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version 1.8, 2011/07/22 07:38:10 version 1.9, 2011/11/21 20:43:02
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   *> \brief \b DPPTRF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DPPTRF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptrf.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptrf.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptrf.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   AP( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] AP
   *> \verbatim
   *>          AP is 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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is 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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  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 ]
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DPPTRF( UPLO, N, AP, INFO )        SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       DOUBLE PRECISION   AP( * )        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 ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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