DPPTRF - compute the Cholesky factorization of a real sym­
       metric positive definite matrix A stored in packed format


SYNOPSIS

       SUBROUTINE DPPTRF( UPLO, N, AP, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, N

           DOUBLE         PRECISION AP( * )


PURPOSE

       DPPTRF computes the Cholesky factorization of a real  sym­
       metric 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  tri­
       angular.


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 sym­
               metric 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 ille­
               gal value
               > 0:  if INFO = i, the leading minor of order i is
               not positive definite, and the factorization could
               not be completed.

       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 ]


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