 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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).
For more information see Further Details in GEQRT.

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 information see Further Details in TPQRT.

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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