'}$, bwork is not referenced.
26:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
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 from element $〈\mathit{\text{value}}〉$.
${\mathbf{info}}={\mathbf{n}}+1$
The $QZ$ iteration failed with an unexpected error, please contact NAG.
${\mathbf{info}}={\mathbf{n}}+2$
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy ${\mathbf{selctg}}=\mathrm{.TRUE.}$. This could also be caused by underflow due to scaling.
${\mathbf{info}}={\mathbf{n}}+3$
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

## 7Accuracy

The computed generalized Schur factorization satisfies
 $A+E = QS ZT , B+F = QT ZT ,$
where
 $E,F F = Oε A,B F$
and $\epsilon$ is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08xbf (dggesx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xbf (dggesx) 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 routine. 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 routine is f08xpf (zggesx).

## 10Example

This example finds the generalized Schur factorization of the matrix pair $\left(A,B\right)$, where
 $A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0 and B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ,$
such that the real positive eigenvalues of $\left(A,B\right)$ correspond to the top left diagonal elements of the generalized Schur form, $\left(S,T\right)$. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
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.

### 10.1Program Text

Program Text (f08xbfe.f90)

### 10.2Program Data

Program Data (f08xbfe.d)

### 10.3Program Results

Program Results (f08xbfe.r)