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

 subroutine dlatsqr ( 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 )

DLATSQR

Purpose:
``` DLATSQR computes a blocked Tall-Skinny QR factorization of
a real M-by-N matrix A for M >= N:

A = Q * ( R ),
( 0 )

where:

Q is a M-by-M orthogonal matrix, stored on exit in an implicit
form in the elements below the diagonal of the array A and in
the elements of the array T;

R is an upper-triangular N-by-N matrix, stored on exit in
the elements on and above the diagonal of the array A.

0 is a (M-N)-by-N zero matrix, and is not stored.```
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 to be used in the blocked QR. MB > 0.``` [in] NB ``` NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] T ``` T is DOUBLE PRECISION array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= NB.``` [out] WORK ` (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= 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```
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 [1].

[1] “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 167 of file dlatsqr.f.

169*
170* -- LAPACK computational routine --
171* -- LAPACK is a software package provided by Univ. of Tennessee, --
172* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
173*
174* .. Scalar Arguments ..
175 INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK
176* ..
177* .. Array Arguments ..
178 DOUBLE PRECISION A( LDA, * ), WORK( * ), T(LDT, *)
179* ..
180*
181* =====================================================================
182*
183* ..
184* .. Local Scalars ..
185 LOGICAL LQUERY
186 INTEGER I, II, KK, CTR
187* ..
188* .. EXTERNAL FUNCTIONS ..
189 LOGICAL LSAME
190 EXTERNAL lsame
191* .. EXTERNAL SUBROUTINES ..
192 EXTERNAL dgeqrt, dtpqrt, xerbla
193* .. INTRINSIC FUNCTIONS ..
194 INTRINSIC max, min, mod
195* ..
196* .. EXECUTABLE STATEMENTS ..
197*
198* TEST THE INPUT ARGUMENTS
199*
200 info = 0
201*
202 lquery = ( lwork.EQ.-1 )
203*
204 IF( m.LT.0 ) THEN
205 info = -1
206 ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
207 info = -2
208 ELSE IF( mb.LT.1 ) THEN
209 info = -3
210 ELSE IF( nb.LT.1 .OR. ( nb.GT.n .AND. n.GT.0 )) THEN
211 info = -4
212 ELSE IF( lda.LT.max( 1, m ) ) THEN
213 info = -6
214 ELSE IF( ldt.LT.nb ) THEN
215 info = -8
216 ELSE IF( lwork.LT.(n*nb) .AND. (.NOT.lquery) ) THEN
217 info = -10
218 END IF
219 IF( info.EQ.0) THEN
220 work(1) = nb*n
221 END IF
222 IF( info.NE.0 ) THEN
223 CALL xerbla( 'DLATSQR', -info )
224 RETURN
225 ELSE IF (lquery) THEN
226 RETURN
227 END IF
228*
229* Quick return if possible
230*
231 IF( min(m,n).EQ.0 ) THEN
232 RETURN
233 END IF
234*
235* The QR Decomposition
236*
237 IF ((mb.LE.n).OR.(mb.GE.m)) THEN
238 CALL dgeqrt( m, n, nb, a, lda, t, ldt, work, info)
239 RETURN
240 END IF
241*
242 kk = mod((m-n),(mb-n))
243 ii=m-kk+1
244*
245* Compute the QR factorization of the first block A(1:MB,1:N)
246*
247 CALL dgeqrt( mb, n, nb, a(1,1), lda, t, ldt, work, info )
248*
249 ctr = 1
250 DO i = mb+1, ii-mb+n , (mb-n)
251*
252* Compute the QR factorization of the current block A(I:I+MB-N,1:N)
253*
254 CALL dtpqrt( mb-n, n, 0, nb, a(1,1), lda, a( i, 1 ), lda,
255 \$ t(1, ctr * n + 1),
256 \$ ldt, work, info )
257 ctr = ctr + 1
258 END DO
259*
260* Compute the QR factorization of the last block A(II:M,1:N)
261*
262 IF (ii.LE.m) THEN
263 CALL dtpqrt( kk, n, 0, nb, a(1,1), lda, a( ii, 1 ), lda,
264 \$ t(1, ctr * n + 1), ldt,
265 \$ work, info )
266 END IF
267*
268 work( 1 ) = n*nb
269 RETURN
270*
271* End of DLATSQR
272*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dgeqrt(m, n, nb, a, lda, t, ldt, work, info)
DGEQRT
Definition dgeqrt.f:141
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dtpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)
DTPQRT
Definition dtpqrt.f:189
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