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

.in +1c
.ti -1c
.RI "subroutine \fBcgghrd\fP (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)"
.br
.RI "\fBCGGHRD\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine cgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer info)"

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

.PP
.nf
 CGGHRD reduces a pair of complex matrices (A,B) to generalized upper
 Hessenberg form using unitary transformations, where A is a
 general matrix and B is upper triangular\&.  The form of the generalized
 eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the unitary matrix Q to the left side
 of the equation\&.

 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**H*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**H*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**H*x\&.

 The unitary matrices Q and Z are determined as products of Givens
 rotations\&.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that
      Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
      Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
 If Q1 is the unitary matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then CGGHRD reduces the original
 problem to generalized Hessenberg form\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fICOMPQ\fP 
.PP
.nf
          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 unitary matrix Q 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': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 unitary matrix Z 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 A and B\&.  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 A which are to be
          reduced\&.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N\&.  ILO and IHI are
          normally set by a previous call to CGGBAL; otherwise they
          should be set to 1 and N respectively\&.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced\&.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B\&.
          On exit, the upper triangular matrix T = Q**H B Z\&.  The
          elements below the diagonal are set to zero\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the unitary matrix Q1, typically
          from the QR factorization of B\&.
          On exit, if COMPQ='I', the unitary matrix Q, and if
          COMPQ = 'V', the product Q1*Q\&.
          Not referenced if COMPQ='N'\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the unitary matrix Z1\&.
          On exit, if COMPZ='I', the unitary matrix Z, and if
          COMPZ = 'V', the product Z1*Z\&.
          Not referenced if COMPZ='N'\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise\&.
.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\&.
.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
  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and van Loan (Johns Hopkins Press)\&.
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB202\fP of file \fBcgghrd\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.