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

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

CLASWLQ

Purpose:
``` CLASWLQ computes a blocked Tall-Skinny LQ factorization of
a complex M-by-N matrix A for M <= N:

A = ( L 0 ) *  Q,

where:

Q is a n-by-N orthogonal matrix, stored on exit in an implicit
form in the elements above the diagonal of the array A and in
the elements of the array T;
L is a lower-triangular M-by-M matrix stored on exit in
the elements on and below the diagonal of the array A.
0 is a M-by-(N-M) zero matrix, if M < N, 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. N >= M >= 0.``` [in] MB ``` MB is INTEGER The row block size to be used in the blocked QR. M >= MB >= 1``` [in] NB ``` NB is INTEGER The column block size to be used in the blocked QR. NB > 0.``` [in,out] A ``` A is COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the N-by-N lower triangular matrix L; the elements above the diagonal represent Q by the rows 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 COMPLEX array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((N-M)/(NB-M)) 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 >= MB.``` [out] WORK ` (workspace) COMPLEX array, dimension (MAX(1,LWORK))` [in] LWORK ``` The dimension of the array WORK. LWORK >= MB*M. 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:
``` Short-Wide LQ (SWLQ) performs LQ 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 upper diagonal entries of a block of NB rows of A:
Q(1) zeros out the upper diagonal entries of rows 1:NB of A
Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
. . .

Q(1) is computed by GELQT, 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 TPLQT, which represents Q(i) by Householder vectors
stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
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 162 of file claswlq.f.

164*
165* -- LAPACK computational routine --
166* -- LAPACK is a software package provided by Univ. of Tennessee, --
167* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
168*
169* .. Scalar Arguments ..
170 INTEGER INFO, LDA, M, N, MB, NB, LWORK, LDT
171* ..
172* .. Array Arguments ..
173 COMPLEX A( LDA, * ), WORK( * ), T( LDT, *)
174* ..
175*
176* =====================================================================
177*
178* ..
179* .. Local Scalars ..
180 LOGICAL LQUERY
181 INTEGER I, II, KK, CTR
182* ..
183* .. EXTERNAL FUNCTIONS ..
184 LOGICAL LSAME
185 EXTERNAL lsame
186* .. EXTERNAL SUBROUTINES ..
187 EXTERNAL cgelqt, ctplqt, xerbla
188* .. INTRINSIC FUNCTIONS ..
189 INTRINSIC max, min, mod
190* ..
191* .. EXTERNAL FUNCTIONS ..
192 INTEGER ILAENV
193 EXTERNAL ilaenv
194* ..
195* .. EXECUTABLE STATEMENTS ..
196*
197* TEST THE INPUT ARGUMENTS
198*
199 info = 0
200*
201 lquery = ( lwork.EQ.-1 )
202*
203 IF( m.LT.0 ) THEN
204 info = -1
205 ELSE IF( n.LT.0 .OR. n.LT.m ) THEN
206 info = -2
207 ELSE IF( mb.LT.1 .OR. ( mb.GT.m .AND. m.GT.0 )) THEN
208 info = -3
209 ELSE IF( nb.LE.0 ) THEN
210 info = -4
211 ELSE IF( lda.LT.max( 1, m ) ) THEN
212 info = -6
213 ELSE IF( ldt.LT.mb ) THEN
214 info = -8
215 ELSE IF( ( lwork.LT.m*mb) .AND. (.NOT.lquery) ) THEN
216 info = -10
217 END IF
218 IF( info.EQ.0) THEN
219 work(1) = mb*m
220 END IF
221*
222 IF( info.NE.0 ) THEN
223 CALL xerbla( 'CLASWLQ', -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 LQ Decomposition
236*
237 IF((m.GE.n).OR.(nb.LE.m).OR.(nb.GE.n)) THEN
238 CALL cgelqt( m, n, mb, a, lda, t, ldt, work, info)
239 RETURN
240 END IF
241*
242 kk = mod((n-m),(nb-m))
243 ii=n-kk+1
244*
245* Compute the LQ factorization of the first block A(1:M,1:NB)
246*
247 CALL cgelqt( m, nb, mb, a(1,1), lda, t, ldt, work, info)
248 ctr = 1
249*
250 DO i = nb+1, ii-nb+m , (nb-m)
251*
252* Compute the QR factorization of the current block A(1:M,I:I+NB-M)
253*
254 CALL ctplqt( m, nb-m, 0, mb, a(1,1), lda, a( 1, i ),
255 \$ lda, t(1,ctr*m+1),
256 \$ ldt, work, info )
257 ctr = ctr + 1
258 END DO
259*
260* Compute the QR factorization of the last block A(1:M,II:N)
261*
262 IF (ii.LE.n) THEN
263 CALL ctplqt( m, kk, 0, mb, a(1,1), lda, a( 1, ii ),
264 \$ lda, t(1,ctr*m+1), ldt,
265 \$ work, info )
266 END IF
267*
268 work( 1 ) = m * mb
269 RETURN
270*
271* End of CLASWLQ
272*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine cgelqt(M, N, MB, A, LDA, T, LDT, WORK, INFO)
CGELQT
Definition: cgelqt.f:124
subroutine ctplqt(M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
CTPLQT
Definition: ctplqt.f:174
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