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version 1.8, 2010/12/21 13:53:42 version 1.9, 2011/11/21 20:43:08
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   *> \brief \b ZGEBAL
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGEBAL + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgebal.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgebal.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebal.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOB
   *       INTEGER            IHI, ILO, INFO, LDA, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   SCALE( * )
   *       COMPLEX*16         A( LDA, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGEBAL balances a general complex matrix A.  This involves, first,
   *> permuting A by a similarity transformation 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 matrix, and improve the
   *> accuracy of the computed eigenvalues and/or eigenvectors.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOB
   *> \verbatim
   *>          JOB is CHARACTER*1
   *>          Specifies the operations to be performed on A:
   *>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.
   *>          See Further Details.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] ILO
   *> \verbatim
   *> \endverbatim
   *>
   *> \param[out] IHI
   *> \verbatim
   *>          ILO and IHI are set to INTEGER such that on exit
   *>          A(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] SCALE
   *> \verbatim
   *>          SCALE is DOUBLE PRECISION array, dimension (N)
   *>          Details of the permutations and scaling factors applied to
   *>          A.  If P(j) is the index of the row and column interchanged
   *>          with row and column j and D(j) is the scaling factor
   *>          applied to row and column j, then
   *>          SCALE(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] 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 complex16GEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The permutations consist of row and column interchanges which put
   *>  the matrix in the form
   *>
   *>             ( T1   X   Y  )
   *>     P A P = (  0   B   Z  )
   *>             (  0   0   T2 )
   *>
   *>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
   *>  along the diagonal.  The column indices ILO and IHI mark the starting
   *>  and ending columns of the submatrix B. Balancing consists of applying
   *>  a diagonal similarity transformation inv(D) * B * D to make the
   *>  1-norms of each row of B and its corresponding column nearly equal.
   *>  The output matrix is
   *>
   *>     ( T1     X*D          Y    )
   *>     (  0  inv(D)*B*D  inv(D)*Z ).
   *>     (  0      0           T2   )
   *>
   *>  Information about the permutations P and the diagonal matrix D is
   *>  returned in the vector SCALE.
   *>
   *>  This subroutine is based on the EISPACK routine CBAL.
   *>
   *>  Modified by Tzu-Yi Chen, Computer Science Division, University of
   *>    California at Berkeley, USA
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )        SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, 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
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       COMPLEX*16         A( LDA, * )        COMPLEX*16         A( LDA, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGEBAL balances a general complex matrix A.  This involves, first,  
 *  permuting A by a similarity transformation 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 matrix, and improve the  
 *  accuracy of the computed eigenvalues and/or eigenvectors.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOB     (input) CHARACTER*1  
 *          Specifies the operations to be performed on A:  
 *          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.  
 *          See Further Details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  ILO     (output) INTEGER  
 *  IHI     (output) INTEGER  
 *          ILO and IHI are set to integers such that on exit  
 *          A(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.  
 *  
 *  SCALE   (output) DOUBLE PRECISION array, dimension (N)  
 *          Details of the permutations and scaling factors applied to  
 *          A.  If P(j) is the index of the row and column interchanged  
 *          with row and column j and D(j) is the scaling factor  
 *          applied to row and column j, then  
 *          SCALE(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.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit.  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The permutations consist of row and column interchanges which put  
 *  the matrix in the form  
 *  
 *             ( T1   X   Y  )  
 *     P A P = (  0   B   Z  )  
 *             (  0   0   T2 )  
 *  
 *  where T1 and T2 are upper triangular matrices whose eigenvalues lie  
 *  along the diagonal.  The column indices ILO and IHI mark the starting  
 *  and ending columns of the submatrix B. Balancing consists of applying  
 *  a diagonal similarity transformation inv(D) * B * D to make the  
 *  1-norms of each row of B and its corresponding column nearly equal.  
 *  The output matrix is  
 *  
 *     ( T1     X*D          Y    )  
 *     (  0  inv(D)*B*D  inv(D)*Z ).  
 *     (  0      0           T2   )  
 *  
 *  Information about the permutations P and the diagonal matrix D is  
 *  returned in the vector SCALE.  
 *  
 *  This subroutine is based on the EISPACK routine CBAL.  
 *  
 *  Modified by Tzu-Yi Chen, Computer Science Division, University of  
 *    California at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
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  Added in v.1.9

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