 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zungtsqr_row()

 subroutine zungtsqr_row ( integer M, integer N, integer MB, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) WORK, integer LWORK, integer INFO )

ZUNGTSQR_ROW

Purpose:
``` ZUNGTSQR_ROW generates an M-by-N complex matrix Q_out with
orthonormal columns from the output of ZLATSQR. 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 ZLATSQR 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 ZLATSQR 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 ZLARFB_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 ZLATSQR 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 ZLATSQR 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 ZLATSQR 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 COMPLEX*16 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 ZLATSQR that defines the input matrices Q_in(k) (ones on the diagonal are not stored). See ZLATSQR 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 COMPLEX*16 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 ZLATSQR 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) COMPLEX*16 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 zungtsqr_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  COMPLEX*16 A( LDA, * ), T( LDT, * ), WORK( * )
199 * ..
200 *
201 * =====================================================================
202 *
203 * .. Parameters ..
204  COMPLEX*16 CONE, CZERO
205  parameter( cone = ( 1.0d+0, 0.0d+0 ),
206  \$ czero = ( 0.0d+0, 0.0d+0 ) )
207 * ..
208 * .. Local Scalars ..
209  LOGICAL LQUERY
210  INTEGER NBLOCAL, MB2, M_PLUS_ONE, ITMP, IB_BOTTOM,
211  \$ LWORKOPT, NUM_ALL_ROW_BLOCKS, JB_T, IB, IMB,
212  \$ KB, KB_LAST, KNB, MB1
213 * ..
214 * .. Local Arrays ..
215  COMPLEX*16 DUMMY( 1, 1 )
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL zlarfb_gett, zlaset, xerbla
219 * ..
220 * .. Intrinsic Functions ..
221  INTRINSIC dcmplx, max, min
222 * ..
223 * .. Executable Statements ..
224 *
225 * Test the input parameters
226 *
227  info = 0
228  lquery = lwork.EQ.-1
229  IF( m.LT.0 ) THEN
230  info = -1
231  ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
232  info = -2
233  ELSE IF( mb.LE.n ) THEN
234  info = -3
235  ELSE IF( nb.LT.1 ) THEN
236  info = -4
237  ELSE IF( lda.LT.max( 1, m ) ) THEN
238  info = -6
239  ELSE IF( ldt.LT.max( 1, min( nb, n ) ) ) THEN
240  info = -8
241  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
242  info = -10
243  END IF
244 *
245  nblocal = min( nb, n )
246 *
247 * Determine the workspace size.
248 *
249  IF( info.EQ.0 ) THEN
250  lworkopt = nblocal * max( nblocal, ( n - nblocal ) )
251  END IF
252 *
253 * Handle error in the input parameters and handle the workspace query.
254 *
255  IF( info.NE.0 ) THEN
256  CALL xerbla( 'ZUNGTSQR_ROW', -info )
257  RETURN
258  ELSE IF ( lquery ) THEN
259  work( 1 ) = dcmplx( lworkopt )
260  RETURN
261  END IF
262 *
263 * Quick return if possible
264 *
265  IF( min( m, n ).EQ.0 ) THEN
266  work( 1 ) = dcmplx( lworkopt )
267  RETURN
268  END IF
269 *
270 * (0) Set the upper-triangular part of the matrix A to zero and
271 * its diagonal elements to one.
272 *
273  CALL zlaset('U', m, n, czero, cone, a, lda )
274 *
275 * KB_LAST is the column index of the last column block reflector
276 * in the matrices T and V.
277 *
278  kb_last = ( ( n-1 ) / nblocal ) * nblocal + 1
279 *
280 *
281 * (1) Bottom-up loop over row blocks of A, except the top row block.
282 * NOTE: If MB>=M, then the loop is never executed.
283 *
284  IF ( mb.LT.m ) THEN
285 *
286 * MB2 is the row blocking size for the row blocks before the
287 * first top row block in the matrix A. IB is the row index for
288 * the row blocks in the matrix A before the first top row block.
289 * IB_BOTTOM is the row index for the last bottom row block
290 * in the matrix A. JB_T is the column index of the corresponding
291 * column block in the matrix T.
292 *
293 * Initialize variables.
294 *
295 * NUM_ALL_ROW_BLOCKS is the number of row blocks in the matrix A
296 * including the first row block.
297 *
298  mb2 = mb - n
299  m_plus_one = m + 1
300  itmp = ( m - mb - 1 ) / mb2
301  ib_bottom = itmp * mb2 + mb + 1
302  num_all_row_blocks = itmp + 2
303  jb_t = num_all_row_blocks * n + 1
304 *
305  DO ib = ib_bottom, mb+1, -mb2
306 *
307 * Determine the block size IMB for the current row block
308 * in the matrix A.
309 *
310  imb = min( m_plus_one - ib, mb2 )
311 *
312 * Determine the column index JB_T for the current column block
313 * in the matrix T.
314 *
315  jb_t = jb_t - n
316 *
317 * Apply column blocks of H in the row block from right to left.
318 *
319 * KB is the column index of the current column block reflector
320 * in the matrices T and V.
321 *
322  DO kb = kb_last, 1, -nblocal
323 *
324 * Determine the size of the current column block KNB in
325 * the matrices T and V.
326 *
327  knb = min( nblocal, n - kb + 1 )
328 *
329  CALL zlarfb_gett( 'I', imb, n-kb+1, knb,
330  \$ t( 1, jb_t+kb-1 ), ldt, a( kb, kb ), lda,
331  \$ a( ib, kb ), lda, work, knb )
332 *
333  END DO
334 *
335  END DO
336 *
337  END IF
338 *
339 * (2) Top row block of A.
340 * NOTE: If MB>=M, then we have only one row block of A of size M
341 * and we work on the entire matrix A.
342 *
343  mb1 = min( mb, m )
344 *
345 * Apply column blocks of H in the top row block from right to left.
346 *
347 * KB is the column index of the current block reflector in
348 * the matrices T and V.
349 *
350  DO kb = kb_last, 1, -nblocal
351 *
352 * Determine the size of the current column block KNB in
353 * the matrices T and V.
354 *
355  knb = min( nblocal, n - kb + 1 )
356 *
357  IF( mb1-kb-knb+1.EQ.0 ) THEN
358 *
359 * In SLARFB_GETT parameters, when M=0, then the matrix B
360 * does not exist, hence we need to pass a dummy array
361 * reference DUMMY(1,1) to B with LDDUMMY=1.
362 *
363  CALL zlarfb_gett( 'N', 0, n-kb+1, knb,
364  \$ t( 1, kb ), ldt, a( kb, kb ), lda,
365  \$ dummy( 1, 1 ), 1, work, knb )
366  ELSE
367  CALL zlarfb_gett( 'N', mb1-kb-knb+1, n-kb+1, knb,
368  \$ t( 1, kb ), ldt, a( kb, kb ), lda,
369  \$ a( kb+knb, kb), lda, work, knb )
370
371  END IF
372 *
373  END DO
374 *
375  work( 1 ) = dcmplx( lworkopt )
376  RETURN
377 *
378 * End of ZUNGTSQR_ROW
379 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zlarfb_gett(IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
ZLARFB_GETT
Definition: zlarfb_gett.f:392
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
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