SRC/zgghrd.f(3)            Library Functions Manual            SRC/zgghrd.f(3)

NAME
       SRC/zgghrd.f

SYNOPSIS
   Functions/Subroutines
       subroutine zgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq,
           z, ldz, info)
           ZGGHRD

Function/Subroutine Documentation
   subroutine zgghrd (character compq, character compz, integer n, integer
       ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda,
       complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension(
       ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer
       ldz, integer info)
       ZGGHRD

       Purpose:


           !>
           !> ZGGHRD 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 ZGGHRD reduces the original
           !> problem to generalized Hessenberg form.
           !>

       Parameters
           COMPQ

           !>          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.
           !>

           COMPZ

           !>          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.
           !>

           N

           !>          N is INTEGER
           !>          The order of the matrices A and B.  N >= 0.
           !>

           ILO

           !>          ILO is INTEGER
           !>

           IHI

           !>          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 ZGGBAL; 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.
           !>

           A

           !>          A is COMPLEX*16 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.
           !>

           LDA

           !>          LDA is INTEGER
           !>          The leading dimension of the array A.  LDA >= max(1,N).
           !>

           B

           !>          B is COMPLEX*16 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.
           !>

           LDB

           !>          LDB is INTEGER
           !>          The leading dimension of the array B.  LDB >= max(1,N).
           !>

           Q

           !>          Q is COMPLEX*16 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'.
           !>

           LDQ

           !>          LDQ is INTEGER
           !>          The leading dimension of the array Q.
           !>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
           !>

           Z

           !>          Z is COMPLEX*16 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'.
           !>

           LDZ

           !>          LDZ is INTEGER
           !>          The leading dimension of the array Z.
           !>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
           !>

           INFO

           !>          INFO is INTEGER
           !>          = 0:  successful exit.
           !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
           !>

       Author
           Univ. of Tennessee


           Univ. of California Berkeley


           Univ. of Colorado Denver


           NAG Ltd.

       Further Details:


           !>
           !>  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).
           !>

       Definition at line 202 of file zgghrd.f.

Author
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LAPACK                          Version 3.12.0                 SRC/zgghrd.f(3)