* * Definition: * =========== * * SUBROUTINE ZLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK, * LWORK, INFO) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), T( LDT, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLASWLQ computes a blocked Short-Wide LQ factorization of a *> M-by-N matrix A, where N >= M: *> A = L * Q *> \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 >= M >= 0. *> \endverbatim *> *> \param[in] MB *> \verbatim *> MB is INTEGER *> The row block size to be used in the blocked QR. *> M >= MB >= 1 *> \endverbatim *> \param[in] NB *> \verbatim *> NB is INTEGER *> The column block size to be used in the blocked QR. *> NB > M. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, the elements on and below the diagonal *> of the array contain the N-by-N lower triangular matrix L; *> the elements above the diagonal represent Q by the rows *> of blocked V (see Further Details). *> *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[out] T *> \verbatim *> T is COMPLEX*16 array, *> dimension (LDT, N * Number_of_row_blocks) *> where Number_of_row_blocks = CEIL((N-M)/(NB-M)) *> The blocked upper triangular block reflectors stored in compact form *> as a sequence of upper triangular blocks. *> See Further Details below. *> \endverbatim *> *> \param[in] LDT *> \verbatim *> LDT is INTEGER *> The leading dimension of the array T. LDT >= MB. *> \endverbatim *> *> *> \param[out] WORK *> \verbatim *> (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) *> *> \endverbatim *> \param[in] LWORK *> \verbatim *> The dimension of the array WORK. LWORK >= MB*M. *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> *> \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. * *> \par Further Details: * ===================== *> *> \verbatim *> Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations, *> representing Q as a product of other orthogonal matrices *> Q = Q(1) * Q(2) * . . . * Q(k) *> where each Q(i) zeros out upper diagonal entries of a block of NB rows of A: *> Q(1) zeros out the upper diagonal entries of rows 1:NB of A *> Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A *> Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A *> . . . *> *> Q(1) is computed by GELQT, which represents Q(1) by Householder vectors *> stored under the diagonal of rows 1:MB of A, and by upper triangular *> block reflectors, stored in array T(1:LDT,1:N). *> For more information see Further Details in GELQT. *> *> Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors *> stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular *> block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M). *> The last Q(k) may use fewer rows. *> For more information see Further Details in TPQRT. *> *> For more details of the overall algorithm, see the description of *> Sequential TSQR in Section 2.2 of [1]. *> *> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” *> J. Demmel, L. Grigori, M. Hoemmen, J. Langou, *> SIAM J. Sci. Comput, vol. 34, no. 1, 2012 *> \endverbatim *> * ===================================================================== SUBROUTINE ZLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, $ INFO) * * -- LAPACK computational routine (version 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. -- * June 2017 * * .. Scalar Arguments .. INTEGER INFO, LDA, M, N, MB, NB, LWORK, LDT * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), WORK( * ), T( LDT, *) * .. * * ===================================================================== * * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, II, KK, CTR * .. * .. EXTERNAL FUNCTIONS .. LOGICAL LSAME EXTERNAL LSAME * .. EXTERNAL SUBROUTINES .. EXTERNAL ZGELQT, ZTPLQT, XERBLA * .. INTRINSIC FUNCTIONS .. INTRINSIC MAX, MIN, MOD * .. * .. EXECUTABLE STATEMENTS .. * * TEST THE INPUT ARGUMENTS * INFO = 0 * LQUERY = ( LWORK.EQ.-1 ) * IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.LT.M ) THEN INFO = -2 ELSE IF( MB.LT.1 .OR. ( MB.GT.M .AND. M.GT.0 )) THEN INFO = -3 ELSE IF( NB.LE.M ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDT.LT.MB ) THEN INFO = -8 ELSE IF( ( LWORK.LT.M*MB) .AND. (.NOT.LQUERY) ) THEN INFO = -10 END IF IF( INFO.EQ.0) THEN WORK(1) = MB*M END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZLASWLQ', -INFO ) RETURN ELSE IF (LQUERY) THEN RETURN END IF * * Quick return if possible * IF( MIN(M,N).EQ.0 ) THEN RETURN END IF * * The LQ Decomposition * IF((M.GE.N).OR.(NB.LE.M).OR.(NB.GE.N)) THEN CALL ZGELQT( M, N, MB, A, LDA, T, LDT, WORK, INFO) RETURN END IF * KK = MOD((N-M),(NB-M)) II=N-KK+1 * * Compute the LQ factorization of the first block A(1:M,1:NB) * CALL ZGELQT( M, NB, MB, A(1,1), LDA, T, LDT, WORK, INFO) CTR = 1 * DO I = NB+1, II-NB+M , (NB-M) * * Compute the QR factorization of the current block A(1:M,I:I+NB-M) * CALL ZTPLQT( M, NB-M, 0, MB, A(1,1), LDA, A( 1, I ), $ LDA, T(1, CTR * M + 1), $ LDT, WORK, INFO ) CTR = CTR + 1 END DO * * Compute the QR factorization of the last block A(1:M,II:N) * IF (II.LE.N) THEN CALL ZTPLQT( M, KK, 0, MB, A(1,1), LDA, A( 1, II ), $ LDA, T(1, CTR * M + 1), LDT, $ WORK, INFO ) END IF * WORK( 1 ) = M * MB RETURN * * End of ZLASWLQ * END