dgeequb.f

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      SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                    INFO )
*
*     -- LAPACK routine (version 3.2)                                 --
*     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
*     -- Jason Riedy of Univ. of California Berkeley.                 --
*     -- November 2008                                                --
*
*     -- LAPACK is a software package provided by Univ. of Tennessee, --
*     -- Univ. of California Berkeley and NAG Ltd.                    --
*
      IMPLICIT NONE
*     ..
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*     ..
*
*  Purpose
*  =======
*
*  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.  Baring over- and underflow, scaling by
*  these factors introduces no additional rounding errors.  However, the
*  scaled entries' magnitured are no longer approximately 1 but lie
*  between sqrt(radix) and 1/sqrt(radix).
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The M-by-N matrix whose equilibration factors are
*          to be computed.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  R       (output) DOUBLE PRECISION array, dimension (M)
*          If INFO = 0 or INFO > M, R contains the row scale factors
*          for A.
*
*  C       (output) DOUBLE PRECISION array, dimension (N)
*          If INFO = 0,  C contains the column scale factors for A.
*
*  ROWCND  (output) 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.
*
*  COLCND  (output) 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.
*
*  AMAX    (output) 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.
*
*  INFO    (output) 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
*
*  =====================================================================
*
*     .. 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