path: root/SRC/zgbequb.f

*> \brief \b ZGBEQUB
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
*                           AMAX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( * ), R( * )
*       COMPLEX*16         AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGBEQUB 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 ZGEEQU 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] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is COMPLEX*16 array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A.  LDAB >= 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.
*
*> \date June 2016
*
*> \ingroup complex16GBcomputational
*
*  =====================================================================
      SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
     $                    AMAX, INFO )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( * ), R( * )
      COMPLEX*16         AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, KD
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
     $                   LOGRDX
      COMPLEX*16         ZDUM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, LOG, REAL, DIMAG
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
*     ..
*     .. 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( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KL+KU+1 ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZGBEQUB', -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.
*
      KD = KU + 1
      DO 30 J = 1, N
         DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
            R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, 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 = MAX( J-KU, 1 ), MIN( J+KL, M )
            C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, 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 ZGBEQUB
*
      END