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    1:       SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, LDA, N
   10:       DOUBLE PRECISION   AMAX, SCOND
   11: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   A( LDA, * ), S( * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  DPOEQU computes row and column scalings intended to equilibrate a
   20: *  symmetric positive definite matrix A and reduce its condition number
   21: *  (with respect to the two-norm).  S contains the scale factors,
   22: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
   23: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
   24: *  choice of S puts the condition number of B within a factor N of the
   25: *  smallest possible condition number over all possible diagonal
   26: *  scalings.
   27: *
   28: *  Arguments
   29: *  =========
   30: *
   31: *  N       (input) INTEGER
   32: *          The order of the matrix A.  N >= 0.
   33: *
   34: *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
   35: *          The N-by-N symmetric positive definite matrix whose scaling
   36: *          factors are to be computed.  Only the diagonal elements of A
   37: *          are referenced.
   38: *
   39: *  LDA     (input) INTEGER
   40: *          The leading dimension of the array A.  LDA >= max(1,N).
   41: *
   42: *  S       (output) DOUBLE PRECISION array, dimension (N)
   43: *          If INFO = 0, S contains the scale factors for A.
   44: *
   45: *  SCOND   (output) DOUBLE PRECISION
   46: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   47: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   48: *          large nor too small, it is not worth scaling by S.
   49: *
   50: *  AMAX    (output) DOUBLE PRECISION
   51: *          Absolute value of largest matrix element.  If AMAX is very
   52: *          close to overflow or very close to underflow, the matrix
   53: *          should be scaled.
   54: *
   55: *  INFO    (output) INTEGER
   56: *          = 0:  successful exit
   57: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   58: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   59: *
   60: *  =====================================================================
   61: *
   62: *     .. Parameters ..
   63:       DOUBLE PRECISION   ZERO, ONE
   64:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
   65: *     ..
   66: *     .. Local Scalars ..
   67:       INTEGER            I
   68:       DOUBLE PRECISION   SMIN
   69: *     ..
   70: *     .. External Subroutines ..
   71:       EXTERNAL           XERBLA
   72: *     ..
   73: *     .. Intrinsic Functions ..
   74:       INTRINSIC          MAX, MIN, SQRT
   75: *     ..
   76: *     .. Executable Statements ..
   77: *
   78: *     Test the input parameters.
   79: *
   80:       INFO = 0
   81:       IF( N.LT.0 ) THEN
   82:          INFO = -1
   83:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
   84:          INFO = -3
   85:       END IF
   86:       IF( INFO.NE.0 ) THEN
   87:          CALL XERBLA( 'DPOEQU', -INFO )
   88:          RETURN
   89:       END IF
   90: *
   91: *     Quick return if possible
   92: *
   93:       IF( N.EQ.0 ) THEN
   94:          SCOND = ONE
   95:          AMAX = ZERO
   96:          RETURN
   97:       END IF
   98: *
   99: *     Find the minimum and maximum diagonal elements.
  100: *
  101:       S( 1 ) = A( 1, 1 )
  102:       SMIN = S( 1 )
  103:       AMAX = S( 1 )
  104:       DO 10 I = 2, N
  105:          S( I ) = A( I, I )
  106:          SMIN = MIN( SMIN, S( I ) )
  107:          AMAX = MAX( AMAX, S( I ) )
  108:    10 CONTINUE
  109: *
  110:       IF( SMIN.LE.ZERO ) THEN
  111: *
  112: *        Find the first non-positive diagonal element and return.
  113: *
  114:          DO 20 I = 1, N
  115:             IF( S( I ).LE.ZERO ) THEN
  116:                INFO = I
  117:                RETURN
  118:             END IF
  119:    20    CONTINUE
  120:       ELSE
  121: *
  122: *        Set the scale factors to the reciprocals
  123: *        of the diagonal elements.
  124: *
  125:          DO 30 I = 1, N
  126:             S( I ) = ONE / SQRT( S( I ) )
  127:    30    CONTINUE
  128: *
  129: *        Compute SCOND = min(S(I)) / max(S(I))
  130: *
  131:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  132:       END IF
  133:       RETURN
  134: *
  135: *     End of DPOEQU
  136: *
  137:       END