* * Definition: * =========== * * SUBROUTINE SLATSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, * LWORK, INFO) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK * .. * .. Array Arguments .. * REAL A( LDA, * ), T( LDT, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLATSQR computes a blocked Tall-Skinny QR factorization of *> an M-by-N matrix A, where M >= N: *> A = Q * R . *> \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. M >= N >= 0. *> \endverbatim *> *> \param[in] MB *> \verbatim *> MB is INTEGER *> The row block size to be used in the blocked QR. *> MB > N. *> \endverbatim *> *> \param[in] NB *> \verbatim *> NB is INTEGER *> The column block size to be used in the blocked QR. *> N >= NB >= 1. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, the elements on and above the diagonal *> of the array contain the N-by-N upper triangular matrix R; *> the elements below the diagonal represent Q by the columns *> 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 REAL array, *> dimension (LDT, N * Number_of_row_blocks) *> where Number_of_row_blocks = CEIL((M-N)/(MB-N)) *> 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 >= NB. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> (workspace) REAL array, dimension (MAX(1,LWORK)) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> The dimension of the array WORK. LWORK >= NB*N. *> 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 *> Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A: *> Q(1) zeros out the subdiagonal entries of rows 1:MB of A *> Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A *> Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A *> . . . *> *> Q(1) is computed by GEQRT, 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 GEQRT. *> *> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors *> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular *> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). *> 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 SLATSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, \$ LWORK, INFO) * * -- LAPACK computational routine (version 3.7.0) -- * -- 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, MB, NB, LDT, LWORK * .. * .. Array Arguments .. REAL A( LDA, * ), WORK( * ), T(LDT, *) * .. * * ===================================================================== * * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, II, KK, CTR * .. * .. EXTERNAL FUNCTIONS .. LOGICAL LSAME EXTERNAL LSAME * .. EXTERNAL SUBROUTINES .. EXTERNAL SGEQRT, STPQRT, 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. M.LT.N ) THEN INFO = -2 ELSE IF( MB.LE.N ) THEN INFO = -3 ELSE IF( NB.LT.1 .OR. ( NB.GT.N .AND. N.GT.0 )) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDT.LT.NB ) THEN INFO = -8 ELSE IF( LWORK.LT.(N*NB) .AND. (.NOT.LQUERY) ) THEN INFO = -10 END IF IF( INFO.EQ.0) THEN WORK(1) = NB*N END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLATSQR', -INFO ) RETURN ELSE IF (LQUERY) THEN RETURN END IF * * Quick return if possible * IF( MIN(M,N).EQ.0 ) THEN RETURN END IF * * The QR Decomposition * IF ((MB.LE.N).OR.(MB.GE.M)) THEN CALL SGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO) RETURN END IF KK = MOD((M-N),(MB-N)) II=M-KK+1 * * Compute the QR factorization of the first block A(1:MB,1:N) * CALL SGEQRT( MB, N, NB, A(1,1), LDA, T, LDT, WORK, INFO ) * CTR = 1 DO I = MB+1, II-MB+N , (MB-N) * * Compute the QR factorization of the current block A(I:I+MB-N,1:N) * CALL STPQRT( MB-N, N, 0, NB, A(1,1), LDA, A( I, 1 ), LDA, \$ T(1, CTR * N + 1), \$ LDT, WORK, INFO ) CTR = CTR + 1 END DO * * Compute the QR factorization of the last block A(II:M,1:N) * IF (II.LE.M) THEN CALL STPQRT( KK, N, 0, NB, A(1,1), LDA, A( II, 1 ), LDA, \$ T(1, CTR * N + 1), LDT, \$ WORK, INFO ) END IF * work( 1 ) = N*NB RETURN * * End of SLATSQR * END