1*> \brief \b DORGTSQR_ROW
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DORGTSQR_ROW + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgtsqr_row.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgtsqr_row.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgtsqr_row.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DORGTSQR_ROW( M, N, MB, NB, A, LDA, T, LDT, WORK,
22*      $                         LWORK, INFO )
23*       IMPLICIT NONE
24*
25*       .. Scalar Arguments ..
26*       INTEGER           INFO, LDA, LDT, LWORK, M, N, MB, NB
27*       ..
28*       .. Array Arguments ..
29*       DOUBLE PRECISION  A( LDA, * ), T( LDT, * ), WORK( * )
30*       ..
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*> DORGTSQR_ROW generates an M-by-N real matrix Q_out with
38*> orthonormal columns from the output of DLATSQR. These N orthonormal
39*> columns are the first N columns of a product of complex unitary
40*> matrices Q(k)_in of order M, which are returned by DLATSQR in
41*> a special format.
42*>
43*>      Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
44*>
45*> The input matrices Q(k)_in are stored in row and column blocks in A.
46*> See the documentation of DLATSQR for more details on the format of
47*> Q(k)_in, where each Q(k)_in is represented by block Householder
48*> transformations. This routine calls an auxiliary routine DLARFB_GETT,
49*> where the computation is performed on each individual block. The
50*> algorithm first sweeps NB-sized column blocks from the right to left
51*> starting in the bottom row block and continues to the top row block
52*> (hence _ROW in the routine name). This sweep is in reverse order of
53*> the order in which DLATSQR generates the output blocks.
54*> \endverbatim
55*
56*  Arguments:
57*  ==========
58*
59*> \param[in] M
60*> \verbatim
61*>          M is INTEGER
62*>          The number of rows of the matrix A.  M >= 0.
63*> \endverbatim
64*>
65*> \param[in] N
66*> \verbatim
67*>          N is INTEGER
68*>          The number of columns of the matrix A. M >= N >= 0.
69*> \endverbatim
70*>
71*> \param[in] MB
72*> \verbatim
73*>          MB is INTEGER
74*>          The row block size used by DLATSQR to return
75*>          arrays A and T. MB > N.
76*>          (Note that if MB > M, then M is used instead of MB
77*>          as the row block size).
78*> \endverbatim
79*>
80*> \param[in] NB
81*> \verbatim
82*>          NB is INTEGER
83*>          The column block size used by DLATSQR to return
84*>          arrays A and T. NB >= 1.
85*>          (Note that if NB > N, then N is used instead of NB
86*>          as the column block size).
87*> \endverbatim
88*>
89*> \param[in,out] A
90*> \verbatim
91*>          A is DOUBLE PRECISION array, dimension (LDA,N)
92*>
93*>          On entry:
94*>
95*>             The elements on and above the diagonal are not used as
96*>             input. The elements below the diagonal represent the unit
97*>             lower-trapezoidal blocked matrix V computed by DLATSQR
98*>             that defines the input matrices Q_in(k) (ones on the
99*>             diagonal are not stored). See DLATSQR for more details.
100*>
101*>          On exit:
102*>
103*>             The array A contains an M-by-N orthonormal matrix Q_out,
104*>             i.e the columns of A are orthogonal unit vectors.
105*> \endverbatim
106*>
107*> \param[in] LDA
108*> \verbatim
109*>          LDA is INTEGER
110*>          The leading dimension of the array A.  LDA >= max(1,M).
111*> \endverbatim
112*>
113*> \param[in] T
114*> \verbatim
115*>          T is DOUBLE PRECISION array,
116*>          dimension (LDT, N * NIRB)
117*>          where NIRB = Number_of_input_row_blocks
118*>                     = MAX( 1, CEIL((M-N)/(MB-N)) )
119*>          Let NICB = Number_of_input_col_blocks
120*>                   = CEIL(N/NB)
121*>
122*>          The upper-triangular block reflectors used to define the
123*>          input matrices Q_in(k), k=(1:NIRB*NICB). The block
124*>          reflectors are stored in compact form in NIRB block
125*>          reflector sequences. Each of the NIRB block reflector
126*>          sequences is stored in a larger NB-by-N column block of T
127*>          and consists of NICB smaller NB-by-NB upper-triangular
128*>          column blocks. See DLATSQR for more details on the format
129*>          of T.
130*> \endverbatim
131*>
132*> \param[in] LDT
133*> \verbatim
134*>          LDT is INTEGER
135*>          The leading dimension of the array T.
136*>          LDT >= max(1,min(NB,N)).
137*> \endverbatim
138*>
139*> \param[out] WORK
140*> \verbatim
141*>          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
142*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
143*> \endverbatim
144*>
145*> \param[in] LWORK
146*> \verbatim
147*>          The dimension of the array WORK.
148*>          LWORK >= NBLOCAL * MAX(NBLOCAL,(N-NBLOCAL)),
149*>          where NBLOCAL=MIN(NB,N).
150*>          If LWORK = -1, then a workspace query is assumed.
151*>          The routine only calculates the optimal size of the WORK
152*>          array, returns this value as the first entry of the WORK
153*>          array, and no error message related to LWORK is issued
154*>          by XERBLA.
155*> \endverbatim
156*>
157*> \param[out] INFO
158*> \verbatim
159*>          INFO is INTEGER
160*>          = 0:  successful exit
161*>          < 0:  if INFO = -i, the i-th argument had an illegal value
162*> \endverbatim
163*>
164*  Authors:
165*  ========
166*
167*> \author Univ. of Tennessee
168*> \author Univ. of California Berkeley
169*> \author Univ. of Colorado Denver
170*> \author NAG Ltd.
171*
172*> \ingroup doubleOTHERcomputational
173*
174*> \par Contributors:
175*  ==================
176*>
177*> \verbatim
178*>
179*> November 2020, Igor Kozachenko,
180*>                Computer Science Division,
181*>                University of California, Berkeley
182*>
183*> \endverbatim
184*>
185*  =====================================================================
186      SUBROUTINE DORGTSQR_ROW( M, N, MB, NB, A, LDA, T, LDT, WORK,
187     $                         LWORK, INFO )
188      IMPLICIT NONE
189*
190*  -- LAPACK computational routine --
191*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
192*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193*
194*     .. Scalar Arguments ..
195      INTEGER           INFO, LDA, LDT, LWORK, M, N, MB, NB
196*     ..
197*     .. Array Arguments ..
198      DOUBLE PRECISION  A( LDA, * ), T( LDT, * ), WORK( * )
199*     ..
200*
201*  =====================================================================
202*
203*     .. Parameters ..
204      DOUBLE PRECISION   ONE, ZERO
205      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
206*     ..
207*     .. Local Scalars ..
208      LOGICAL            LQUERY
209      INTEGER            NBLOCAL, MB2, M_PLUS_ONE, ITMP, IB_BOTTOM,
210     $                   LWORKOPT, NUM_ALL_ROW_BLOCKS, JB_T, IB, IMB,
211     $                   KB, KB_LAST, KNB, MB1
212*     ..
213*     .. Local Arrays ..
214      DOUBLE PRECISION   DUMMY( 1, 1 )
215*     ..
216*     .. External Subroutines ..
217      EXTERNAL           DLARFB_GETT, DLASET, XERBLA
218*     ..
219*     .. Intrinsic Functions ..
220      INTRINSIC          DBLE, MAX, MIN
221*     ..
222*     .. Executable Statements ..
223*
224*     Test the input parameters
225*
226      INFO = 0
227      LQUERY  = LWORK.EQ.-1
228      IF( M.LT.0 ) THEN
229         INFO = -1
230      ELSE IF( N.LT.0 .OR. M.LT.N ) THEN
231         INFO = -2
232      ELSE IF( MB.LE.N ) THEN
233         INFO = -3
234      ELSE IF( NB.LT.1 ) THEN
235         INFO = -4
236      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
237         INFO = -6
238      ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
239         INFO = -8
240      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
241         INFO = -10
242      END IF
243*
244      NBLOCAL = MIN( NB, N )
245*
246*     Determine the workspace size.
247*
248      IF( INFO.EQ.0 ) THEN
249         LWORKOPT = NBLOCAL * MAX( NBLOCAL, ( N - NBLOCAL ) )
250      END IF
251*
252*     Handle error in the input parameters and handle the workspace query.
253*
254      IF( INFO.NE.0 ) THEN
255         CALL XERBLA( 'DORGTSQR_ROW', -INFO )
256         RETURN
257      ELSE IF ( LQUERY ) THEN
258         WORK( 1 ) = DBLE( LWORKOPT )
259         RETURN
260      END IF
261*
262*     Quick return if possible
263*
264      IF( MIN( M, N ).EQ.0 ) THEN
265         WORK( 1 ) = DBLE( LWORKOPT )
266         RETURN
267      END IF
268*
269*     (0) Set the upper-triangular part of the matrix A to zero and
270*     its diagonal elements to one.
271*
272      CALL DLASET('U', M, N, ZERO, ONE, A, LDA )
273*
274*     KB_LAST is the column index of the last column block reflector
275*     in the matrices T and V.
276*
277      KB_LAST = ( ( N-1 ) / NBLOCAL ) * NBLOCAL + 1
278*
279*
280*     (1) Bottom-up loop over row blocks of A, except the top row block.
281*     NOTE: If MB>=M, then the loop is never executed.
282*
283      IF ( MB.LT.M ) THEN
284*
285*        MB2 is the row blocking size for the row blocks before the
286*        first top row block in the matrix A. IB is the row index for
287*        the row blocks in the matrix A before the first top row block.
288*        IB_BOTTOM is the row index for the last bottom row block
289*        in the matrix A. JB_T is the column index of the corresponding
290*        column block in the matrix T.
291*
292*        Initialize variables.
293*
294*        NUM_ALL_ROW_BLOCKS is the number of row blocks in the matrix A
295*        including the first row block.
296*
297         MB2 = MB - N
298         M_PLUS_ONE = M + 1
299         ITMP = ( M - MB - 1 ) / MB2
300         IB_BOTTOM = ITMP * MB2 + MB + 1
301         NUM_ALL_ROW_BLOCKS = ITMP + 2
302         JB_T = NUM_ALL_ROW_BLOCKS * N + 1
303*
304         DO IB = IB_BOTTOM, MB+1, -MB2
305*
306*           Determine the block size IMB for the current row block
307*           in the matrix A.
308*
309            IMB = MIN( M_PLUS_ONE - IB, MB2 )
310*
311*           Determine the column index JB_T for the current column block
312*           in the matrix T.
313*
314            JB_T = JB_T - N
315*
316*           Apply column blocks of H in the row block from right to left.
317*
318*           KB is the column index of the current column block reflector
319*           in the matrices T and V.
320*
321            DO KB = KB_LAST, 1, -NBLOCAL
322*
323*              Determine the size of the current column block KNB in
324*              the matrices T and V.
325*
326               KNB = MIN( NBLOCAL, N - KB + 1 )
327*
328               CALL DLARFB_GETT( 'I', IMB, N-KB+1, KNB,
329     $                     T( 1, JB_T+KB-1 ), LDT, A( KB, KB ), LDA,
330     $                     A( IB, KB ), LDA, WORK, KNB )
331*
332            END DO
333*
334         END DO
335*
336      END IF
337*
338*     (2) Top row block of A.
339*     NOTE: If MB>=M, then we have only one row block of A of size M
340*     and we work on the entire matrix A.
341*
342      MB1 = MIN( MB, M )
343*
344*     Apply column blocks of H in the top row block from right to left.
345*
346*     KB is the column index of the current block reflector in
347*     the matrices T and V.
348*
349      DO KB = KB_LAST, 1, -NBLOCAL
350*
351*        Determine the size of the current column block KNB in
352*        the matrices T and V.
353*
354         KNB = MIN( NBLOCAL, N - KB + 1 )
355*
356         IF( MB1-KB-KNB+1.EQ.0 ) THEN
357*
358*           In SLARFB_GETT parameters, when M=0, then the matrix B
359*           does not exist, hence we need to pass a dummy array
360*           reference DUMMY(1,1) to B with LDDUMMY=1.
361*
362            CALL DLARFB_GETT( 'N', 0, N-KB+1, KNB,
363     $                        T( 1, KB ), LDT, A( KB, KB ), LDA,
364     $                        DUMMY( 1, 1 ), 1, WORK, KNB )
365         ELSE
366            CALL DLARFB_GETT( 'N', MB1-KB-KNB+1, N-KB+1, KNB,
367     $                        T( 1, KB ), LDT, A( KB, KB ), LDA,
368     $                        A( KB+KNB, KB), LDA, WORK, KNB )
369
370         END IF
371*
372      END DO
373*
374      WORK( 1 ) = DBLE( LWORKOPT )
375      RETURN
376*
377*     End of DORGTSQR_ROW
378*
379      END
380