443 lines
13 KiB
Fortran
443 lines
13 KiB
Fortran
*> \brief \b DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
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*
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DPSTRF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpstrf.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstrf.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstrf.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* DOUBLE PRECISION TOL
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* INTEGER INFO, LDA, N, RANK
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* CHARACTER UPLO
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
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* INTEGER PIV( N )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DPSTRF computes the Cholesky factorization with complete
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*> pivoting of a real symmetric positive semidefinite matrix A.
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*>
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*> The factorization has the form
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*> P**T * A * P = U**T * U , if UPLO = 'U',
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*> P**T * A * P = L * L**T, if UPLO = 'L',
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*> where U is an upper triangular matrix and L is lower triangular, and
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*> P is stored as vector PIV.
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*>
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*> This algorithm does not attempt to check that A is positive
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*> semidefinite. This version of the algorithm calls level 3 BLAS.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> Specifies whether the upper or lower triangular part of the
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*> symmetric matrix A is stored.
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*> = 'U': Upper triangular
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*> = 'L': Lower triangular
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA,N)
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*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
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*> n by n upper triangular part of A contains the upper
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*> triangular part of the matrix A, and the strictly lower
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*> triangular part of A is not referenced. If UPLO = 'L', the
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*> leading n by n lower triangular part of A contains the lower
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*> triangular part of the matrix A, and the strictly upper
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*> triangular part of A is not referenced.
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*>
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*> On exit, if INFO = 0, the factor U or L from the Cholesky
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*> factorization as above.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] PIV
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*> \verbatim
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*> PIV is INTEGER array, dimension (N)
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*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
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*> \endverbatim
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*>
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*> \param[out] RANK
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*> \verbatim
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*> RANK is INTEGER
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*> The rank of A given by the number of steps the algorithm
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*> completed.
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*> \endverbatim
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*>
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*> \param[in] TOL
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*> \verbatim
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*> TOL is DOUBLE PRECISION
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*> User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
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*> will be used. The algorithm terminates at the (K-1)st step
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*> if the pivot <= TOL.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (2*N)
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*> Work space.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> < 0: If INFO = -K, the K-th argument had an illegal value,
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*> = 0: algorithm completed successfully, and
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*> > 0: the matrix A is either rank deficient with computed rank
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*> as returned in RANK, or is not positive semidefinite. See
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*> Section 7 of LAPACK Working Note #161 for further
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*> information.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup doubleOTHERcomputational
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*
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* =====================================================================
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SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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DOUBLE PRECISION TOL
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INTEGER INFO, LDA, N, RANK
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CHARACTER UPLO
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
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INTEGER PIV( N )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ONE, ZERO
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PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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DOUBLE PRECISION AJJ, DSTOP, DTEMP
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INTEGER I, ITEMP, J, JB, K, NB, PVT
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LOGICAL UPPER
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH
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INTEGER ILAENV
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LOGICAL LSAME, DISNAN
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EXTERNAL DLAMCH, ILAENV, LSAME, DISNAN
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEMV, DPSTF2, DSCAL, DSWAP, DSYRK, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT, MAXLOC
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -4
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DPSTRF', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Get block size
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*
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NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
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IF( NB.LE.1 .OR. NB.GE.N ) THEN
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*
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* Use unblocked code
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*
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CALL DPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK,
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$ INFO )
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GO TO 200
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*
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ELSE
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*
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* Initialize PIV
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*
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DO 100 I = 1, N
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PIV( I ) = I
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100 CONTINUE
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*
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* Compute stopping value
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*
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PVT = 1
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AJJ = A( PVT, PVT )
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DO I = 2, N
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IF( A( I, I ).GT.AJJ ) THEN
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PVT = I
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AJJ = A( PVT, PVT )
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END IF
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END DO
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IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
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RANK = 0
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INFO = 1
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GO TO 200
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END IF
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*
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* Compute stopping value if not supplied
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*
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IF( TOL.LT.ZERO ) THEN
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DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
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ELSE
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DSTOP = TOL
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END IF
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*
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*
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IF( UPPER ) THEN
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*
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* Compute the Cholesky factorization P**T * A * P = U**T * U
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*
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DO 140 K = 1, N, NB
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*
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* Account for last block not being NB wide
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*
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JB = MIN( NB, N-K+1 )
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*
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* Set relevant part of first half of WORK to zero,
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* holds dot products
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*
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DO 110 I = K, N
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WORK( I ) = 0
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110 CONTINUE
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*
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DO 130 J = K, K + JB - 1
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*
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* Find pivot, test for exit, else swap rows and columns
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* Update dot products, compute possible pivots which are
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* stored in the second half of WORK
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*
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DO 120 I = J, N
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*
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IF( J.GT.K ) THEN
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WORK( I ) = WORK( I ) + A( J-1, I )**2
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END IF
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WORK( N+I ) = A( I, I ) - WORK( I )
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*
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120 CONTINUE
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*
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IF( J.GT.1 ) THEN
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ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
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PVT = ITEMP + J - 1
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AJJ = WORK( N+PVT )
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IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
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A( J, J ) = AJJ
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GO TO 190
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END IF
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END IF
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*
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IF( J.NE.PVT ) THEN
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*
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* Pivot OK, so can now swap pivot rows and columns
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*
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A( PVT, PVT ) = A( J, J )
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CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
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IF( PVT.LT.N )
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$ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
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$ A( PVT, PVT+1 ), LDA )
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CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA,
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$ A( J+1, PVT ), 1 )
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*
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* Swap dot products and PIV
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*
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DTEMP = WORK( J )
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WORK( J ) = WORK( PVT )
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WORK( PVT ) = DTEMP
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ITEMP = PIV( PVT )
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PIV( PVT ) = PIV( J )
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PIV( J ) = ITEMP
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END IF
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*
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AJJ = SQRT( AJJ )
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A( J, J ) = AJJ
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*
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* Compute elements J+1:N of row J.
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*
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IF( J.LT.N ) THEN
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CALL DGEMV( 'Trans', J-K, N-J, -ONE, A( K, J+1 ),
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$ LDA, A( K, J ), 1, ONE, A( J, J+1 ),
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$ LDA )
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CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
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END IF
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*
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130 CONTINUE
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*
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* Update trailing matrix, J already incremented
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*
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IF( K+JB.LE.N ) THEN
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CALL DSYRK( 'Upper', 'Trans', N-J+1, JB, -ONE,
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$ A( K, J ), LDA, ONE, A( J, J ), LDA )
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END IF
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*
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140 CONTINUE
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*
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ELSE
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*
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* Compute the Cholesky factorization P**T * A * P = L * L**T
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*
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DO 180 K = 1, N, NB
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*
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* Account for last block not being NB wide
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*
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JB = MIN( NB, N-K+1 )
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*
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* Set relevant part of first half of WORK to zero,
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* holds dot products
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*
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DO 150 I = K, N
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WORK( I ) = 0
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150 CONTINUE
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*
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DO 170 J = K, K + JB - 1
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*
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* Find pivot, test for exit, else swap rows and columns
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* Update dot products, compute possible pivots which are
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* stored in the second half of WORK
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*
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DO 160 I = J, N
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*
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IF( J.GT.K ) THEN
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WORK( I ) = WORK( I ) + A( I, J-1 )**2
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END IF
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WORK( N+I ) = A( I, I ) - WORK( I )
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*
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160 CONTINUE
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*
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IF( J.GT.1 ) THEN
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ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
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PVT = ITEMP + J - 1
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AJJ = WORK( N+PVT )
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IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
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A( J, J ) = AJJ
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GO TO 190
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END IF
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END IF
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*
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IF( J.NE.PVT ) THEN
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*
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* Pivot OK, so can now swap pivot rows and columns
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*
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A( PVT, PVT ) = A( J, J )
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CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
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IF( PVT.LT.N )
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$ CALL DSWAP( N-PVT, A( PVT+1, J ), 1,
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$ A( PVT+1, PVT ), 1 )
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CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ),
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$ LDA )
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*
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* Swap dot products and PIV
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*
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DTEMP = WORK( J )
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WORK( J ) = WORK( PVT )
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WORK( PVT ) = DTEMP
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ITEMP = PIV( PVT )
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PIV( PVT ) = PIV( J )
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PIV( J ) = ITEMP
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END IF
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*
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AJJ = SQRT( AJJ )
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A( J, J ) = AJJ
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*
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* Compute elements J+1:N of column J.
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*
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IF( J.LT.N ) THEN
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CALL DGEMV( 'No Trans', N-J, J-K, -ONE,
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$ A( J+1, K ), LDA, A( J, K ), LDA, ONE,
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$ A( J+1, J ), 1 )
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CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
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END IF
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*
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170 CONTINUE
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*
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* Update trailing matrix, J already incremented
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*
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IF( K+JB.LE.N ) THEN
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CALL DSYRK( 'Lower', 'No Trans', N-J+1, JB, -ONE,
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$ A( J, K ), LDA, ONE, A( J, J ), LDA )
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END IF
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*
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180 CONTINUE
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*
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END IF
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END IF
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*
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* Ran to completion, A has full rank
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*
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RANK = N
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*
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GO TO 200
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190 CONTINUE
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*
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* Rank is the number of steps completed. Set INFO = 1 to signal
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* that the factorization cannot be used to solve a system.
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*
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RANK = J - 1
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INFO = 1
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*
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200 CONTINUE
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RETURN
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*
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* End of DPSTRF
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*
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END
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