# NAG Library Routine Document

## 1Purpose

f08xnf (zgges) computes the generalized eigenvalues, the generalized Schur form $\left(S,T\right)$ and, optionally, the left and/or right generalized Schur vectors for a pair of $n$ by $n$ complex nonsymmetric matrices $\left(A,B\right)$. f08xnf (zgges) is marked as deprecated by LAPACK; the replacement routine is f08xqf (zgges3) which makes better use of level $3$ BLAS.

## 2Specification

Fortran Interface
 Subroutine f08xnf ( sort, n, a, lda, b, ldb, sdim, beta, vsl, vsr, work, info)
 Integer, Intent (In) :: n, lda, ldb, ldvsl, ldvsr, lwork Integer, Intent (Out) :: sdim, info Real (Kind=nag_wp), Intent (Out) :: rwork(max(1,8*n)) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), vsl(ldvsl,*), vsr(ldvsr,*) Complex (Kind=nag_wp), Intent (Out) :: alpha(n), beta(n), work(max(1,lwork)) Logical, External :: selctg Logical, Intent (Inout) :: bwork(*) Character (1), Intent (In) :: jobvsl, jobvsr, sort
#include <nagmk26.h>
 void f08xnf_ (const char *jobvsl, const char *jobvsr, const char *sort, logical (NAG_CALL *selctg)(const Complex *a, const Complex *b),const Integer *n, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Integer *sdim, Complex alpha[], Complex beta[], Complex vsl[], const Integer *ldvsl, Complex vsr[], const Integer *ldvsr, Complex work[], const Integer *lwork, double rwork[], logical bwork[], Integer *info, const Charlen length_jobvsl, const Charlen length_jobvsr, const Charlen length_sort)
The routine may be called by its LAPACK name zgges.

## 3Description

The generalized Schur factorization for a pair of complex matrices $\left(A,B\right)$ is given by
 $A = QSZH , B = QTZH ,$
where $Q$ and $Z$ are unitary, $T$ and $S$ are upper triangular. 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, f08xnf (zgges) 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.
f08xnf (zgges) computes $T$ to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the $QZ$ algorithm.

## 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 f08xnf (zgges), and may be called with the dummy function f08xnz.
The specification of selctg is:
Fortran Interface
 Function selctg ( a, b)
 Logical :: selctg Complex (Kind=nag_wp), Intent (In) :: a, b
#include <nagmk26.h>
 Nag_Boolean selctg (const Complex *a, const Complex *b)
1:     $\mathbf{a}$ – Complex (Kind=nag_wp)Input
2:     $\mathbf{b}$ – Complex (Kind=nag_wp)Input
On entry: an eigenvalue ${\mathbf{a}}\left(j\right)/{\mathbf{b}}\left(j\right)$ is selected if ${\mathbf{selctg}}\left({\mathbf{a}}\left(j\right),{\mathbf{b}}\left(j\right)\right)$ is .TRUE..
Note that in the ill-conditioned case, a selected generalized eigenvalue may no longer satisfy ${\mathbf{selctg}}\left({\mathbf{a}}\left(j\right),{\mathbf{b}}\left(j\right)\right)=\mathrm{.TRUE.}$ 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 f08xnf (zgges) is called. Arguments denoted as Input must not be changed by this procedure.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (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 $S$.
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08xnf (zgges) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (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 $T$.
9:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08xnf (zgges) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   $\mathbf{sdim}$ – 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..
11:   $\mathbf{alpha}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: see the description of beta.
12:   $\mathbf{beta}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{alpha}}\left(\mathit{j}\right)/{\mathbf{beta}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues. ${\mathbf{alpha}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$ and ${\mathbf{beta}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, are the diagonals of the complex Schur form $\left(A,B\right)$ output by f08xnf (zgges). The ${\mathbf{beta}}\left(j\right)$ will be non-negative real.
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 $\alpha /\beta$. However, alpha will always be less than and usually comparable with ${‖A‖}_{2}$ in magnitude, and beta will always be less than and usually comparable with ${‖B‖}_{2}$.
13:   $\mathbf{vsl}\left({\mathbf{ldvsl}},*\right)$ – Complex (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 ${\mathbf{jobvsl}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvsl}}=\text{'V'}$, vsl will contain the left Schur vectors, $Q$.
If ${\mathbf{jobvsl}}=\text{'N'}$, vsl is not referenced.
14:   $\mathbf{ldvsl}$ – IntegerInput
On entry: the first dimension of the array vsl as declared in the (sub)program from which f08xnf (zgges) is called.
Constraints:
• if ${\mathbf{jobvsl}}=\text{'V'}$, ${\mathbf{ldvsl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvsl}}\ge 1$.
15:   $\mathbf{vsr}\left({\mathbf{ldvsr}},*\right)$ – Complex (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 ${\mathbf{jobvsr}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvsr}}=\text{'V'}$, vsr will contain the right Schur vectors, $Z$.
If ${\mathbf{jobvsr}}=\text{'N'}$, vsr is not referenced.
16:   $\mathbf{ldvsr}$ – IntegerInput
On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xnf (zgges) is called.
Constraints:
• if ${\mathbf{jobvsr}}=\text{'V'}$, ${\mathbf{ldvsr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvsr}}\ge 1$.
17:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
18:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08xnf (zgges) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork must generally be larger than the minimum, say $2×{\mathbf{n}}+\mathit{nb}×{\mathbf{n}}$, where $\mathit{nb}$ is the optimal block size for f08nsf (zgehrd).
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$.
19:   $\mathbf{rwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,8×{\mathbf{n}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
20:   $\mathbf{bwork}\left(*\right)$ – 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.
21:   $\mathbf{info}$ – 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 did not converge and the matrix pair $\left(A,B\right)$ is not in the generalized Schur form. The computed ${\alpha }_{i}$ and ${\beta }_{i}$ should be correct for $i=〈\mathit{\text{value}}〉,\dots ,〈\mathit{\text{value}}〉$.
${\mathbf{info}}={\mathbf{n}}+1$
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
 $A+E = QS ZH , B+F = QT ZH ,$
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

f08xnf (zgges) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xnf (zgges) 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 real analogue of this routine is f08xaf (dgges).

## 10Example

This example finds the generalized Schur factorization 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
 $B = 1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i 0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i 1.00+0.00i 2.40+1.80i -4.00-5.00i 0.00-3.00i 0.00+1.00i -1.80+2.40i 0.00-4.00i 4.00-5.00i .$
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 (f08xnfe.f90)

### 10.2Program Data

Program Data (f08xnfe.d)

### 10.3Program Results

Program Results (f08xnfe.r)