spoequb.f

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      SUBROUTINE SPOEQUB( N, A, LDA, S, SCOND, 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, N
      REAL               AMAX, SCOND
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), S( * )
*     ..
*
*  Purpose
*  =======
*
*  SPOEQU computes row and column scalings intended to equilibrate a
*  symmetric positive definite 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.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) REAL array, dimension (LDA,N)
*          The N-by-N symmetric positive definite matrix whose scaling
*          factors are to be computed.  Only the diagonal elements of A
*          are referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  S       (output) REAL array, dimension (N)
*          If INFO = 0, S contains the scale factors for A.
*
*  SCOND   (output) 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.
*
*  AMAX    (output) REAL
*          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, the i-th diagonal element is nonpositive.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      REAL               SMIN, BASE, TMP
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT, LOG, INT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
*     Positive definite only performs 1 pass of equilibration.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SPOEQUB', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF

      BASE = SLAMCH( 'B' )
      TMP = -0.5 / LOG ( BASE )
*
*     Find the minimum and maximum diagonal elements.
*
      S( 1 ) = A( 1, 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
      DO 10 I = 2, N
         S( I ) = A( I, 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 ) = BASE ** INT( TMP * LOG( S( I ) ) )
   30    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I)).
*
         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
*
      RETURN
*
*     End of SPOEQUB
*
      END