dlaswlq()

subroutine dlaswlq	(	integer	m,
		integer	n,
		integer	mb,
		integer	nb,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( ldt, * )	t,
		integer	ldt,
		double precision, dimension( * )	work,
		integer	lwork,
		integer	info )

DLASWLQ

Purpose:

!>
!> DLASWLQ computes a blocked Tall-Skinny LQ factorization of
!> a real M-by-N matrix A for M <= N:
!>
!>    A = ( L 0 ) *  Q,
!>
!> where:
!>
!>    Q is a n-by-N orthogonal matrix, stored on exit in an implicit
!>    form in the elements above the diagonal of the array A and in
!>    the elements of the array T;
!>    L is a lower-triangular M-by-M matrix stored on exit in
!>    the elements on and below the diagonal of the array A.
!>    0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.
!>
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. N >= M >= 0. !>
[in]	MB	!> MB is INTEGER !> The row block size to be used in the blocked QR. !> M >= MB >= 1 !>
[in]	NB	!> NB is INTEGER !> The column block size to be used in the blocked QR. !> NB > 0. !>
[in,out]	A	!> A is DOUBLE PRECISION 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). !> !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[out]	T	!> T is DOUBLE PRECISION 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. !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of the array T. LDT >= MB. !>
[out]	WORK	!> (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the minimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. !> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the minimal 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. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!> 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
!> 

Definition at line 168 of file dlaswlq.f.

*
*  -- 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, MB, NB, LWORK, LDT
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, II, KK, CTR, MINMN, LWMIN
*     ..
*     .. EXTERNAL FUNCTIONS ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. EXTERNAL SUBROUTINES ..
      EXTERNAL           dgelqt, dtplqt, xerbla
*     ..
*     .. INTRINSIC FUNCTIONS ..
      INTRINSIC          max, min, mod
*     ..
*     .. EXECUTABLE STATEMENTS ..
*
*     TEST THE INPUT ARGUMENTS
*
      info = 0
*
      lquery = ( lwork.EQ.-1 )
*
      minmn = min( m, n )
      IF( minmn.EQ.0 ) THEN
         lwmin = 1
      ELSE
         lwmin = m*mb
      END IF
*
      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.LT.0 ) THEN
        info = -4
      ELSE IF( lda.LT.max( 1, m ) ) THEN
        info = -6
      ELSE IF( ldt.LT.mb ) THEN
        info = -8
      ELSE IF( lwork.LT.lwmin .AND. (.NOT.lquery) ) THEN
        info = -10
      END IF
*
      IF( info.EQ.0 ) THEN
        work( 1 ) = lwmin
      END IF
*
      IF( info.NE.0 ) THEN
        CALL xerbla( 'DLASWLQ', -info )
        RETURN
      ELSE IF( lquery ) THEN
        RETURN
      END IF
*
*     Quick return if possible
*
      IF( minmn.EQ.0 ) THEN
        RETURN
      END IF
*
*     The LQ Decomposition
*
      IF( (m.GE.n) .OR. (nb.LE.m) .OR. (nb.GE.n) ) THEN
        CALL dgelqt( 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 dgelqt( 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 dtplqt( 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 dtplqt( m, kk, 0, mb, a(1,1), lda, a( 1, ii ),
     $                 lda, t(1, ctr * m + 1), ldt,
     $                 work, info )
      END IF
*
      work( 1 ) = lwmin
*
      RETURN
*
*     End of DLASWLQ
*

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