*> \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))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> 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 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
* ..
* .. 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 )
*
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.(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( 'ZLATSQR', -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 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 ) = N*NB
RETURN
*
* End of ZLATSQR
*
END