.TH "SRC/zhgeqz.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/zhgeqz.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.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 "Function/Subroutine Documentation"
.PP 
.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\&.