*> \brief \b SORGTSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE SORGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
* $ INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
*> which are the first N columns of a product of real orthogonal
*> matrices of order M which are returned by SLATSQR
*>
*> Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
*>
*> See the documentation for SLATSQR.
*> \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 used by SLATSQR to return
*> arrays A and T. MB > N.
*> (Note that if MB > M, then M is used instead of MB
*> as the row block size).
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The column block size used by SLATSQR to return
*> arrays A and T. NB >= 1.
*> (Note that if NB > N, then N is used instead of NB
*> as the column block size).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*>
*> On entry:
*>
*> The elements on and above the diagonal are not accessed.
*> The elements below the diagonal represent the unit
*> lower-trapezoidal blocked matrix V computed by SLATSQR
*> that defines the input matrices Q_in(k) (ones on the
*> diagonal are not stored) (same format as the output A
*> below the diagonal in SLATSQR).
*>
*> On exit:
*>
*> The array A contains an M-by-N orthonormal matrix Q_out,
*> i.e the columns of A are orthogonal unit vectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is REAL array,
*> dimension (LDT, N * NIRB)
*> where NIRB = Number_of_input_row_blocks
*> = MAX( 1, CEIL((M-N)/(MB-N)) )
*> Let NICB = Number_of_input_col_blocks
*> = CEIL(N/NB)
*>
*> The upper-triangular block reflectors used to define the
*> input matrices Q_in(k), k=(1:NIRB*NICB). The block
*> reflectors are stored in compact form in NIRB block
*> reflector sequences. Each of NIRB block reflector sequences
*> is stored in a larger NB-by-N column block of T and consists
*> of NICB smaller NB-by-NB upper-triangular column blocks.
*> (same format as the output T in SLATSQR).
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB1,N)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> (workspace) REAL array, dimension (MAX(2,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> The dimension of the array WORK. LWORK >= (M+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.
*
*> \ingroup singleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE SORGTSQR( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
$ INFO )
IMPLICIT NONE
*
* -- 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, LDT, LWORK, M, N, MB, NB
* ..
* .. Array Arguments ..
REAL A( LDA, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER IINFO, LDC, LWORKOPT, LC, LW, NBLOCAL, J
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAMTSQR, SLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC REAL, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
LQUERY = LWORK.EQ.-1
INFO = 0
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 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -8
ELSE
*
* Test the input LWORK for the dimension of the array WORK.
* This workspace is used to store array C(LDC, N) and WORK(LWORK)
* in the call to DLAMTSQR. See the documentation for DLAMTSQR.
*
IF( LWORK.LT.2 .AND. (.NOT.LQUERY) ) THEN
INFO = -10
ELSE
*
* Set block size for column blocks
*
NBLOCAL = MIN( NB, N )
*
* LWORK = -1, then set the size for the array C(LDC,N)
* in DLAMTSQR call and set the optimal size of the work array
* WORK(LWORK) in DLAMTSQR call.
*
LDC = M
LC = LDC*N
LW = N * NBLOCAL
*
LWORKOPT = LC+LW
*
IF( ( LWORK.LT.MAX( 1, LWORKOPT ) ).AND.(.NOT.LQUERY) ) THEN
INFO = -10
END IF
END IF
*
END IF
*
* Handle error in the input parameters and return workspace query.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORGTSQR', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
WORK( 1 ) = REAL( LWORKOPT )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
WORK( 1 ) = REAL( LWORKOPT )
RETURN
END IF
*
* (1) Form explicitly the tall-skinny M-by-N left submatrix Q1_in
* of M-by-M orthogonal matrix Q_in, which is implicitly stored in
* the subdiagonal part of input array A and in the input array T.
* Perform by the following operation using the routine DLAMTSQR.
*
* Q1_in = Q_in * ( I ), where I is a N-by-N identity matrix,
* ( 0 ) 0 is a (M-N)-by-N zero matrix.
*
* (1a) Form M-by-N matrix in the array WORK(1:LDC*N) with ones
* on the diagonal and zeros elsewhere.
*
CALL SLASET( 'F', M, N, ZERO, ONE, WORK, LDC )
*
* (1b) On input, WORK(1:LDC*N) stores ( I );
* ( 0 )
*
* On output, WORK(1:LDC*N) stores Q1_in.
*
CALL SLAMTSQR( 'L', 'N', M, N, N, MB, NBLOCAL, A, LDA, T, LDT,
$ WORK, LDC, WORK( LC+1 ), LW, IINFO )
*
* (2) Copy the result from the part of the work array (1:M,1:N)
* with the leading dimension LDC that starts at WORK(1) into
* the output array A(1:M,1:N) column-by-column.
*
DO J = 1, N
CALL SCOPY( M, WORK( (J-1)*LDC + 1 ), 1, A( 1, J ), 1 )
END DO
*
WORK( 1 ) = REAL( LWORKOPT )
RETURN
*
* End of SORGTSQR
*
END