 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dlamtsqr()

 subroutine dlamtsqr ( character SIDE, character TRANS, integer M, integer N, integer K, integer MB, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension(ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DLAMTSQR

Purpose:
```      DLAMTSQR overwrites the general real M-by-N matrix C with

SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'T':      Q**T * C       C * Q**T
where Q is a real orthogonal matrix defined as the product
of blocked elementary reflectors computed by tall skinny
QR factorization (DLATSQR)```
Parameters
 [in] SIDE ``` SIDE is CHARACTER*1 = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right.``` [in] TRANS ``` TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.``` [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 C. N >= 0.``` [in] K ``` K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0;``` [in] MB ``` MB is INTEGER The block size to be used in the blocked QR. MB > N. (must be the same as DLATSQR)``` [in] NB ``` NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,K) The i-th column must contain the vector which defines the blockedelementary reflector H(i), for i = 1,2,...,k, as returned by DLATSQR in the first k columns of its array argument A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N).``` [in] T ``` T is DOUBLE PRECISION array, dimension ( N * Number of blocks(CEIL(M-K/MB-K)), The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= NB.``` [in,out] C ``` C is DOUBLE PRECISION array, dimension (LDC,N) On entry, the M-by-N matrix C. On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.``` [in] LDC ``` LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).``` [out] WORK ` (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N)*NB; if SIDE = 'R', LWORK >= max(1,MB)*NB. 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```
Further Details:
``` 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).

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 details of the overall algorithm, see the description of
Sequential TSQR in Section 2.2 of .

 “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```

Definition at line 195 of file dlamtsqr.f.

197*
198* -- LAPACK computational routine --
199* -- LAPACK is a software package provided by Univ. of Tennessee, --
200* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
201*
202* .. Scalar Arguments ..
203 CHARACTER SIDE, TRANS
204 INTEGER INFO, LDA, M, N, K, MB, NB, LDT, LWORK, LDC
205* ..
206* .. Array Arguments ..
207 DOUBLE PRECISION A( LDA, * ), WORK( * ), C(LDC, * ),
208 \$ T( LDT, * )
209* ..
210*
211* =====================================================================
212*
213* ..
214* .. Local Scalars ..
215 LOGICAL LEFT, RIGHT, TRAN, NOTRAN, LQUERY
216 INTEGER I, II, KK, LW, CTR, Q
217* ..
218* .. External Functions ..
219 LOGICAL LSAME
220 EXTERNAL lsame
221* .. External Subroutines ..
222 EXTERNAL dgemqrt, dtpmqrt, xerbla
223* ..
224* .. Executable Statements ..
225*
226* Test the input arguments
227*
228 lquery = lwork.LT.0
229 notran = lsame( trans, 'N' )
230 tran = lsame( trans, 'T' )
231 left = lsame( side, 'L' )
232 right = lsame( side, 'R' )
233 IF (left) THEN
234 lw = n * nb
235 q = m
236 ELSE
237 lw = mb * nb
238 q = n
239 END IF
240*
241 info = 0
242 IF( .NOT.left .AND. .NOT.right ) THEN
243 info = -1
244 ELSE IF( .NOT.tran .AND. .NOT.notran ) THEN
245 info = -2
246 ELSE IF( m.LT.k ) THEN
247 info = -3
248 ELSE IF( n.LT.0 ) THEN
249 info = -4
250 ELSE IF( k.LT.0 ) THEN
251 info = -5
252 ELSE IF( k.LT.nb .OR. nb.LT.1 ) THEN
253 info = -7
254 ELSE IF( lda.LT.max( 1, q ) ) THEN
255 info = -9
256 ELSE IF( ldt.LT.max( 1, nb) ) THEN
257 info = -11
258 ELSE IF( ldc.LT.max( 1, m ) ) THEN
259 info = -13
260 ELSE IF(( lwork.LT.max(1,lw)).AND.(.NOT.lquery)) THEN
261 info = -15
262 END IF
263*
264* Determine the block size if it is tall skinny or short and wide
265*
266 IF( info.EQ.0) THEN
267 work(1) = lw
268 END IF
269*
270 IF( info.NE.0 ) THEN
271 CALL xerbla( 'DLAMTSQR', -info )
272 RETURN
273 ELSE IF (lquery) THEN
274 RETURN
275 END IF
276*
277* Quick return if possible
278*
279 IF( min(m,n,k).EQ.0 ) THEN
280 RETURN
281 END IF
282*
283 IF((mb.LE.k).OR.(mb.GE.max(m,n,k))) THEN
284 CALL dgemqrt( side, trans, m, n, k, nb, a, lda,
285 \$ t, ldt, c, ldc, work, info)
286 RETURN
287 END IF
288*
289 IF(left.AND.notran) THEN
290*
291* Multiply Q to the last block of C
292*
293 kk = mod((m-k),(mb-k))
294 ctr = (m-k)/(mb-k)
295 IF (kk.GT.0) THEN
296 ii=m-kk+1
297 CALL dtpmqrt('L','N',kk , n, k, 0, nb, a(ii,1), lda,
298 \$ t(1,ctr*k+1),ldt , c(1,1), ldc,
299 \$ c(ii,1), ldc, work, info )
300 ELSE
301 ii=m+1
302 END IF
303*
304 DO i=ii-(mb-k),mb+1,-(mb-k)
305*
306* Multiply Q to the current block of C (I:I+MB,1:N)
307*
308 ctr = ctr - 1
309 CALL dtpmqrt('L','N',mb-k , n, k, 0,nb, a(i,1), lda,
310 \$ t(1,ctr*k+1),ldt, c(1,1), ldc,
311 \$ c(i,1), ldc, work, info )
312*
313 END DO
314*
315* Multiply Q to the first block of C (1:MB,1:N)
316*
317 CALL dgemqrt('L','N',mb , n, k, nb, a(1,1), lda, t
318 \$ ,ldt ,c(1,1), ldc, work, info )
319*
320 ELSE IF (left.AND.tran) THEN
321*
322* Multiply Q to the first block of C
323*
324 kk = mod((m-k),(mb-k))
325 ii=m-kk+1
326 ctr = 1
327 CALL dgemqrt('L','T',mb , n, k, nb, a(1,1), lda, t
328 \$ ,ldt ,c(1,1), ldc, work, info )
329*
330 DO i=mb+1,ii-mb+k,(mb-k)
331*
332* Multiply Q to the current block of C (I:I+MB,1:N)
333*
334 CALL dtpmqrt('L','T',mb-k , n, k, 0,nb, a(i,1), lda,
335 \$ t(1,ctr * k + 1),ldt, c(1,1), ldc,
336 \$ c(i,1), ldc, work, info )
337 ctr = ctr + 1
338*
339 END DO
340 IF(ii.LE.m) THEN
341*
342* Multiply Q to the last block of C
343*
344 CALL dtpmqrt('L','T',kk , n, k, 0,nb, a(ii,1), lda,
345 \$ t(1,ctr * k + 1), ldt, c(1,1), ldc,
346 \$ c(ii,1), ldc, work, info )
347*
348 END IF
349*
350 ELSE IF(right.AND.tran) THEN
351*
352* Multiply Q to the last block of C
353*
354 kk = mod((n-k),(mb-k))
355 ctr = (n-k)/(mb-k)
356 IF (kk.GT.0) THEN
357 ii=n-kk+1
358 CALL dtpmqrt('R','T',m , kk, k, 0, nb, a(ii,1), lda,
359 \$ t(1,ctr*k+1), ldt, c(1,1), ldc,
360 \$ c(1,ii), ldc, work, info )
361 ELSE
362 ii=n+1
363 END IF
364*
365 DO i=ii-(mb-k),mb+1,-(mb-k)
366*
367* Multiply Q to the current block of C (1:M,I:I+MB)
368*
369 ctr = ctr - 1
370 CALL dtpmqrt('R','T',m , mb-k, k, 0,nb, a(i,1), lda,
371 \$ t(1,ctr*k+1), ldt, c(1,1), ldc,
372 \$ c(1,i), ldc, work, info )
373*
374 END DO
375*
376* Multiply Q to the first block of C (1:M,1:MB)
377*
378 CALL dgemqrt('R','T',m , mb, k, nb, a(1,1), lda, t
379 \$ ,ldt ,c(1,1), ldc, work, info )
380*
381 ELSE IF (right.AND.notran) THEN
382*
383* Multiply Q to the first block of C
384*
385 kk = mod((n-k),(mb-k))
386 ii=n-kk+1
387 ctr = 1
388 CALL dgemqrt('R','N', m, mb , k, nb, a(1,1), lda, t
389 \$ ,ldt ,c(1,1), ldc, work, info )
390*
391 DO i=mb+1,ii-mb+k,(mb-k)
392*
393* Multiply Q to the current block of C (1:M,I:I+MB)
394*
395 CALL dtpmqrt('R','N', m, mb-k, k, 0,nb, a(i,1), lda,
396 \$ t(1, ctr * k + 1),ldt, c(1,1), ldc,
397 \$ c(1,i), ldc, work, info )
398 ctr = ctr + 1
399*
400 END DO
401 IF(ii.LE.n) THEN
402*
403* Multiply Q to the last block of C
404*
405 CALL dtpmqrt('R','N', m, kk , k, 0,nb, a(ii,1), lda,
406 \$ t(1, ctr * k + 1),ldt, c(1,1), ldc,
407 \$ c(1,ii), ldc, work, info )
408*
409 END IF
410*
411 END IF
412*
413 work(1) = lw
414 RETURN
415*
416* End of DLAMTSQR
417*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dgemqrt(SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
DGEMQRT
Definition: dgemqrt.f:168
subroutine dtpmqrt(SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
DTPMQRT
Definition: dtpmqrt.f:216
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