spbequ.f

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*> \brief \b SPBEQU
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       REAL               AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       REAL               AB( LDAB, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SPBEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite band matrix A and reduce its condition
*> number (with respect to the two-norm).  S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangular of A is stored;
*>          = 'L':  Lower triangular of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is REAL array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first KD+1 rows of the array.  The j-th column
*>          of A is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is REAL array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is REAL
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is REAL
*>          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, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbequ
*
*  =====================================================================
      SUBROUTINE spbequ( UPLO, N, KD, AB, LDAB, S, SCOND, 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 ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
      REAL               AMAX, SCOND
*     ..
*     .. Array Arguments ..
      REAL               AB( LDAB, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, J
      REAL               SMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( kd.LT.0 ) THEN
         info = -3
      ELSE IF( ldab.LT.kd+1 ) THEN
         info = -5
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPBEQU', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 ) THEN
         scond = one
         amax = zero
         RETURN
      END IF
*
      IF( upper ) THEN
         j = kd + 1
      ELSE
         j = 1
      END IF
*
*     Initialize SMIN and AMAX.
*
      s( 1 ) = ab( j, 1 )
      smin = s( 1 )
      amax = s( 1 )
*
*     Find the minimum and maximum diagonal elements.
*
      DO 10 i = 2, n
         s( i ) = ab( j, i )
         smin = min( smin, s( i ) )
         amax = max( amax, s( i ) )
   10 CONTINUE
*
      IF( smin.LE.zero ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 20 i = 1, n
            IF( s( i ).LE.zero ) THEN
               info = i
               RETURN
            END IF
   20    CONTINUE
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 30 i = 1, n
            s( i ) = one / sqrt( s( i ) )
   30    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I))
*
         scond = sqrt( smin ) / sqrt( amax )
      END IF
      RETURN
*
*     End of SPBEQU
*
      END