.TH "hgeqz" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hgeqz \- hgeqz: generalized Hessenberg eig
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchgeqz\fP (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)"
.br
.RI "\fBCHGEQZ\fP "
.ti -1c
.RI "subroutine \fBdhgeqz\fP (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)"
.br
.RI "\fBDHGEQZ\fP "
.ti -1c
.RI "subroutine \fBshgeqz\fP (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)"
.br
.RI "\fBSHGEQZ\fP "
.ti -1c
.RI "subroutine \fBzhgeqz\fP (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)"
.br
.RI "\fBZHGEQZ\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine chgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)"

.PP
\fBCHGEQZ\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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, , SIAM J\&. Numer\&. Anal\&., 10(1973),
!>      pp\&. 241--256\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          JOB is CHARACTER*1
!>          = 'E': Compute eigenvalues only;
!>          = 'S': Computer eigenvalues and the Schur form\&.
!> 
.fi
.PP
.br
\fICOMPQ\fP 
.PP
.nf
!>          COMPQ is 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\&.
!> 
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
!>          COMPZ is 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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrices H, T, Q, and Z\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          IHI is 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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          H is 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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>          The leading dimension of the array H\&.  LDH >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is 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\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&.  LDT >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
!>          ALPHA is COMPLEX array, dimension (N)
!>          The complex scalars alpha that define the eigenvalues of
!>          GNEP\&.  ALPHA(i) = S(i,i) in the generalized Schur
!>          factorization\&.
!> 
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
!>          BETA is 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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is COMPLEX array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
!>          reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPQ = 'I', the unitary matrix of left Schur
!>          vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
!>          left Schur vectors of (A,B)\&.
!>          Not referenced if COMPQ = 'N'\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.  LDQ >= 1\&.
!>          If COMPQ='V' or 'I', then LDQ >= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is 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'\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1\&.
!>          If COMPZ='V' or 'I', then LDZ >= N\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is 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\&.
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is REAL array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  We assume that complex ABS works as long as its value is less than
!>  overflow\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB281\fP of file \fBchgeqz\&.f\fP\&.
.SS "subroutine dhgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDHGEQZ\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
!> where H is an upper Hessenberg matrix and T is upper triangular,
!> using the double-shift QZ method\&.
!> Matrix pairs of this type are produced by the reduction to
!> generalized upper Hessenberg form of a real matrix pair (A,B):
!>
!>    A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
!>
!> as computed by DGGHRD\&.
!>
!> If JOB='S', then the Hessenberg-triangular pair (H,T) is
!> also reduced to generalized Schur form,
!>
!>    H = Q*S*Z**T,  T = Q*P*Z**T,
!>
!> where Q and Z are orthogonal matrices, P is an upper triangular
!> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
!> diagonal blocks\&.
!>
!> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
!> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
!> eigenvalues\&.
!>
!> Additionally, the 2-by-2 upper triangular diagonal blocks of P
!> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
!> form, i\&.e\&., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
!> P(j,j) > 0, and P(j+1,j+1) > 0\&.
!>
!> Optionally, the orthogonal matrix Q from the generalized Schur
!> factorization may be postmultiplied into an input matrix Q1, and the
!> orthogonal matrix Z may be postmultiplied into an input matrix Z1\&.
!> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
!> the matrix pair (A,B) to generalized upper Hessenberg form, then the
!> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
!> generalized Schur factorization of (A,B):
!>
!>    A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T\&.
!>
!> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
!> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
!> complex and beta real\&.
!> 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\&.
!> Real eigenvalues can be read directly from the generalized Schur
!> form:
!>   alpha = S(i,i), beta = P(i,i)\&.
!>
!> Ref: C\&.B\&. Moler & G\&.W\&. Stewart, , SIAM J\&. Numer\&. Anal\&., 10(1973),
!>      pp\&. 241--256\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          JOB is CHARACTER*1
!>          = 'E': Compute eigenvalues only;
!>          = 'S': Compute eigenvalues and the Schur form\&.
!> 
.fi
.PP
.br
\fICOMPQ\fP 
.PP
.nf
!>          COMPQ is 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 an orthogonal matrix Q1 on entry and
!>                 the product Q1*Q is returned\&.
!> 
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
!>          COMPZ is CHARACTER*1
!>          = 'N': Right Schur vectors (Z) are not computed;
!>          = 'I': Z is initialized to the unit matrix and the matrix Z
!>                 of right Schur vectors of (H,T) is returned;
!>          = 'V': Z must contain an orthogonal matrix Z1 on entry and
!>                 the product Z1*Z is returned\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrices H, T, Q, and Z\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          IHI is 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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          H is DOUBLE PRECISION array, dimension (LDH, N)
!>          On entry, the N-by-N upper Hessenberg matrix H\&.
!>          On exit, if JOB = 'S', H contains the upper quasi-triangular
!>          matrix S from the generalized Schur factorization\&.
!>          If JOB = 'E', the diagonal blocks of H match those of S, but
!>          the rest of H is unspecified\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>          The leading dimension of the array H\&.  LDH >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is DOUBLE PRECISION 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;
!>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
!>          are reduced to positive diagonal form, i\&.e\&., if H(j+1,j) is
!>          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
!>          T(j+1,j+1) > 0\&.
!>          If JOB = 'E', the diagonal blocks of T match those of P, but
!>          the rest of T is unspecified\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&.  LDT >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!>          The real parts of each scalar alpha defining an eigenvalue
!>          of GNEP\&.
!> 
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!>          The imaginary parts of each scalar alpha defining an
!>          eigenvalue of GNEP\&.
!>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
!>          positive, then the j-th and (j+1)-st eigenvalues are a
!>          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j)\&.
!> 
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
!>          BETA is DOUBLE PRECISION array, dimension (N)
!>          The scalars beta that define the eigenvalues of GNEP\&.
!>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
!>          the reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
!>          vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
!>          of left Schur vectors of (A,B)\&.
!>          Not referenced if COMPQ = 'N'\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.  LDQ >= 1\&.
!>          If COMPQ='V' or 'I', then LDQ >= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
!>          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
!>          the reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPZ = 'I', the orthogonal matrix of
!>          right Schur vectors of (H,T), and if COMPZ = 'V', the
!>          orthogonal matrix of right Schur vectors of (A,B)\&.
!>          Not referenced if COMPZ = 'N'\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1\&.
!>          If COMPZ='V' or 'I', then LDZ >= N\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is 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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
!>                     BETA(i), i=INFO-N+1,\&.\&.\&.,N should be correct\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  Iteration counters:
!>
!>  JITER  -- counts iterations\&.
!>  IITER  -- counts iterations run since ILAST was last
!>            changed\&.  This is therefore reset only when a 1-by-1 or
!>            2-by-2 block deflates off the bottom\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB301\fP of file \fBdhgeqz\&.f\fP\&.
.SS "subroutine shgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSHGEQZ\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
!> where H is an upper Hessenberg matrix and T is upper triangular,
!> using the double-shift QZ method\&.
!> Matrix pairs of this type are produced by the reduction to
!> generalized upper Hessenberg form of a real matrix pair (A,B):
!>
!>    A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
!>
!> as computed by SGGHRD\&.
!>
!> If JOB='S', then the Hessenberg-triangular pair (H,T) is
!> also reduced to generalized Schur form,
!>
!>    H = Q*S*Z**T,  T = Q*P*Z**T,
!>
!> where Q and Z are orthogonal matrices, P is an upper triangular
!> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
!> diagonal blocks\&.
!>
!> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
!> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
!> eigenvalues\&.
!>
!> Additionally, the 2-by-2 upper triangular diagonal blocks of P
!> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
!> form, i\&.e\&., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
!> P(j,j) > 0, and P(j+1,j+1) > 0\&.
!>
!> Optionally, the orthogonal matrix Q from the generalized Schur
!> factorization may be postmultiplied into an input matrix Q1, and the
!> orthogonal matrix Z may be postmultiplied into an input matrix Z1\&.
!> If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
!> the matrix pair (A,B) to generalized upper Hessenberg form, then the
!> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
!> generalized Schur factorization of (A,B):
!>
!>    A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T\&.
!>
!> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
!> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
!> complex and beta real\&.
!> 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\&.
!> Real eigenvalues can be read directly from the generalized Schur
!> form:
!>   alpha = S(i,i), beta = P(i,i)\&.
!>
!> Ref: C\&.B\&. Moler & G\&.W\&. Stewart, , SIAM J\&. Numer\&. Anal\&., 10(1973),
!>      pp\&. 241--256\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          JOB is CHARACTER*1
!>          = 'E': Compute eigenvalues only;
!>          = 'S': Compute eigenvalues and the Schur form\&.
!> 
.fi
.PP
.br
\fICOMPQ\fP 
.PP
.nf
!>          COMPQ is 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 an orthogonal matrix Q1 on entry and
!>                 the product Q1*Q is returned\&.
!> 
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
!>          COMPZ is CHARACTER*1
!>          = 'N': Right Schur vectors (Z) are not computed;
!>          = 'I': Z is initialized to the unit matrix and the matrix Z
!>                 of right Schur vectors of (H,T) is returned;
!>          = 'V': Z must contain an orthogonal matrix Z1 on entry and
!>                 the product Z1*Z is returned\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrices H, T, Q, and Z\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          IHI is 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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          H is REAL array, dimension (LDH, N)
!>          On entry, the N-by-N upper Hessenberg matrix H\&.
!>          On exit, if JOB = 'S', H contains the upper quasi-triangular
!>          matrix S from the generalized Schur factorization\&.
!>          If JOB = 'E', the diagonal blocks of H match those of S, but
!>          the rest of H is unspecified\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>          The leading dimension of the array H\&.  LDH >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is REAL 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;
!>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
!>          are reduced to positive diagonal form, i\&.e\&., if H(j+1,j) is
!>          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
!>          T(j+1,j+1) > 0\&.
!>          If JOB = 'E', the diagonal blocks of T match those of P, but
!>          the rest of T is unspecified\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&.  LDT >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
!>          ALPHAR is REAL array, dimension (N)
!>          The real parts of each scalar alpha defining an eigenvalue
!>          of GNEP\&.
!> 
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
!>          ALPHAI is REAL array, dimension (N)
!>          The imaginary parts of each scalar alpha defining an
!>          eigenvalue of GNEP\&.
!>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
!>          positive, then the j-th and (j+1)-st eigenvalues are a
!>          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j)\&.
!> 
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
!>          BETA is REAL array, dimension (N)
!>          The scalars beta that define the eigenvalues of GNEP\&.
!>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is REAL array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
!>          the reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
!>          vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
!>          of left Schur vectors of (A,B)\&.
!>          Not referenced if COMPQ = 'N'\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.  LDQ >= 1\&.
!>          If COMPQ='V' or 'I', then LDQ >= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is REAL array, dimension (LDZ, N)
!>          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
!>          the reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPZ = 'I', the orthogonal matrix of
!>          right Schur vectors of (H,T), and if COMPZ = 'V', the
!>          orthogonal matrix of right Schur vectors of (A,B)\&.
!>          Not referenced if COMPZ = 'N'\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1\&.
!>          If COMPZ='V' or 'I', then LDZ >= N\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is 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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
!>                     BETA(i), i=INFO-N+1,\&.\&.\&.,N should be correct\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  Iteration counters:
!>
!>  JITER  -- counts iterations\&.
!>  IITER  -- counts iterations run since ILAST was last
!>            changed\&.  This is therefore reset only when a 1-by-1 or
!>            2-by-2 block deflates off the bottom\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB301\fP of file \fBshgeqz\&.f\fP\&.
.SS "subroutine zhgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)"

.PP
\fBZHGEQZ\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZHGEQZ 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 ZGGHRD\&.
!>
!> 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 ZGGHRD 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, , SIAM J\&. Numer\&. Anal\&., 10(1973),
!>      pp\&. 241--256\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          JOB is CHARACTER*1
!>          = 'E': Compute eigenvalues only;
!>          = 'S': Computer eigenvalues and the Schur form\&.
!> 
.fi
.PP
.br
\fICOMPQ\fP 
.PP
.nf
!>          COMPQ is 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\&.
!> 
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
!>          COMPZ is 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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrices H, T, Q, and Z\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          IHI is 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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          H is COMPLEX*16 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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>          The leading dimension of the array H\&.  LDH >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is COMPLEX*16 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\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&.  LDT >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
!>          ALPHA is COMPLEX*16 array, dimension (N)
!>          The complex scalars alpha that define the eigenvalues of
!>          GNEP\&.  ALPHA(i) = S(i,i) in the generalized Schur
!>          factorization\&.
!> 
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
!>          BETA is COMPLEX*16 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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is COMPLEX*16 array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
!>          reduction of (A,B) to generalized Hessenberg form\&.
!>          On exit, if COMPQ = 'I', the unitary matrix of left Schur
!>          vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
!>          left Schur vectors of (A,B)\&.
!>          Not referenced if COMPQ = 'N'\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.  LDQ >= 1\&.
!>          If COMPQ='V' or 'I', then LDQ >= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is COMPLEX*16 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'\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1\&.
!>          If COMPZ='V' or 'I', then LDZ >= N\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is 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\&.
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is DOUBLE PRECISION array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  We assume that complex ABS works as long as its value is less than
!>  overflow\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB281\fP of file \fBzhgeqz\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.