          SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AP( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DPPTRF computes the Cholesky factorization of a real symmetric
    *  positive definite matrix A stored in packed format.
    *
    *  The factorization has the form
    *     A = U**T * U,  if UPLO = 'U', or
    *     A = L  * L**T,  if UPLO = 'L',
    *  where U is an upper triangular matrix and L is lower triangular.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored;
    *          = 'L':  Lower triangle of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          On entry, the upper or lower triangle of the symmetric matrix
    *          A, packed columnwise in a linear array.  The j-th column of A
    *          is stored in the array AP as follows:
    *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    *          See below for further details.
    *
    *          On exit, if INFO = 0, the triangular factor U or L from the
    *          Cholesky factorization A = U**T*U or A = L*L**T, in the same
    *          storage format as A.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  if INFO = i, the leading minor of order i is not
    *                positive definite, and the factorization could not be
    *                completed.
    *
    *  Further Details
    *  ======= =======
    *
    *  The packed storage scheme is illustrated by the following example
    *  when N = 4, UPLO = 'U':
    *
    *  Two-dimensional storage of the symmetric matrix A:
    *
    *     a11 a12 a13 a14
    *         a22 a23 a24
    *             a33 a34     (aij = aji)
    *                 a44
    *
    *  Packed storage of the upper triangle of A:
    *
    *  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            J, JC, JJ
          DOUBLE PRECISION   AJJ
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DDOT
          EXTERNAL           LSAME, DDOT
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DSCAL, DSPR, DTPSV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          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
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DPPTRF', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
          IF( UPPER ) THEN
    *
    *        Compute the Cholesky factorization A = U'*U.
    *
             JJ = 0
             DO 10 J = 1, N
                JC = JJ + 1
                JJ = JJ + J
    *
    *           Compute elements 1:J-1 of column J.
    *
                IF( J.GT.1 )
         $         CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP,
         $                     AP( JC ), 1 )
    *
    *           Compute U(J,J) and test for non-positive-definiteness.
    *
                AJJ = AP( JJ ) - DDOT( J-1, AP( JC ), 1, AP( JC ), 1 )
                IF( AJJ.LE.ZERO ) THEN
                   AP( JJ ) = AJJ
                   GO TO 30
                END IF
                AP( JJ ) = SQRT( AJJ )
       10    CONTINUE
          ELSE
    *
    *        Compute the Cholesky factorization A = L*L'.
    *
             JJ = 1
             DO 20 J = 1, N
    *
    *           Compute L(J,J) and test for non-positive-definiteness.
    *
                AJJ = AP( JJ )
                IF( AJJ.LE.ZERO ) THEN
                   AP( JJ ) = AJJ
                   GO TO 30
                END IF
                AJJ = SQRT( AJJ )
                AP( JJ ) = AJJ
    *
    *           Compute elements J+1:N of column J and update the trailing
    *           submatrix.
    *
                IF( J.LT.N ) THEN
                   CALL DSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 )
                   CALL DSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1,
         $                    AP( JJ+N-J+1 ) )
                   JJ = JJ + N - J + 1
                END IF
       20    CONTINUE
          END IF
          GO TO 40
    *
       30 CONTINUE
          INFO = J
    *
       40 CONTINUE
          RETURN
    *
    *     End of DPPTRF
    *
          END