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dgghrd.f(3)				      LAPACK				      dgghrd.f(3)

NAME
       dgghrd.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
	   DGGHRD

Function/Subroutine Documentation
   subroutine dgghrd (characterCOMPQ, characterCOMPZ, integerN, integerILO, integerIHI, double
       precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B,
       integerLDB, double precision, dimension( ldq, * )Q, integerLDQ, double precision,
       dimension( ldz, * )Z, integerLDZ, integerINFO)
       DGGHRD

       Purpose:

	    DGGHRD reduces a pair of real matrices (A,B) to generalized upper
	    Hessenberg form using orthogonal 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 orthogonal matrix Q to the left side
	    of the equation.

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

	    The orthogonal 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**T = (Q1*Q) * H * (Z1*Z)**T

		 Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

	    If Q1 is the orthogonal matrix from the QR factorization of B in the
	    original equation A*x = lambda*B*x, then DGGHRD 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
			    orthogonal matrix Q is returned;
		     = 'V': Q must contain an orthogonal 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
			    orthogonal matrix Z is returned;
		     = 'V': Z must contain an orthogonal 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 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

		     A is 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 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 DOUBLE PRECISION array, dimension (LDB, N)
		     On entry, the N-by-N upper triangular matrix B.
		     On exit, the upper triangular matrix T = Q**T 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 DOUBLE PRECISION array, dimension (LDQ, N)
		     On entry, if COMPQ = 'V', the orthogonal matrix Q1,
		     typically from the QR factorization of B.
		     On exit, if COMPQ='I', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N)
		     On entry, if COMPZ = 'V', the orthogonal matrix Z1.
		     On exit, if COMPZ='I', the orthogonal 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.

       Date:
	   November 2011

       Further Details:

	     This routine reduces A to Hessenberg and B to triangular form by
	     an unblocked reduction, as described in <em>Matrix_Computations</em>,
	     by Golub and Van Loan (Johns Hopkins Press.)

       Definition at line 207 of file dgghrd.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dgghrd.f(3)
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