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DGGHRD(l)			       )			     DGGHRD(l)

NAME
       DGGHRD - reduce a pair of real matrices (A,B) to generalized upper Hes‐
       senberg form using orthogonal transformations, where  A	is  a  general
       matrix and B is upper triangular

SYNOPSIS
       SUBROUTINE DGGHRD( 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

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

PURPOSE
       DGGHRD  reduces a pair of real matrices (A,B) to generalized upper Hes‐
       senberg form using orthogonal 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
       orthogonal, and ' means transpose.

       The  orthogonal	matrices  Q and Z are determined as products of Givens
       rotations.  They may either be formed explicitly, or they may be	 post‐
       multiplied 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 orthogonal
	       matrix Q is returned; =	'V':  Q	 must  contain	an  orthogonal
	       matrix Q1 on entry, and the product Q1*Q is returned.

       COMPZ   (input) CHARACTER*1
	       = 'N': do not compute Z;
	       =  'I': Z is initialized to the unit matrix, and the orthogonal
	       matrix Z is returned; =	'V':  Z	 must  contain	an  orthogonal
	       matrix Z1 on entry, and the product Z1*Z 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 DGGBAL; 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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  orthogonal 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 an orthogonal
	       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) DOUBLE PRECISION 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 orthogonal 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	an  orthogonal
	       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			     DGGHRD(l)
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