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version 1.4, 2010/08/06 15:32:39 version 1.18, 2018/05/29 07:18:16
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   *> \brief \b ZGGBAL
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZGGBAL + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggbal.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggbal.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggbal.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGGBAL( 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   LSCALE( * ), RSCALE( * ), WORK( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGGBAL balances a pair of general complex 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 COMPLEX*16 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 COMPLEX*16 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
   *>            RSCALE(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 June 2016
   *
   *> \ingroup complex16GBcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  See R.C. WARD, Balancing the generalized eigenvalue problem,
   *>                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,        SUBROUTINE ZGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
      $                   RSCALE, WORK, INFO )       $                   RSCALE, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.7.0) --
 *  -- 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  *     June 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOB        CHARACTER          JOB
Line 15 Line 191
       COMPLEX*16         A( LDA, * ), B( LDB, * )        COMPLEX*16         A( LDA, * ), B( LDB, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGGBAL balances a pair of general complex 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) COMPLEX*16 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) COMPLEX*16 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  
 *            RSCALE(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) REAL 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 ..
Line 449 Line 538
          IRAB = IZAMAX( N-ILO+1, B( I, ILO ), LDB )           IRAB = IZAMAX( N-ILO+1, B( I, ILO ), LDB )
          RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )           RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )
          LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE )           LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE )
          IR = LSCALE( I ) + SIGN( HALF, LSCALE( I ) )           IR = INT(LSCALE( I ) + SIGN( HALF, LSCALE( I ) ))
          IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )           IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )
          LSCALE( I ) = SCLFAC**IR           LSCALE( I ) = SCLFAC**IR
          ICAB = IZAMAX( IHI, A( 1, I ), 1 )           ICAB = IZAMAX( IHI, A( 1, I ), 1 )
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          ICAB = IZAMAX( IHI, B( 1, I ), 1 )           ICAB = IZAMAX( IHI, B( 1, I ), 1 )
          CAB = MAX( CAB, ABS( B( ICAB, I ) ) )           CAB = MAX( CAB, ABS( B( ICAB, I ) ) )
          LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE )           LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE )
          JC = RSCALE( I ) + SIGN( HALF, RSCALE( I ) )           JC = INT(RSCALE( I ) + SIGN( HALF, RSCALE( I ) ))
          JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )           JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )
          RSCALE( I ) = SCLFAC**JC           RSCALE( I ) = SCLFAC**JC
   360 CONTINUE    360 CONTINUE
Removed from v.1.4  
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  Added in v.1.18

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