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    1:       SUBROUTINE DPPEQU( UPLO, N, AP, 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:       CHARACTER          UPLO
   10:       INTEGER            INFO, N
   11:       DOUBLE PRECISION   AMAX, SCOND
   12: *     ..
   13: *     .. Array Arguments ..
   14:       DOUBLE PRECISION   AP( * ), S( * )
   15: *     ..
   16: *
   17: *  Purpose
   18: *  =======
   19: *
   20: *  DPPEQU computes row and column scalings intended to equilibrate a
   21: *  symmetric positive definite matrix A in packed storage and reduce
   22: *  its condition number (with respect to the two-norm).  S contains the
   23: *  scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
   24: *  B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
   25: *  This choice of S puts the condition number of B within a factor N of
   26: *  the smallest possible condition number over all possible diagonal
   27: *  scalings.
   28: *
   29: *  Arguments
   30: *  =========
   31: *
   32: *  UPLO    (input) CHARACTER*1
   33: *          = 'U':  Upper triangle of A is stored;
   34: *          = 'L':  Lower triangle of A is stored.
   35: *
   36: *  N       (input) INTEGER
   37: *          The order of the matrix A.  N >= 0.
   38: *
   39: *  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
   40: *          The upper or lower triangle of the symmetric matrix A, packed
   41: *          columnwise in a linear array.  The j-th column of A is stored
   42: *          in the array AP as follows:
   43: *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   44: *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
   45: *
   46: *  S       (output) DOUBLE PRECISION array, dimension (N)
   47: *          If INFO = 0, S contains the scale factors for A.
   48: *
   49: *  SCOND   (output) DOUBLE PRECISION
   50: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   51: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   52: *          large nor too small, it is not worth scaling by S.
   53: *
   54: *  AMAX    (output) DOUBLE PRECISION
   55: *          Absolute value of largest matrix element.  If AMAX is very
   56: *          close to overflow or very close to underflow, the matrix
   57: *          should be scaled.
   58: *
   59: *  INFO    (output) INTEGER
   60: *          = 0:  successful exit
   61: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   62: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   63: *
   64: *  =====================================================================
   65: *
   66: *     .. Parameters ..
   67:       DOUBLE PRECISION   ONE, ZERO
   68:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
   69: *     ..
   70: *     .. Local Scalars ..
   71:       LOGICAL            UPPER
   72:       INTEGER            I, JJ
   73:       DOUBLE PRECISION   SMIN
   74: *     ..
   75: *     .. External Functions ..
   76:       LOGICAL            LSAME
   77:       EXTERNAL           LSAME
   78: *     ..
   79: *     .. External Subroutines ..
   80:       EXTERNAL           XERBLA
   81: *     ..
   82: *     .. Intrinsic Functions ..
   83:       INTRINSIC          MAX, MIN, SQRT
   84: *     ..
   85: *     .. Executable Statements ..
   86: *
   87: *     Test the input parameters.
   88: *
   89:       INFO = 0
   90:       UPPER = LSAME( UPLO, 'U' )
   91:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
   92:          INFO = -1
   93:       ELSE IF( N.LT.0 ) THEN
   94:          INFO = -2
   95:       END IF
   96:       IF( INFO.NE.0 ) THEN
   97:          CALL XERBLA( 'DPPEQU', -INFO )
   98:          RETURN
   99:       END IF
  100: *
  101: *     Quick return if possible
  102: *
  103:       IF( N.EQ.0 ) THEN
  104:          SCOND = ONE
  105:          AMAX = ZERO
  106:          RETURN
  107:       END IF
  108: *
  109: *     Initialize SMIN and AMAX.
  110: *
  111:       S( 1 ) = AP( 1 )
  112:       SMIN = S( 1 )
  113:       AMAX = S( 1 )
  114: *
  115:       IF( UPPER ) THEN
  116: *
  117: *        UPLO = 'U':  Upper triangle of A is stored.
  118: *        Find the minimum and maximum diagonal elements.
  119: *
  120:          JJ = 1
  121:          DO 10 I = 2, N
  122:             JJ = JJ + I
  123:             S( I ) = AP( JJ )
  124:             SMIN = MIN( SMIN, S( I ) )
  125:             AMAX = MAX( AMAX, S( I ) )
  126:    10    CONTINUE
  127: *
  128:       ELSE
  129: *
  130: *        UPLO = 'L':  Lower triangle of A is stored.
  131: *        Find the minimum and maximum diagonal elements.
  132: *
  133:          JJ = 1
  134:          DO 20 I = 2, N
  135:             JJ = JJ + N - I + 2
  136:             S( I ) = AP( JJ )
  137:             SMIN = MIN( SMIN, S( I ) )
  138:             AMAX = MAX( AMAX, S( I ) )
  139:    20    CONTINUE
  140:       END IF
  141: *
  142:       IF( SMIN.LE.ZERO ) THEN
  143: *
  144: *        Find the first non-positive diagonal element and return.
  145: *
  146:          DO 30 I = 1, N
  147:             IF( S( I ).LE.ZERO ) THEN
  148:                INFO = I
  149:                RETURN
  150:             END IF
  151:    30    CONTINUE
  152:       ELSE
  153: *
  154: *        Set the scale factors to the reciprocals
  155: *        of the diagonal elements.
  156: *
  157:          DO 40 I = 1, N
  158:             S( I ) = ONE / SQRT( S( I ) )
  159:    40    CONTINUE
  160: *
  161: *        Compute SCOND = min(S(I)) / max(S(I))
  162: *
  163:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  164:       END IF
  165:       RETURN
  166: *
  167: *     End of DPPEQU
  168: *
  169:       END