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version 1.7, 2010/12/21 13:53:27 version 1.8, 2011/11/21 20:42:52
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   *> \brief \b DGGHRD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGGHRD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghrd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgghrd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgghrd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
   *                          LDQ, Z, LDZ, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          COMPQ, COMPZ
   *       INTEGER            IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
   *      $                   Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] COMPQ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] COMPZ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] ILO
   *> \verbatim
   *>          ILO is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IHI
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] Q
   *> \verbatim
   *>          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'.
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>          The leading dimension of the array Q.
   *>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[in,out] Z
   *> \verbatim
   *>          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'.
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>          The leading dimension of the array Z.
   *>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit.
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  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.)
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,        SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
      $                   LDQ, Z, LDZ, INFO )       $                   LDQ, Z, LDZ, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          COMPQ, COMPZ        CHARACTER          COMPQ, COMPZ
Line 15 Line 221
      $                   Z( LDZ, * )       $                   Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  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.  
 *  
 *  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  
 *          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 SGGBAL; 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 overwritten 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**T 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)  
 *          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     (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)  
 *          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     (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.)  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
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  Added in v.1.8

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