C> \brief \b SGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SGETRF( M, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* REAL A( LDA, * )
* ..
*
* Purpose
* =======
*
C>\details \b Purpose:
C>\verbatim
C>
C> SGETRF computes an LU factorization of a general M-by-N matrix A
C> using partial pivoting with row interchanges.
C>
C> The factorization has the form
C> A = P * L * U
C> where P is a permutation matrix, L is lower triangular with unit
C> diagonal elements (lower trapezoidal if m > n), and U is upper
C> triangular (upper trapezoidal if m < n).
C>
C> This code implements an iterative version of Sivan Toledo's recursive
C> LU algorithm[1]. For square matrices, this iterative versions should
C> be within a factor of two of the optimum number of memory transfers.
C>
C> The pattern is as follows, with the large blocks of U being updated
C> in one call to STRSM, and the dotted lines denoting sections that
C> have had all pending permutations applied:
C>
C> 1 2 3 4 5 6 7 8
C> +-+-+---+-------+------
C> | |1| | |
C> |.+-+ 2 | |
C> | | | | |
C> |.|.+-+-+ 4 |
C> | | | |1| |
C> | | |.+-+ |
C> | | | | | |
C> |.|.|.|.+-+-+---+ 8
C> | | | | | |1| |
C> | | | | |.+-+ 2 |
C> | | | | | | | |
C> | | | | |.|.+-+-+
C> | | | | | | | |1|
C> | | | | | | |.+-+
C> | | | | | | | | |
C> |.|.|.|.|.|.|.|.+-----
C> | | | | | | | | |
C>
C> The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
C> the binary expansion of the current column. Each Schur update is
C> applied as soon as the necessary portion of U is available.
C>
C> [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
C> Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
C> 1065-1081. http://dx.doi.org/10.1137/S0895479896297744
C>
C>\endverbatim
*
* Arguments:
* ==========
*
C> \param[in] M
C> \verbatim
C> M is INTEGER
C> The number of rows of the matrix A. M >= 0.
C> \endverbatim
C>
C> \param[in] N
C> \verbatim
C> N is INTEGER
C> The number of columns of the matrix A. N >= 0.
C> \endverbatim
C>
C> \param[in,out] A
C> \verbatim
C> A is REAL array, dimension (LDA,N)
C> On entry, the M-by-N matrix to be factored.
C> On exit, the factors L and U from the factorization
C> A = P*L*U; the unit diagonal elements of L are not stored.
C> \endverbatim
C>
C> \param[in] LDA
C> \verbatim
C> LDA is INTEGER
C> The leading dimension of the array A. LDA >= max(1,M).
C> \endverbatim
C>
C> \param[out] IPIV
C> \verbatim
C> IPIV is INTEGER array, dimension (min(M,N))
C> The pivot indices; for 1 <= i <= min(M,N), row i of the
C> matrix was interchanged with row IPIV(i).
C> \endverbatim
C>
C> \param[out] INFO
C> \verbatim
C> INFO is INTEGER
C> = 0: successful exit
C> < 0: if INFO = -i, the i-th argument had an illegal value
C> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
C> has been completed, but the factor U is exactly
C> singular, and division by zero will occur if it is used
C> to solve a system of equations.
C> \endverbatim
C>
*
* Authors:
* ========
*
C> \author Univ. of Tennessee
C> \author Univ. of California Berkeley
C> \author Univ. of Colorado Denver
C> \author NAG Ltd.
*
C> \date December 2016
*
C> \ingroup variantsGEcomputational
*
* =====================================================================
SUBROUTINE SGETRF( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK computational routine (version 3.X) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
REAL A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
PARAMETER ( NEGONE = -1.0E+0 )
* ..
* .. Local Scalars ..
REAL SFMIN, TMP
INTEGER I, J, JP, NSTEP, NTOPIV, NPIVED, KAHEAD
INTEGER KSTART, IPIVSTART, JPIVSTART, KCOLS
* ..
* .. External Functions ..
REAL SLAMCH
INTEGER ISAMAX
LOGICAL SISNAN
EXTERNAL SLAMCH, ISAMAX, SISNAN
* ..
* .. External Subroutines ..
EXTERNAL STRSM, SSCAL, XERBLA, SLASWP
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, IAND
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGETRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Compute machine safe minimum
*
SFMIN = SLAMCH( 'S' )
*
NSTEP = MIN( M, N )
DO J = 1, NSTEP
KAHEAD = IAND( J, -J )
KSTART = J + 1 - KAHEAD
KCOLS = MIN( KAHEAD, M-J )
*
* Find pivot.
*
JP = J - 1 + ISAMAX( M-J+1, A( J, J ), 1 )
IPIV( J ) = JP
! Permute just this column.
IF (JP .NE. J) THEN
TMP = A( J, J )
A( J, J ) = A( JP, J )
A( JP, J ) = TMP
END IF
! Apply pending permutations to L
NTOPIV = 1
IPIVSTART = J
JPIVSTART = J - NTOPIV
DO WHILE ( NTOPIV .LT. KAHEAD )
CALL SLASWP( NTOPIV, A( 1, JPIVSTART ), LDA, IPIVSTART, J,
$ IPIV, 1 )
IPIVSTART = IPIVSTART - NTOPIV;
NTOPIV = NTOPIV * 2;
JPIVSTART = JPIVSTART - NTOPIV;
END DO
! Permute U block to match L
CALL SLASWP( KCOLS, A( 1,J+1 ), LDA, KSTART, J, IPIV, 1 )
! Factor the current column
IF( A( J, J ).NE.ZERO .AND. .NOT.SISNAN( A( J, J ) ) ) THEN
IF( ABS(A( J, J )) .GE. SFMIN ) THEN
CALL SSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 )
ELSE
DO I = 1, M-J
A( J+I, J ) = A( J+I, J ) / A( J, J )
END DO
END IF
ELSE IF( A( J,J ) .EQ. ZERO .AND. INFO .EQ. 0 ) THEN
INFO = J
END IF
! Solve for U block.
CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit', KAHEAD,
$ KCOLS, ONE, A( KSTART, KSTART ), LDA,
$ A( KSTART, J+1 ), LDA )
! Schur complement.
CALL SGEMM( 'No transpose', 'No transpose', M-J,
$ KCOLS, KAHEAD, NEGONE, A( J+1, KSTART ), LDA,
$ A( KSTART, J+1 ), LDA, ONE, A( J+1, J+1 ), LDA )
END DO
! Handle pivot permutations on the way out of the recursion
NPIVED = IAND( NSTEP, -NSTEP )
J = NSTEP - NPIVED
DO WHILE ( J .GT. 0 )
NTOPIV = IAND( J, -J )
CALL SLASWP( NTOPIV, A( 1, J-NTOPIV+1 ), LDA, J+1, NSTEP,
$ IPIV, 1 )
J = J - NTOPIV
END DO
! If short and wide, handle the rest of the columns.
IF ( M .LT. N ) THEN
CALL SLASWP( N-M, A( 1, M+KCOLS+1 ), LDA, 1, M, IPIV, 1 )
CALL STRSM( 'Left', 'Lower', 'No transpose', 'Unit', M,
$ N-M, ONE, A, LDA, A( 1,M+KCOLS+1 ), LDA )
END IF
RETURN
*
* End of SGETRF
*
END