 CHGEQZ(3) computes the eigenvalues of a complex matrix pair (H,T),

## SYNOPSIS

SUBROUTINE CHGEQZ(
JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )

CHARACTER COMPQ, COMPZ, JOB

INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N

REAL RWORK( * )

COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ), Q( LDQ, * ), T( LDT, * ), WORK( * ), Z( LDZ, * )

## PURPOSE

CHGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method.
Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B):

A = Q1*H*Z1**H,  B = Q1*T*Z1**H,

as computed by CGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,

H = Q*S*Z**H,  T = Q*P*Z**H,

where Q and Z are unitary matrices and S and P are upper triangular.
Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.

To avoid overflow, eigenvalues of the matrix pair (H,T)
(equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP

mu*A*y = B*y.
The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

## ARGUMENTS

JOB (input) CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Computer eigenvalues and the Schur form.
COMPQ (input) CHARACTER*1

= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (H,T) is returned; = 'V': Q must contain a unitary matrix Q1 on entry and the product Q1*Q is returned.
COMPZ (input) CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Z of right Schur vectors of (H,T) is returned; = 'V': Z must contain a unitary matrix Z1 on entry and the product Z1*Z is returned.
N (input) INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER ILO and IHI mark the rows and columns of H which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H (input/output) COMPLEX array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H. On exit, if JOB = 'S', H contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal of H matches that of S, but the rest of H is unspecified.
LDH (input) INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T (input/output) COMPLEX array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T. On exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization. If JOB = 'E', the diagonal of T matches that of P, but the rest of T is unspecified.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHA (output) COMPLEX array, dimension (N)
The complex scalars alpha that define the eigenvalues of GNEP. ALPHA(i) = S(i,i) in the generalized Schur factorization.
BETA (output) COMPLEX array, dimension (N)
The real non-negative scalars beta that define the eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized Schur factorization. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.
Q (input/output) COMPLEX array, dimension (LDQ, N)
On entry, if COMPZ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of left Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of left Schur vectors of (A,B). Not referenced if COMPZ = 'N'.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N.
Z (input/output) COMPLEX array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of right Schur vectors of (A,B). Not referenced if COMPZ = 'N'.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.
WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N). If LWORK = -1, then 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 by XERBLA.
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct. = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N should be correct.

## FURTHER DETAILS

We assume that complex ABS works as long as its value is less than overflow.