numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
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| lapack/SRC/VARIANTS/lu/REC/zgetrf.f | 8236B | -rw-r--r-- |
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C> \brief \b ZGETRF 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 ZGETRF( M, N, A, LDA, IPIV, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 A( LDA, * ) * .. * * Purpose * ======= * C>\details \b Purpose: C>\verbatim C> C> ZGETRF 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 DTRSM, 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 COMPLEX*16 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 ZGETRF( 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..-- * * .. Scalar Arguments .. INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 A( LDA, * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 ONE, NEGONE DOUBLE PRECISION ZERO PARAMETER ( ONE = (1.0D+0, 0.0D+0) ) PARAMETER ( NEGONE = (-1.0D+0, 0.0D+0) ) PARAMETER ( ZERO = 0.0D+0 ) * .. * .. Local Scalars .. DOUBLE PRECISION SFMIN, PIVMAG COMPLEX*16 TMP INTEGER I, J, JP, NSTEP, NTOPIV, NPIVED, KAHEAD INTEGER KSTART, IPIVSTART, JPIVSTART, KCOLS * .. * .. External Functions .. DOUBLE PRECISION DLAMCH INTEGER IZAMAX LOGICAL DISNAN EXTERNAL DLAMCH, IZAMAX, DISNAN * .. * .. External Subroutines .. EXTERNAL ZTRSM, ZSCAL, XERBLA, ZLASWP * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, IAND, ABS * .. * .. 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( 'ZGETRF', -INFO ) RETURN END IF * * Quick return if possible * IF( M.EQ.0 .OR. N.EQ.0 ) $ RETURN * * Compute machine safe minimum * SFMIN = DLAMCH( '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 + IZAMAX( 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 ZLASWP( 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 ZLASWP( KCOLS, A( 1,J+1 ), LDA, KSTART, J, IPIV, 1 ) ! Factor the current column PIVMAG = ABS( A( J, J ) ) IF( PIVMAG.NE.ZERO .AND. .NOT.DISNAN( PIVMAG ) ) THEN IF( PIVMAG .GE. SFMIN ) THEN CALL ZSCAL( 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( PIVMAG .EQ. ZERO .AND. INFO .EQ. 0 ) THEN INFO = J END IF ! Solve for U block. CALL ZTRSM( 'Left', 'Lower', 'No transpose', 'Unit', KAHEAD, $ KCOLS, ONE, A( KSTART, KSTART ), LDA, $ A( KSTART, J+1 ), LDA ) ! Schur complement. CALL ZGEMM( '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 ZLASWP( 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 ZLASWP( N-M, A( 1, M+KCOLS+1 ), LDA, 1, M, IPIV, 1 ) CALL ZTRSM( 'Left', 'Lower', 'No transpose', 'Unit', M, $ N-M, ONE, A, LDA, A( 1,M+KCOLS+1 ), LDA ) END IF RETURN * * End of ZGETRF * END