*> \brief \b CLAUNHR_COL_GETRFNP2 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CLAUNHR_COL_GETRFNP2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * RECURSIVE SUBROUTINE CLAUNHR_COL_GETRFNP2( M, N, A, LDA, D, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. * COMPLEX A( LDA, * ), D( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CLAUNHR_COL_GETRFNP2 computes the modified LU factorization without *> pivoting of a complex general M-by-N matrix A. The factorization has *> the form: *> *> A - S = L * U, *> *> where: *> S is a m-by-n diagonal sign matrix with the diagonal D, so that *> D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed *> as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing *> i-1 steps of Gaussian elimination. This means that the diagonal *> element at each step of "modified" Gaussian elimination is at *> least one in absolute value (so that division-by-zero not *> possible during the division by the diagonal element); *> *> L is a M-by-N lower triangular matrix with unit diagonal elements *> (lower trapezoidal if M > N); *> *> and U is a M-by-N upper triangular matrix *> (upper trapezoidal if M < N). *> *> This routine is an auxiliary routine used in the Householder *> reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is *> applied to an M-by-N matrix A with orthonormal columns, where each *> element is bounded by one in absolute value. With the choice of *> the matrix S above, one can show that the diagonal element at each *> step of Gaussian elimination is the largest (in absolute value) in *> the column on or below the diagonal, so that no pivoting is required *> for numerical stability . *> *> For more details on the Householder reconstruction algorithm, *> including the modified LU factorization, see . *> *> This is the recursive version of the LU factorization algorithm. *> Denote A - S by B. The algorithm divides the matrix B into four *> submatrices: *> *> [ B11 | B12 ] where B11 is n1 by n1, *> B = [ -----|----- ] B21 is (m-n1) by n1, *> [ B21 | B22 ] B12 is n1 by n2, *> B22 is (m-n1) by n2, *> with n1 = min(m,n)/2, n2 = n-n1. *> *> *> The subroutine calls itself to factor B11, solves for B21, *> solves for B12, updates B22, then calls itself to factor B22. *> *> For more details on the recursive LU algorithm, see . *> *> CLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked *> routine CLAUNHR_COL_GETRFNP, which uses blocked code calling *> Level 3 BLAS to update the submatrix. However, CLAUNHR_COL_GETRFNP2 *> is self-sufficient and can be used without CLAUNHR_COL_GETRFNP. *> *>  "Reconstructing Householder vectors from tall-skinny QR", *> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, *> E. Solomonik, J. Parallel Distrib. Comput., *> vol. 85, pp. 3-31, 2015. *> *>  "Recursion leads to automatic variable blocking for dense linear *> algebra algorithms", F. Gustavson, IBM J. of Res. and Dev., *> vol. 41, no. 6, pp. 737-755, 1997. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX array, dimension (LDA,N) *> On entry, the M-by-N matrix to be factored. *> On exit, the factors L and U from the factorization *> A-S=L*U; the unit diagonal elements of L are not stored. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[out] D *> \verbatim *> D is COMPLEX array, dimension min(M,N) *> The diagonal elements of the diagonal M-by-N sign matrix S, *> D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be *> only ( +1.0, 0.0 ) or (-1.0, 0.0 ). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim *> * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complexGEcomputational * *> \par Contributors: * ================== *> *> \verbatim *> *> November 2019, Igor Kozachenko, *> Computer Science Division, *> University of California, Berkeley *> *> \endverbatim * * ===================================================================== RECURSIVE SUBROUTINE CLAUNHR_COL_GETRFNP2( M, N, A, LDA, D, INFO ) IMPLICIT NONE * * -- LAPACK computational routine -- * -- 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 .. COMPLEX A( LDA, * ), D( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE PARAMETER ( ONE = 1.0E+0 ) COMPLEX CONE PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. REAL SFMIN INTEGER I, IINFO, N1, N2 COMPLEX Z * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL CGEMM, CSCAL, CTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, REAL, CMPLX, AIMAG, SIGN, MAX, MIN * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) ) * .. * .. 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( 'CLAUNHR_COL_GETRFNP2', -INFO ) RETURN END IF * * Quick return if possible * IF( MIN( M, N ).EQ.0 ) \$ RETURN IF ( M.EQ.1 ) THEN * * One row case, (also recursion termination case), * use unblocked code * * Transfer the sign * D( 1 ) = CMPLX( -SIGN( ONE, REAL( A( 1, 1 ) ) ) ) * * Construct the row of U * A( 1, 1 ) = A( 1, 1 ) - D( 1 ) * ELSE IF( N.EQ.1 ) THEN * * One column case, (also recursion termination case), * use unblocked code * * Transfer the sign * D( 1 ) = CMPLX( -SIGN( ONE, REAL( A( 1, 1 ) ) ) ) * * Construct the row of U * A( 1, 1 ) = A( 1, 1 ) - D( 1 ) * * Scale the elements 2:M of the column * * Determine machine safe minimum * SFMIN = SLAMCH('S') * * Construct the subdiagonal elements of L * IF( CABS1( A( 1, 1 ) ) .GE. SFMIN ) THEN CALL CSCAL( M-1, CONE / A( 1, 1 ), A( 2, 1 ), 1 ) ELSE DO I = 2, M A( I, 1 ) = A( I, 1 ) / A( 1, 1 ) END DO END IF * ELSE * * Divide the matrix B into four submatrices * N1 = MIN( M, N ) / 2 N2 = N-N1 * * Factor B11, recursive call * CALL CLAUNHR_COL_GETRFNP2( N1, N1, A, LDA, D, IINFO ) * * Solve for B21 * CALL CTRSM( 'R', 'U', 'N', 'N', M-N1, N1, CONE, A, LDA, \$ A( N1+1, 1 ), LDA ) * * Solve for B12 * CALL CTRSM( 'L', 'L', 'N', 'U', N1, N2, CONE, A, LDA, \$ A( 1, N1+1 ), LDA ) * * Update B22, i.e. compute the Schur complement * B22 := B22 - B21*B12 * CALL CGEMM( 'N', 'N', M-N1, N2, N1, -CONE, A( N1+1, 1 ), LDA, \$ A( 1, N1+1 ), LDA, CONE, A( N1+1, N1+1 ), LDA ) * * Factor B22, recursive call * CALL CLAUNHR_COL_GETRFNP2( M-N1, N2, A( N1+1, N1+1 ), LDA, \$ D( N1+1 ), IINFO ) * END IF RETURN * * End of CLAUNHR_COL_GETRFNP2 * END