*> \brief \b DGEEQUB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
*                           INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEEQUB computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number.  R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
*> the radix.
*>
*> R(i) and C(j) are restricted to be a power of the radix between
*> SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
*> of these scaling factors is not guaranteed to reduce the condition
*> number of A but works well in practice.
*>
*> This routine differs from DGEEQU by restricting the scaling factors
*> to a power of the radix.  Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors.  However, the
*> scaled entries' magnitudes are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The M-by-N matrix whose equilibration factors are
*>          to be computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          If INFO = 0 or INFO > M, R contains the row scale factors
*>          for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0,  C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*>          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
*>          AMAX is neither too large nor too small, it is not worth
*>          scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          If INFO = 0, COLCND contains the ratio of the smallest
*>          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
*>          worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i,  and i is
*>                <= M:  the i-th row of A is exactly zero
*>                >  M:  the (i-M)-th column of A is exactly zero
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geequb
*
*  =====================================================================
      SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                    INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, LOG
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEEQUB', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         ROWCND = ONE
         COLCND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Get machine constants.  Assume SMLNUM is a power of the radix.
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      RADIX = DLAMCH( 'B' )
      LOGRDX = LOG( RADIX )
*
*     Compute row scale factors.
*
      DO 10 I = 1, M
         R( I ) = ZERO
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      DO 30 J = 1, N
         DO 20 I = 1, M
            R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
   20    CONTINUE
   30 CONTINUE
      DO I = 1, M
         IF( R( I ).GT.ZERO ) THEN
            R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
         END IF
      END DO
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = 1, M
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      AMAX = RCMAX
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = 1, M
            IF( R( I ).EQ.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   50    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = 1, M
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I)).
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
*     Compute column scale factors
*
      DO 70 J = 1, N
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      DO 90 J = 1, N
         DO 80 I = 1, M
            C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
   80    CONTINUE
         IF( C( J ).GT.ZERO ) THEN
            C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
         END IF
   90 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = 1, N
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 J = 1, N
            IF( C( J ).EQ.ZERO ) THEN
               INFO = M + J
               RETURN
            END IF
  110    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 120 J = 1, N
            C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J)).
*
         COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
      RETURN
*
*     End of DGEEQUB
*
      END
