LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dorgtsqr_row()

 subroutine dorgtsqr_row ( 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 )

DORGTSQR_ROW

Purpose:
``` DORGTSQR_ROW generates an M-by-N real matrix Q_out with
orthonormal columns from the output of DLATSQR. These N orthonormal
columns are the first N columns of a product of complex unitary
matrices Q(k)_in of order M, which are returned by DLATSQR in
a special format.

Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).

The input matrices Q(k)_in are stored in row and column blocks in A.
See the documentation of DLATSQR for more details on the format of
Q(k)_in, where each Q(k)_in is represented by block Householder
transformations. This routine calls an auxiliary routine DLARFB_GETT,
where the computation is performed on each individual block. The
algorithm first sweeps NB-sized column blocks from the right to left
starting in the bottom row block and continues to the top row block
(hence _ROW in the routine name). This sweep is in reverse order of
the order in which DLATSQR generates the output blocks.```
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. M >= N >= 0.``` [in] MB ``` MB is INTEGER The row block size used by DLATSQR 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).``` [in] NB ``` NB is INTEGER The column block size used by DLATSQR 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).``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry: The elements on and above the diagonal are not used as input. The elements below the diagonal represent the unit lower-trapezoidal blocked matrix V computed by DLATSQR that defines the input matrices Q_in(k) (ones on the diagonal are not stored). See DLATSQR for more details. On exit: The array A contains an M-by-N orthonormal matrix Q_out, i.e the columns of A are orthogonal unit vectors.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in] T ``` T is DOUBLE PRECISION 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 the 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. See DLATSQR for more details on the format of T.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB,N)).``` [out] WORK ``` (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` The dimension of the array WORK. LWORK >= NBLOCAL * MAX(NBLOCAL,(N-NBLOCAL)), where NBLOCAL=MIN(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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Contributors:
``` November 2020, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 186 of file dorgtsqr_row.f.

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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dlarfb_gett(ident, m, n, k, t, ldt, a, lda, b, ldb, work, ldwork)
DLARFB_GETT
subroutine dlaset(uplo, m, n, alpha, beta, a, lda)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition dlaset.f:110
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