387 lines
11 KiB
Fortran
387 lines
11 KiB
Fortran
*> \brief \b SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
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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 SPSTF2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spstf2.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/spstf2.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/spstf2.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 SPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* REAL 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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* REAL 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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*> SPSTF2 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 2 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 REAL 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[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 REAL
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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[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] WORK
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*> \verbatim
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*> WORK is REAL 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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*> \date November 2015
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*
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*> \ingroup realOTHERcomputational
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*
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* =====================================================================
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SUBROUTINE SPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
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*
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* -- LAPACK computational routine (version 3.6.0) --
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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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* November 2015
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*
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* .. Scalar Arguments ..
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REAL 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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REAL 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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REAL ONE, ZERO
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PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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REAL AJJ, SSTOP, STEMP
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INTEGER I, ITEMP, J, PVT
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LOGICAL UPPER
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* ..
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* .. External Functions ..
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REAL SLAMCH
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LOGICAL LSAME, SISNAN
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EXTERNAL SLAMCH, LSAME, SISNAN
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEMV, SSCAL, SSWAP, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, 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( 'SPSTF2', -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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* 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.SISNAN( AJJ ) ) THEN
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RANK = 0
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INFO = 1
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GO TO 170
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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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SSTOP = N * SLAMCH( 'Epsilon' ) * AJJ
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ELSE
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SSTOP = TOL
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END IF
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*
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* Set first half of WORK to zero, holds dot products
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*
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DO 110 I = 1, N
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WORK( I ) = 0
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110 CONTINUE
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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 130 J = 1, N
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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.1 ) 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.SSTOP.OR.SISNAN( AJJ ) ) THEN
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A( J, J ) = AJJ
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GO TO 160
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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 SSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
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IF( PVT.LT.N )
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$ CALL SSWAP( N-PVT, A( J, PVT+1 ), LDA,
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$ A( PVT, PVT+1 ), LDA )
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CALL SSWAP( PVT-J-1, A( J, J+1 ), LDA, 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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STEMP = WORK( J )
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WORK( J ) = WORK( PVT )
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WORK( PVT ) = STEMP
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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 SGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
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$ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
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CALL SSCAL( 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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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 150 J = 1, N
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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 140 I = J, N
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*
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IF( J.GT.1 ) 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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140 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.SSTOP.OR.SISNAN( AJJ ) ) THEN
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A( J, J ) = AJJ
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GO TO 160
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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 SSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
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IF( PVT.LT.N )
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$ CALL SSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
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$ 1 )
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CALL SSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
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*
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* Swap dot products and PIV
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*
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STEMP = WORK( J )
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WORK( J ) = WORK( PVT )
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WORK( PVT ) = STEMP
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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 SGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ), LDA,
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$ A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
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CALL SSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
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END IF
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*
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150 CONTINUE
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*
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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 170
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160 CONTINUE
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*
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* Rank is 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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170 CONTINUE
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RETURN
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*
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* End of SPSTF2
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*
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END
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