dsyequb.f

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*> \brief \b DSYEQUB
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/ 
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   AMAX, SCOND
*       CHARACTER          UPLO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEQUB computes row and column scalings intended to equilibrate a
*> symmetric 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
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The N-by-N symmetric matrix whose scaling
*>          factors are to be computed.  Only the diagonal elements of A
*>          are referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          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 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] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \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. 
*
*> \date November 2011
*
*> \ingroup doubleSYcomputational
*
*> \par References:
*  ================
*>
*>  Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
*>  Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
*>  DOI 10.1023/B:NUMA.0000016606.32820.69 \n
*>  Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
*>
*  =====================================================================
      SUBROUTINE dsyequb( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
*  -- LAPACK computational routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   AMAX, SCOND
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( lda, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      parameter                ( one = 1.0d+0, zero = 0.0d+0 )
      INTEGER            MAX_ITER
      parameter                ( max_iter = 100 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, ITER
      DOUBLE PRECISION   AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
     $                   smin, smax, smlnum, bignum, scale, sumsq
      LOGICAL            UP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      LOGICAL            LSAME
      EXTERNAL           dlamch, lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlassq
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, int, log, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test input parameters.
*
      info = 0
      IF ( .NOT. ( lsame( uplo, 'U' ) .OR. lsame( uplo, 'L' ) ) ) THEN
        info = -1
      ELSE IF ( n .LT. 0 ) THEN
        info = -2
      ELSE IF ( lda .LT. max( 1, n ) ) THEN
        info = -4
      END IF
      IF ( info .NE. 0 ) THEN
        CALL xerbla( 'DSYEQUB', -info )
        RETURN
      END IF

      up = lsame( uplo, 'U' )
      amax = zero
*
*     Quick return if possible.
*
      IF ( n .EQ. 0 ) THEN
        scond = one
        RETURN
      END IF

      DO i = 1, n
        s( i ) = zero
      END DO

      amax = zero
      IF ( up ) THEN
         DO j = 1, n
            DO i = 1, j-1
               s( i ) = max( s( i ), abs( a( i, j ) ) )
               s( j ) = max( s( j ), abs( a( i, j ) ) )
               amax = max( amax, abs( a(i, j) ) )
            END DO
            s( j ) = max( s( j ), abs( a( j, j ) ) )
            amax = max( amax, abs( a( j, j ) ) )
         END DO
      ELSE
         DO j = 1, n
            s( j ) = max( s( j ), abs( a( j, j ) ) )
            amax = max( amax, abs( a( j, j ) ) )
            DO i = j+1, n
               s( i ) = max( s( i ), abs( a( i, j ) ) )
               s( j ) = max( s( j ), abs( a( i, j ) ) )
               amax = max( amax, abs( a( i, j ) ) )
            END DO
         END DO
      END IF
      DO j = 1, n
         s( j ) = 1.0d+0 / s( j )
      END DO

      tol = one / sqrt(2.0d0 * n)

      DO iter = 1, max_iter
         scale = 0.0d+0
         sumsq = 0.0d+0
*       BETA = |A|S
        DO i = 1, n
           work(i) = zero
        END DO
        IF ( up ) THEN
           DO j = 1, n
              DO i = 1, j-1
                 t = abs( a( i, j ) )
                 work( i ) = work( i ) + abs( a( i, j ) ) * s( j )
                 work( j ) = work( j ) + abs( a( i, j ) ) * s( i )
              END DO
              work( j ) = work( j ) + abs( a( j, j ) ) * s( j )
           END DO
        ELSE
           DO j = 1, n
              work( j ) = work( j ) + abs( a( j, j ) ) * s( j )
              DO i = j+1, n
                 t = abs( a( i, j ) )
                 work( i ) = work( i ) + abs( a( i, j ) ) * s( j )
                 work( j ) = work( j ) + abs( a( i, j ) ) * s( i )
              END DO
           END DO
        END IF

*       avg = s^T beta / n
        avg = 0.0d+0
        DO i = 1, n
          avg = avg + s( i )*work( i )
        END DO
        avg = avg / n

        std = 0.0d+0
        DO i = 2*n+1, 3*n
           work( i ) = s( i-2*n ) * work( i-2*n ) - avg
        END DO
        CALL dlassq( n, work( 2*n+1 ), 1, scale, sumsq )
        std = scale * sqrt( sumsq / n )

        IF ( std .LT. tol * avg ) GOTO 999

        DO i = 1, n
          t = abs( a( i, i ) )
          si = s( i )
          c2 = ( n-1 ) * t
          c1 = ( n-2 ) * ( work( i ) - t*si )
          c0 = -(t*si)*si + 2*work( i )*si - n*avg
          d = c1*c1 - 4*c0*c2

          IF ( d .LE. 0 ) THEN
            info = -1
            RETURN
          END IF
          si = -2*c0 / ( c1 + sqrt( d ) )

          d = si - s( i )
          u = zero
          IF ( up ) THEN
            DO j = 1, i
              t = abs( a( j, i ) )
              u = u + s( j )*t
              work( j ) = work( j ) + d*t
            END DO
            DO j = i+1,n
              t = abs( a( i, j ) )
              u = u + s( j )*t
              work( j ) = work( j ) + d*t
            END DO
          ELSE
            DO j = 1, i
              t = abs( a( i, j ) )
              u = u + s( j )*t
              work( j ) = work( j ) + d*t
            END DO
            DO j = i+1,n
              t = abs( a( j, i ) )
              u = u + s( j )*t
              work( j ) = work( j ) + d*t
            END DO
          END IF

          avg = avg + ( u + work( i ) ) * d / n
          s( i ) = si

        END DO

      END DO

 999  CONTINUE

      smlnum = dlamch( 'SAFEMIN' )
      bignum = one / smlnum
      smin = bignum
      smax = zero
      t = one / sqrt(avg)
      base = dlamch( 'B' )
      u = one / log( base )
      DO i = 1, n
        s( i ) = base ** int( u * log( s( i ) * t ) )
        smin = min( smin, s( i ) )
        smax = max( smax, s( i ) )
      END DO
      scond = max( smin, smlnum ) / min( smax, bignum )
*
      END