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

 subroutine slatsqr ( integer m, integer n, integer mb, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension(ldt, *) t, integer ldt, real, dimension( * ) work, integer lwork, integer info )

SLATSQR

Purpose:
SLATSQR 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 > N. [in] NB NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1. [in,out] A A is REAL 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 REAL 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) REAL 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 slatsqr.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 REAL 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 sgeqrt, stpqrt, 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( 'SLATSQR', -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 sgeqrt( m, n, nb, a, lda, t, ldt, work, info)
239 RETURN
240 END IF
241 kk = mod((m-n),(mb-n))
242 ii=m-kk+1
243*
244* Compute the QR factorization of the first block A(1:MB,1:N)
245*
246 CALL sgeqrt( mb, n, nb, a(1,1), lda, t, ldt, work, info )
247*
248 ctr = 1
249 DO i = mb+1, ii-mb+n , (mb-n)
250*
251* Compute the QR factorization of the current block A(I:I+MB-N,1:N)
252*
253 CALL stpqrt( mb-n, n, 0, nb, a(1,1), lda, a( i, 1 ), lda,
254 \$ t(1, ctr * n + 1),
255 \$ ldt, work, info )
256 ctr = ctr + 1
257 END DO
258*
259* Compute the QR factorization of the last block A(II:M,1:N)
260*
261 IF (ii.LE.m) THEN
262 CALL stpqrt( kk, n, 0, nb, a(1,1), lda, a( ii, 1 ), lda,
263 \$ t(1, ctr * n + 1), ldt,
264 \$ work, info )
265 END IF
266*
267 work( 1 ) = n*nb
268 RETURN
269*
270* End of SLATSQR
271*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine sgeqrt(m, n, nb, a, lda, t, ldt, work, info)
SGEQRT
Definition sgeqrt.f:141
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine stpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)
STPQRT
Definition stpqrt.f:189
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