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    1:       SUBROUTINE DPPTRF( UPLO, N, AP, 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: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   AP( * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  DPPTRF computes the Cholesky factorization of a real symmetric
   20: *  positive definite matrix A stored in packed format.
   21: *
   22: *  The factorization has the form
   23: *     A = U**T * U,  if UPLO = 'U', or
   24: *     A = L  * L**T,  if UPLO = 'L',
   25: *  where U is an upper triangular matrix and L is lower triangular.
   26: *
   27: *  Arguments
   28: *  =========
   29: *
   30: *  UPLO    (input) CHARACTER*1
   31: *          = 'U':  Upper triangle of A is stored;
   32: *          = 'L':  Lower triangle of A is stored.
   33: *
   34: *  N       (input) INTEGER
   35: *          The order of the matrix A.  N >= 0.
   36: *
   37: *  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
   38: *          On entry, the upper or lower triangle of the symmetric matrix
   39: *          A, packed columnwise in a linear array.  The j-th column of A
   40: *          is stored in the array AP as follows:
   41: *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   42: *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
   43: *          See below for further details.
   44: *
   45: *          On exit, if INFO = 0, the triangular factor U or L from the
   46: *          Cholesky factorization A = U**T*U or A = L*L**T, in the same
   47: *          storage format as A.
   48: *
   49: *  INFO    (output) INTEGER
   50: *          = 0:  successful exit
   51: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   52: *          > 0:  if INFO = i, the leading minor of order i is not
   53: *                positive definite, and the factorization could not be
   54: *                completed.
   55: *
   56: *  Further Details
   57: *  ======= =======
   58: *
   59: *  The packed storage scheme is illustrated by the following example
   60: *  when N = 4, UPLO = 'U':
   61: *
   62: *  Two-dimensional storage of the symmetric matrix A:
   63: *
   64: *     a11 a12 a13 a14
   65: *         a22 a23 a24
   66: *             a33 a34     (aij = aji)
   67: *                 a44
   68: *
   69: *  Packed storage of the upper triangle of A:
   70: *
   71: *  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
   72: *
   73: *  =====================================================================
   74: *
   75: *     .. Parameters ..
   76:       DOUBLE PRECISION   ONE, ZERO
   77:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
   78: *     ..
   79: *     .. Local Scalars ..
   80:       LOGICAL            UPPER
   81:       INTEGER            J, JC, JJ
   82:       DOUBLE PRECISION   AJJ
   83: *     ..
   84: *     .. External Functions ..
   85:       LOGICAL            LSAME
   86:       DOUBLE PRECISION   DDOT
   87:       EXTERNAL           LSAME, DDOT
   88: *     ..
   89: *     .. External Subroutines ..
   90:       EXTERNAL           DSCAL, DSPR, DTPSV, XERBLA
   91: *     ..
   92: *     .. Intrinsic Functions ..
   93:       INTRINSIC          SQRT
   94: *     ..
   95: *     .. Executable Statements ..
   96: *
   97: *     Test the input parameters.
   98: *
   99:       INFO = 0
  100:       UPPER = LSAME( UPLO, 'U' )
  101:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  102:          INFO = -1
  103:       ELSE IF( N.LT.0 ) THEN
  104:          INFO = -2
  105:       END IF
  106:       IF( INFO.NE.0 ) THEN
  107:          CALL XERBLA( 'DPPTRF', -INFO )
  108:          RETURN
  109:       END IF
  110: *
  111: *     Quick return if possible
  112: *
  113:       IF( N.EQ.0 )
  114:      $   RETURN
  115: *
  116:       IF( UPPER ) THEN
  117: *
  118: *        Compute the Cholesky factorization A = U'*U.
  119: *
  120:          JJ = 0
  121:          DO 10 J = 1, N
  122:             JC = JJ + 1
  123:             JJ = JJ + J
  124: *
  125: *           Compute elements 1:J-1 of column J.
  126: *
  127:             IF( J.GT.1 )
  128:      $         CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP,
  129:      $                     AP( JC ), 1 )
  130: *
  131: *           Compute U(J,J) and test for non-positive-definiteness.
  132: *
  133:             AJJ = AP( JJ ) - DDOT( J-1, AP( JC ), 1, AP( JC ), 1 )
  134:             IF( AJJ.LE.ZERO ) THEN
  135:                AP( JJ ) = AJJ
  136:                GO TO 30
  137:             END IF
  138:             AP( JJ ) = SQRT( AJJ )
  139:    10    CONTINUE
  140:       ELSE
  141: *
  142: *        Compute the Cholesky factorization A = L*L'.
  143: *
  144:          JJ = 1
  145:          DO 20 J = 1, N
  146: *
  147: *           Compute L(J,J) and test for non-positive-definiteness.
  148: *
  149:             AJJ = AP( JJ )
  150:             IF( AJJ.LE.ZERO ) THEN
  151:                AP( JJ ) = AJJ
  152:                GO TO 30
  153:             END IF
  154:             AJJ = SQRT( AJJ )
  155:             AP( JJ ) = AJJ
  156: *
  157: *           Compute elements J+1:N of column J and update the trailing
  158: *           submatrix.
  159: *
  160:             IF( J.LT.N ) THEN
  161:                CALL DSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 )
  162:                CALL DSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1,
  163:      $                    AP( JJ+N-J+1 ) )
  164:                JJ = JJ + N - J + 1
  165:             END IF
  166:    20    CONTINUE
  167:       END IF
  168:       GO TO 40
  169: *
  170:    30 CONTINUE
  171:       INFO = J
  172: *
  173:    40 CONTINUE
  174:       RETURN
  175: *
  176: *     End of DPPTRF
  177: *
  178:       END