zlatsqr.f

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*> \brief \b ZLATSQR
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLATSQR( 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
*>
*> ZLATSQR computes a blocked Tall-Skinny QR factorization of
*> a complex M-by-N matrix A for M >= N:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
*>    form in the elements below the diagonal of the array A and in
*>    the elements of the array T;
*>
*>    R is an upper-triangular N-by-N matrix, stored on exit in
*>    the elements on and above the diagonal of the array A.
*>
*>    0 is a (M-N)-by-N zero matrix, and is not stored.
*>
*> \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 COMPLEX*16 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 COMPLEX*16 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) COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, 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.
*> \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
*>
*> \ingroup latsqr
*>
*  =====================================================================
      SUBROUTINE zlatsqr( M, N, MB, NB, A, LDA, T, LDT, WORK,
     $                    LWORK, INFO )
*
*  -- 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, LDT, LWORK
*     ..
*     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), WORK( * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, II, KK, CTR, LWMIN, MINMN
*     ..
*     .. EXTERNAL FUNCTIONS ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. EXTERNAL SUBROUTINES ..
      EXTERNAL           zgeqrt, ztpqrt, 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 = n*nb
      END IF
*
      IF( m.LT.0 ) THEN
        info = -1
      ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
        info = -2
      ELSE IF( mb.LT.1 ) 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 = -6
      ELSE IF( ldt.LT.nb ) 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( 'ZLATSQR', -info )
        RETURN
      ELSE IF( lquery ) THEN
        RETURN
      END IF
*
*     Quick return if possible
*
      IF( minmn.EQ.0 ) THEN
        RETURN
      END IF
*
*     The QR Decomposition
*
      IF( (mb.LE.n) .OR. (mb.GE.m) ) THEN
        CALL zgeqrt( 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 zgeqrt( 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 ztpqrt( 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 ztpqrt( kk, n, 0, nb, a(1,1), lda, a( ii, 1 ), lda,
     $                t(1,ctr * n + 1), ldt,
     $                work, info )
      END IF
*
      work( 1 ) = lwmin
      RETURN
*
*     End of ZLATSQR
*
      SUBROUTINE zlatsqr( M, N, MB, NB, A, LDA, T, LDT, WORK, …
      END