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version 1.8, 2010/12/21 13:53:26 version 1.9, 2011/11/21 20:42:52
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   *> \brief \b DGGBAL
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGGBAL + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggbal.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggbal.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggbal.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
   *                          RSCALE, WORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOB
   *       INTEGER            IHI, ILO, INFO, LDA, LDB, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), LSCALE( * ),
   *      $                   RSCALE( * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGGBAL balances a pair of general real matrices (A,B).  This
   *> involves, first, permuting A and B by similarity transformations to
   *> isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
   *> elements on the diagonal; and second, applying a diagonal similarity
   *> transformation to rows and columns ILO to IHI to make the rows
   *> and columns as close in norm as possible. Both steps are optional.
   *>
   *> Balancing may reduce the 1-norm of the matrices, and improve the
   *> accuracy of the computed eigenvalues and/or eigenvectors in the
   *> generalized eigenvalue problem A*x = lambda*B*x.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOB
   *> \verbatim
   *>          JOB is CHARACTER*1
   *>          Specifies the operations to be performed on A and B:
   *>          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
   *>                  and RSCALE(I) = 1.0 for i = 1,...,N.
   *>          = 'P':  permute only;
   *>          = 'S':  scale only;
   *>          = 'B':  both permute and scale.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the input matrix A.
   *>          On exit,  A is overwritten by the balanced matrix.
   *>          If JOB = 'N', A is not referenced.
   *> \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 input matrix B.
   *>          On exit,  B is overwritten by the balanced matrix.
   *>          If JOB = 'N', B is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] ILO
   *> \verbatim
   *>          ILO is INTEGER
   *> \endverbatim
   *>
   *> \param[out] IHI
   *> \verbatim
   *>          IHI is INTEGER
   *>          ILO and IHI are set to integers such that on exit
   *>          A(i,j) = 0 and B(i,j) = 0 if i > j and
   *>          j = 1,...,ILO-1 or i = IHI+1,...,N.
   *>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
   *> \endverbatim
   *>
   *> \param[out] LSCALE
   *> \verbatim
   *>          LSCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied
   *>          to the left side of A and B.  If P(j) is the index of the
   *>          row interchanged with row j, and D(j)
   *>          is the scaling factor applied to row j, then
   *>            LSCALE(j) = P(j)    for J = 1,...,ILO-1
   *>                      = D(j)    for J = ILO,...,IHI
   *>                      = P(j)    for J = IHI+1,...,N.
   *>          The order in which the interchanges are made is N to IHI+1,
   *>          then 1 to ILO-1.
   *> \endverbatim
   *>
   *> \param[out] RSCALE
   *> \verbatim
   *>          RSCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied
   *>          to the right side of A and B.  If P(j) is the index of the
   *>          column interchanged with column j, and D(j)
   *>          is the scaling factor applied to column j, then
   *>            LSCALE(j) = P(j)    for J = 1,...,ILO-1
   *>                      = D(j)    for J = ILO,...,IHI
   *>                      = P(j)    for J = IHI+1,...,N.
   *>          The order in which the interchanges are made is N to IHI+1,
   *>          then 1 to ILO-1.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (lwork)
   *>          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
   *>          at least 1 when JOB = 'N' or 'P'.
   *> \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 doubleGBcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  See R.C. WARD, Balancing the generalized eigenvalue problem,
   *>                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,        SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
      $                   RSCALE, WORK, INFO )       $                   RSCALE, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2.2) --  *  -- 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..--
 *     June 2010  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOB        CHARACTER          JOB
Line 15 Line 191
      $                   RSCALE( * ), WORK( * )       $                   RSCALE( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGGBAL balances a pair of general real matrices (A,B).  This  
 *  involves, first, permuting A and B by similarity transformations to  
 *  isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N  
 *  elements on the diagonal; and second, applying a diagonal similarity  
 *  transformation to rows and columns ILO to IHI to make the rows  
 *  and columns as close in norm as possible. Both steps are optional.  
 *  
 *  Balancing may reduce the 1-norm of the matrices, and improve the  
 *  accuracy of the computed eigenvalues and/or eigenvectors in the  
 *  generalized eigenvalue problem A*x = lambda*B*x.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOB     (input) CHARACTER*1  
 *          Specifies the operations to be performed on A and B:  
 *          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0  
 *                  and RSCALE(I) = 1.0 for i = 1,...,N.  
 *          = 'P':  permute only;  
 *          = 'S':  scale only;  
 *          = 'B':  both permute and scale.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrices A and B.  N >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the input matrix A.  
 *          On exit,  A is overwritten by the balanced matrix.  
 *          If JOB = 'N', A is not referenced.  
 *  
 *  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 input matrix B.  
 *          On exit,  B is overwritten by the balanced matrix.  
 *          If JOB = 'N', B is not referenced.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,N).  
 *  
 *  ILO     (output) INTEGER  
 *  IHI     (output) INTEGER  
 *          ILO and IHI are set to integers such that on exit  
 *          A(i,j) = 0 and B(i,j) = 0 if i > j and  
 *          j = 1,...,ILO-1 or i = IHI+1,...,N.  
 *          If JOB = 'N' or 'S', ILO = 1 and IHI = N.  
 *  
 *  LSCALE  (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied  
 *          to the left side of A and B.  If P(j) is the index of the  
 *          row interchanged with row j, and D(j)  
 *          is the scaling factor applied to row j, then  
 *            LSCALE(j) = P(j)    for J = 1,...,ILO-1  
 *                      = D(j)    for J = ILO,...,IHI  
 *                      = P(j)    for J = IHI+1,...,N.  
 *          The order in which the interchanges are made is N to IHI+1,  
 *          then 1 to ILO-1.  
 *  
 *  RSCALE  (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied  
 *          to the right side of A and B.  If P(j) is the index of the  
 *          column interchanged with column j, and D(j)  
 *          is the scaling factor applied to column j, then  
 *            LSCALE(j) = P(j)    for J = 1,...,ILO-1  
 *                      = D(j)    for J = ILO,...,IHI  
 *                      = P(j)    for J = IHI+1,...,N.  
 *          The order in which the interchanges are made is N to IHI+1,  
 *          then 1 to ILO-1.  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (lwork)  
 *          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and  
 *          at least 1 when JOB = 'N' or 'P'.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  See R.C. WARD, Balancing the generalized eigenvalue problem,  
 *                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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