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version 1.7, 2010/08/13 21:03:43 version 1.21, 2023/08/07 08:38:47
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   *> \brief \b DGEBAL
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DGEBAL + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebal.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebal.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebal.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOB
   *       INTEGER            IHI, ILO, INFO, LDA, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), SCALE( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGEBAL balances a general real 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 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.
   *>          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
   *>          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 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.
   *
   *> \ingroup doubleGEcomputational
   *
   *> \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 BALANC.
   *>
   *>  Modified by Tzu-Yi Chen, Computer Science Division, University of
   *>    California at Berkeley, USA
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )        SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
 *  *
 *  -- LAPACK routine (version 3.2.2) --  *  -- LAPACK computational routine --
 *  -- 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  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOB        CHARACTER          JOB
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       DOUBLE PRECISION   A( LDA, * ), SCALE( * )        DOUBLE PRECISION   A( LDA, * ), SCALE( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGEBAL balances a general real 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) 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.  
 *          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 BALANC.  
 *  
 *  Modified by Tzu-Yi Chen, Computer Science Division, University of  
 *    California at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *     .. External Functions ..  *     .. External Functions ..
       LOGICAL            DISNAN, LSAME        LOGICAL            DISNAN, LSAME
       INTEGER            IDAMAX        INTEGER            IDAMAX
       DOUBLE PRECISION   DLAMCH        DOUBLE PRECISION   DLAMCH, DNRM2
       EXTERNAL           DISNAN, LSAME, IDAMAX, DLAMCH        EXTERNAL           DISNAN, LSAME, IDAMAX, DLAMCH, DNRM2
 *     ..  *     ..
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL           DSCAL, DSWAP, XERBLA        EXTERNAL           DSCAL, DSWAP, XERBLA
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 *     .. Intrinsic Functions ..  *     .. Intrinsic Functions ..
       INTRINSIC          ABS, MAX, MIN        INTRINSIC          ABS, MAX, MIN
 *     ..  *     ..
 *     .. Executable Statements ..  
 *  
 *     Test the input parameters  *     Test the input parameters
 *  *
       INFO = 0        INFO = 0
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       SFMAX1 = ONE / SFMIN1        SFMAX1 = ONE / SFMIN1
       SFMIN2 = SFMIN1*SCLFAC        SFMIN2 = SFMIN1*SCLFAC
       SFMAX2 = ONE / SFMIN2        SFMAX2 = ONE / SFMIN2
   *
   140 CONTINUE    140 CONTINUE
       NOCONV = .FALSE.        NOCONV = .FALSE.
 *  *
       DO 200 I = K, L        DO 200 I = K, L
          C = ZERO  
          R = ZERO  
 *  *
          DO 150 J = K, L           C = DNRM2( L-K+1, A( K, I ), 1 )
             IF( J.EQ.I )           R = DNRM2( L-K+1, A( I, K ), LDA )
      $         GO TO 150  
             C = C + ABS( A( J, I ) )  
             R = R + ABS( A( I, J ) )  
   150    CONTINUE  
          ICA = IDAMAX( L, A( 1, I ), 1 )           ICA = IDAMAX( L, A( 1, I ), 1 )
          CA = ABS( A( ICA, I ) )           CA = ABS( A( ICA, I ) )
          IRA = IDAMAX( N-K+1, A( I, K ), LDA )           IRA = IDAMAX( N-K+1, A( I, K ), LDA )
Removed from v.1.7  
changed lines
  Added in v.1.21

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