*> \brief \b ZLAUNHR_COL_GETRFNP * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLAUNHR_COL_GETRFNP + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLAUNHR_COL_GETRFNP( M, N, A, LDA, D, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), D( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLAUNHR_COL_GETRFNP 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 *> 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 ZUNHR_COL. In ZUNHR_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 blocked right-looking version of the algorithm, *> calling Level 3 BLAS to update the submatrix. To factorize a block, *> this routine calls the recursive routine ZLAUNHR_COL_GETRFNP2. *> *>  "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. *> \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*16 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*16 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 complex16GEcomputational * *> \par Contributors: * ================== *> *> \verbatim *> *> November 2019, Igor Kozachenko, *> Computer Science Division, *> University of California, Berkeley *> *> \endverbatim * * ===================================================================== SUBROUTINE ZLAUNHR_COL_GETRFNP( 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*16 A( LDA, * ), D( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER IINFO, J, JB, NB * .. * .. External Subroutines .. EXTERNAL ZGEMM, ZLAUNHR_COL_GETRFNP2, ZTRSM, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. 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( 'ZLAUNHR_COL_GETRFNP', -INFO ) RETURN END IF * * Quick return if possible * IF( MIN( M, N ).EQ.0 ) \$ RETURN * * Determine the block size for this environment. * NB = ILAENV( 1, 'ZLAUNHR_COL_GETRFNP', ' ', M, N, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN * * Use unblocked code. * CALL ZLAUNHR_COL_GETRFNP2( M, N, A, LDA, D, INFO ) ELSE * * Use blocked code. * DO J = 1, MIN( M, N ), NB JB = MIN( MIN( M, N )-J+1, NB ) * * Factor diagonal and subdiagonal blocks. * CALL ZLAUNHR_COL_GETRFNP2( M-J+1, JB, A( J, J ), LDA, \$ D( J ), IINFO ) * IF( J+JB.LE.N ) THEN * * Compute block row of U. * CALL ZTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB, \$ N-J-JB+1, CONE, A( J, J ), LDA, A( J, J+JB ), \$ LDA ) IF( J+JB.LE.M ) THEN * * Update trailing submatrix. * CALL ZGEMM( 'No transpose', 'No transpose', M-J-JB+1, \$ N-J-JB+1, JB, -CONE, A( J+JB, J ), LDA, \$ A( J, J+JB ), LDA, CONE, A( J+JB, J+JB ), \$ LDA ) END IF END IF END DO END IF RETURN * * End of ZLAUNHR_COL_GETRFNP * END