278 lines
7.9 KiB
Fortran
278 lines
7.9 KiB
Fortran
C> \brief \b SGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm
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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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* Definition:
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* ===========
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*
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* SUBROUTINE SGETRF( M, N, A, LDA, IPIV, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* REAL A( LDA, * )
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* ..
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*
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* Purpose
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* =======
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*
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C>\details \b Purpose:
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C>\verbatim
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C>
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C> SGETRF computes an LU factorization of a general M-by-N matrix A
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C> using partial pivoting with row interchanges.
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C>
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C> The factorization has the form
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C> A = P * L * U
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C> where P is a permutation matrix, L is lower triangular with unit
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C> diagonal elements (lower trapezoidal if m > n), and U is upper
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C> triangular (upper trapezoidal if m < n).
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C>
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C> This code implements an iterative version of Sivan Toledo's recursive
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C> LU algorithm[1]. For square matrices, this iterative versions should
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C> be within a factor of two of the optimum number of memory transfers.
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C>
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C> The pattern is as follows, with the large blocks of U being updated
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C> in one call to STRSM, and the dotted lines denoting sections that
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C> have had all pending permutations applied:
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C>
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C> 1 2 3 4 5 6 7 8
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C> +-+-+---+-------+------
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C> | |1| | |
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C> |.+-+ 2 | |
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C> | | | | |
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C> |.|.+-+-+ 4 |
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C> | | | |1| |
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C> | | |.+-+ |
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C> | | | | | |
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C> |.|.|.|.+-+-+---+ 8
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C> | | | | | |1| |
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C> | | | | |.+-+ 2 |
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C> | | | | | | | |
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C> | | | | |.|.+-+-+
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C> | | | | | | | |1|
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C> | | | | | | |.+-+
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C> | | | | | | | | |
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C> |.|.|.|.|.|.|.|.+-----
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C> | | | | | | | | |
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C>
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C> The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
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C> the binary expansion of the current column. Each Schur update is
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C> applied as soon as the necessary portion of U is available.
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C>
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C> [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
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C> Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
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C> 1065-1081. http://dx.doi.org/10.1137/S0895479896297744
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C>
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C>\endverbatim
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*
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* Arguments:
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* ==========
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*
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C> \param[in] M
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C> \verbatim
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C> M is INTEGER
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C> The number of rows of the matrix A. M >= 0.
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C> \endverbatim
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C>
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C> \param[in] N
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C> \verbatim
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C> N is INTEGER
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C> The number of columns of the matrix A. N >= 0.
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C> \endverbatim
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C>
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C> \param[in,out] A
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C> \verbatim
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C> A is REAL array, dimension (LDA,N)
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C> On entry, the M-by-N matrix to be factored.
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C> On exit, the factors L and U from the factorization
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C> A = P*L*U; the unit diagonal elements of L are not stored.
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C> \endverbatim
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C>
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C> \param[in] LDA
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C> \verbatim
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C> LDA is INTEGER
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C> The leading dimension of the array A. LDA >= max(1,M).
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C> \endverbatim
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C>
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C> \param[out] IPIV
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C> \verbatim
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C> IPIV is INTEGER array, dimension (min(M,N))
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C> The pivot indices; for 1 <= i <= min(M,N), row i of the
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C> matrix was interchanged with row IPIV(i).
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C> \endverbatim
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C>
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C> \param[out] INFO
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C> \verbatim
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C> INFO is INTEGER
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C> = 0: successful exit
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C> < 0: if INFO = -i, the i-th argument had an illegal value
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C> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
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C> has been completed, but the factor U is exactly
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C> singular, and division by zero will occur if it is used
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C> to solve a system of equations.
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C> \endverbatim
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C>
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*
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* Authors:
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* ========
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*
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C> \author Univ. of Tennessee
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C> \author Univ. of California Berkeley
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C> \author Univ. of Colorado Denver
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C> \author NAG Ltd.
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*
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C> \date November 2011
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*
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C> \ingroup variantsGEcomputational
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*
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* =====================================================================
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SUBROUTINE SGETRF( M, N, A, LDA, IPIV, INFO )
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*
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* -- LAPACK computational routine (version 3.X) --
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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 2011
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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REAL A( LDA, * )
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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, NEGONE
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PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
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PARAMETER ( NEGONE = -1.0E+0 )
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* ..
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* .. Local Scalars ..
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REAL SFMIN, TMP
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INTEGER I, J, JP, NSTEP, NTOPIV, NPIVED, KAHEAD
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INTEGER KSTART, IPIVSTART, JPIVSTART, KCOLS
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* ..
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* .. External Functions ..
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REAL SLAMCH
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INTEGER ISAMAX
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LOGICAL SISNAN
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EXTERNAL SLAMCH, ISAMAX, SISNAN
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* ..
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* .. External Subroutines ..
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EXTERNAL STRSM, SSCAL, XERBLA, SLASWP
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, IAND
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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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IF( M.LT.0 ) 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, M ) ) 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( 'SGETRF', -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( M.EQ.0 .OR. N.EQ.0 )
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$ RETURN
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*
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* Compute machine safe minimum
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*
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SFMIN = SLAMCH( 'S' )
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*
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NSTEP = MIN( M, N )
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DO J = 1, NSTEP
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KAHEAD = IAND( J, -J )
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KSTART = J + 1 - KAHEAD
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KCOLS = MIN( KAHEAD, M-J )
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*
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* Find pivot.
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*
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JP = J - 1 + ISAMAX( M-J+1, A( J, J ), 1 )
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IPIV( J ) = JP
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! Permute just this column.
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IF (JP .NE. J) THEN
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TMP = A( J, J )
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A( J, J ) = A( JP, J )
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A( JP, J ) = TMP
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END IF
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! Apply pending permutations to L
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NTOPIV = 1
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IPIVSTART = J
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JPIVSTART = J - NTOPIV
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DO WHILE ( NTOPIV .LT. KAHEAD )
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CALL SLASWP( NTOPIV, A( 1, JPIVSTART ), LDA, IPIVSTART, J,
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$ IPIV, 1 )
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IPIVSTART = IPIVSTART - NTOPIV;
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NTOPIV = NTOPIV * 2;
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JPIVSTART = JPIVSTART - NTOPIV;
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END DO
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! Permute U block to match L
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CALL SLASWP( KCOLS, A( 1,J+1 ), LDA, KSTART, J, IPIV, 1 )
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! Factor the current column
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IF( A( J, J ).NE.ZERO .AND. .NOT.SISNAN( A( J, J ) ) ) THEN
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IF( ABS(A( J, J )) .GE. SFMIN ) THEN
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CALL SSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 )
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ELSE
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DO I = 1, M-J
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A( J+I, J ) = A( J+I, J ) / A( J, J )
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END DO
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END IF
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ELSE IF( A( J,J ) .EQ. ZERO .AND. INFO .EQ. 0 ) THEN
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INFO = J
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END IF
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! Solve for U block.
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CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit', KAHEAD,
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$ KCOLS, ONE, A( KSTART, KSTART ), LDA,
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$ A( KSTART, J+1 ), LDA )
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! Schur complement.
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CALL SGEMM( 'No transpose', 'No transpose', M-J,
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$ KCOLS, KAHEAD, NEGONE, A( J+1, KSTART ), LDA,
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$ A( KSTART, J+1 ), LDA, ONE, A( J+1, J+1 ), LDA )
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END DO
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! Handle pivot permutations on the way out of the recursion
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NPIVED = IAND( NSTEP, -NSTEP )
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J = NSTEP - NPIVED
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DO WHILE ( J .GT. 0 )
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NTOPIV = IAND( J, -J )
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CALL SLASWP( NTOPIV, A( 1, J-NTOPIV+1 ), LDA, J+1, NSTEP,
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$ IPIV, 1 )
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J = J - NTOPIV
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END DO
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! If short and wide, handle the rest of the columns.
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IF ( M .LT. N ) THEN
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CALL SLASWP( N-M, A( 1, M+KCOLS+1 ), LDA, 1, M, IPIV, 1 )
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CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit', M,
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$ N-M, ONE, A, LDA, A( 1,M+KCOLS+1 ), LDA )
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END IF
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RETURN
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*
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* End of SGETRF
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*
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END
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