zgbequb.f

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*> \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.
*
*> \ingroup gbequb
*
*  =====================================================================
      SUBROUTINE zgbequb( M, N, KL, KU, AB, LDAB, 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, 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