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CGGHRD(l)			       )			     CGGHRD(l)

NAME
       CGGHRD  -  reduce 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

SYNOPSIS
       SUBROUTINE CGGHRD( COMPQ,  COMPZ,  N, ILO, IHI, A, LDA, B, LDB, Q, LDQ,
			  Z, LDZ, INFO )

	   CHARACTER	  COMPQ, COMPZ

	   INTEGER	  IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N

	   COMPLEX	  A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )

PURPOSE
       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: Q' * A * Z = H and Q' * B *  Z	 =  T,
       where  H	 is  upper  Hessenberg, T is upper triangular, and Q and Z are
       unitary, and ' means conjugate transpose.

       The unitary matrices Q and Z are determined as products of Givens rota‐
       tions.  They may either be formed explicitly, or they may be postmulti‐
       plied into input matrices Q1 and Z1, so that

	    Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
	    Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'

ARGUMENTS
       COMPQ   (input) 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   (input) 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.

       N       (input) INTEGER
	       The order of the matrices A and B.  N >= 0.

       ILO     (input) INTEGER
	       IHI     (input) INTEGER 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.

       A       (input/output) 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 overwrit‐
	       ten with the upper Hessenberg matrix H, and the rest is set  to
	       zero.

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

       B       (input/output) COMPLEX array, dimension (LDB, N)
	       On  entry,  the N-by-N upper triangular matrix B.  On exit, the
	       upper triangular matrix T = Q' B Z.   The  elements  below  the
	       diagonal are set to zero.

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

       Q       (input/output) COMPLEX array, dimension (LDQ, N)
	       If COMPQ='N':  Q is not referenced.
	       If COMPQ='I':  on entry, Q need not be set, and on exit it con‐
	       tains the unitary matrix Q, where Q'  is	 the  product  of  the
	       Givens  transformations	which  are  applied  to A and B on the
	       left.  If COMPQ='V':  on entry, Q must contain a unitary matrix
	       Q1, and on exit this is overwritten by Q1*Q.

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

       Z       (input/output) COMPLEX array, dimension (LDZ, N)
	       If COMPZ='N':  Z is not referenced.
	       If COMPZ='I':  on entry, Z need not be set, and on exit it con‐
	       tains  the unitary matrix Z, which is the product of the Givens
	       transformations which are applied to A and B on the right.   If
	       COMPZ='V':   on	entry, Z must contain a unitary matrix Z1, and
	       on exit this is overwritten by Z1*Z.

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

       INFO    (output) INTEGER
	       = 0:  successful exit.
	       < 0:  if INFO = -i, the i-th argument had an illegal value.

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).

LAPACK version 3.0		 15 June 2000			     CGGHRD(l)
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