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version 1.8, 2010/12/21 13:53:34 version 1.9, 2011/11/21 20:42:59
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   *> \brief \b DLATDF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLATDF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
   *                          JPIV )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            IJOB, LDZ, N
   *       DOUBLE PRECISION   RDSCAL, RDSUM
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * ), JPIV( * )
   *       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLATDF uses the LU factorization of the n-by-n matrix Z computed by
   *> DGETC2 and computes a contribution to the reciprocal Dif-estimate
   *> by solving Z * x = b for x, and choosing the r.h.s. b such that
   *> the norm of x is as large as possible. On entry RHS = b holds the
   *> contribution from earlier solved sub-systems, and on return RHS = x.
   *>
   *> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
   *> where P and Q are permutation matrices. L is lower triangular with
   *> unit diagonal elements and U is upper triangular.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] IJOB
   *> \verbatim
   *>          IJOB is INTEGER
   *>          IJOB = 2: First compute an approximative null-vector e
   *>              of Z using DGECON, e is normalized and solve for
   *>              Zx = +-e - f with the sign giving the greater value
   *>              of 2-norm(x). About 5 times as expensive as Default.
   *>          IJOB .ne. 2: Local look ahead strategy where all entries of
   *>              the r.h.s. b is choosen as either +1 or -1 (Default).
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the matrix Z.
   *> \endverbatim
   *>
   *> \param[in] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
   *>          On entry, the LU part of the factorization of the n-by-n
   *>          matrix Z computed by DGETC2:  Z = P * L * U * Q
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>          The leading dimension of the array Z.  LDA >= max(1, N).
   *> \endverbatim
   *>
   *> \param[in,out] RHS
   *> \verbatim
   *>          RHS is DOUBLE PRECISION array, dimension (N)
   *>          On entry, RHS contains contributions from other subsystems.
   *>          On exit, RHS contains the solution of the subsystem with
   *>          entries acoording to the value of IJOB (see above).
   *> \endverbatim
   *>
   *> \param[in,out] RDSUM
   *> \verbatim
   *>          RDSUM is DOUBLE PRECISION
   *>          On entry, the sum of squares of computed contributions to
   *>          the Dif-estimate under computation by DTGSYL, where the
   *>          scaling factor RDSCAL (see below) has been factored out.
   *>          On exit, the corresponding sum of squares updated with the
   *>          contributions from the current sub-system.
   *>          If TRANS = 'T' RDSUM is not touched.
   *>          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
   *> \endverbatim
   *>
   *> \param[in,out] RDSCAL
   *> \verbatim
   *>          RDSCAL is DOUBLE PRECISION
   *>          On entry, scaling factor used to prevent overflow in RDSUM.
   *>          On exit, RDSCAL is updated w.r.t. the current contributions
   *>          in RDSUM.
   *>          If TRANS = 'T', RDSCAL is not touched.
   *>          NOTE: RDSCAL only makes sense when DTGSY2 is called by
   *>                DTGSYL.
   *> \endverbatim
   *>
   *> \param[in] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N).
   *>          The pivot indices; for 1 <= i <= N, row i of the
   *>          matrix has been interchanged with row IPIV(i).
   *> \endverbatim
   *>
   *> \param[in] JPIV
   *> \verbatim
   *>          JPIV is INTEGER array, dimension (N).
   *>          The pivot indices; for 1 <= j <= N, column j of the
   *>          matrix has been interchanged with column JPIV(j).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERauxiliary
   *
   *> \par Further Details:
   *  =====================
   *>
   *>  This routine is a further developed implementation of algorithm
   *>  BSOLVE in [1] using complete pivoting in the LU factorization.
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   *>     Umea University, S-901 87 Umea, Sweden.
   *
   *> \par References:
   *  ================
   *>
   *> \verbatim
   *>
   *>
   *>  [1] Bo Kagstrom and Lars Westin,
   *>      Generalized Schur Methods with Condition Estimators for
   *>      Solving the Generalized Sylvester Equation, IEEE Transactions
   *>      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
   *>
   *>  [2] Peter Poromaa,
   *>      On Efficient and Robust Estimators for the Separation
   *>      between two Regular Matrix Pairs with Applications in
   *>      Condition Estimation. Report IMINF-95.05, Departement of
   *>      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,        SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
      $                   JPIV )       $                   JPIV )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2.2) --  *  -- LAPACK auxiliary 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..--
 *     June 2010  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IJOB, LDZ, N        INTEGER            IJOB, LDZ, N
Line 15 Line 185
       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )        DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLATDF uses the LU factorization of the n-by-n matrix Z computed by  
 *  DGETC2 and computes a contribution to the reciprocal Dif-estimate  
 *  by solving Z * x = b for x, and choosing the r.h.s. b such that  
 *  the norm of x is as large as possible. On entry RHS = b holds the  
 *  contribution from earlier solved sub-systems, and on return RHS = x.  
 *  
 *  The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,  
 *  where P and Q are permutation matrices. L is lower triangular with  
 *  unit diagonal elements and U is upper triangular.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  IJOB    (input) INTEGER  
 *          IJOB = 2: First compute an approximative null-vector e  
 *              of Z using DGECON, e is normalized and solve for  
 *              Zx = +-e - f with the sign giving the greater value  
 *              of 2-norm(x). About 5 times as expensive as Default.  
 *          IJOB .ne. 2: Local look ahead strategy where all entries of  
 *              the r.h.s. b is choosen as either +1 or -1 (Default).  
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the matrix Z.  
 *  
 *  Z       (input) DOUBLE PRECISION array, dimension (LDZ, N)  
 *          On entry, the LU part of the factorization of the n-by-n  
 *          matrix Z computed by DGETC2:  Z = P * L * U * Q  
 *  
 *  LDZ     (input) INTEGER  
 *          The leading dimension of the array Z.  LDA >= max(1, N).  
 *  
 *  RHS     (input/output) DOUBLE PRECISION array, dimension (N)  
 *          On entry, RHS contains contributions from other subsystems.  
 *          On exit, RHS contains the solution of the subsystem with  
 *          entries acoording to the value of IJOB (see above).  
 *  
 *  RDSUM   (input/output) DOUBLE PRECISION  
 *          On entry, the sum of squares of computed contributions to  
 *          the Dif-estimate under computation by DTGSYL, where the  
 *          scaling factor RDSCAL (see below) has been factored out.  
 *          On exit, the corresponding sum of squares updated with the  
 *          contributions from the current sub-system.  
 *          If TRANS = 'T' RDSUM is not touched.  
 *          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.  
 *  
 *  RDSCAL  (input/output) DOUBLE PRECISION  
 *          On entry, scaling factor used to prevent overflow in RDSUM.  
 *          On exit, RDSCAL is updated w.r.t. the current contributions  
 *          in RDSUM.  
 *          If TRANS = 'T', RDSCAL is not touched.  
 *          NOTE: RDSCAL only makes sense when DTGSY2 is called by  
 *                DTGSYL.  
 *  
 *  IPIV    (input) INTEGER array, dimension (N).  
 *          The pivot indices; for 1 <= i <= N, row i of the  
 *          matrix has been interchanged with row IPIV(i).  
 *  
 *  JPIV    (input) INTEGER array, dimension (N).  
 *          The pivot indices; for 1 <= j <= N, column j of the  
 *          matrix has been interchanged with column JPIV(j).  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,  
 *     Umea University, S-901 87 Umea, Sweden.  
 *  
 *  This routine is a further developed implementation of algorithm  
 *  BSOLVE in [1] using complete pivoting in the LU factorization.  
 *  
 *  [1] Bo Kagstrom and Lars Westin,  
 *      Generalized Schur Methods with Condition Estimators for  
 *      Solving the Generalized Sylvester Equation, IEEE Transactions  
 *      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.  
 *  
 *  [2] Peter Poromaa,  
 *      On Efficient and Robust Estimators for the Separation  
 *      between two Regular Matrix Pairs with Applications in  
 *      Condition Estimation. Report IMINF-95.05, Departement of  
 *      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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