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

 subroutine zlaswlq ( integer M, integer N, integer MB, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, *) T, integer LDT, complex*16, dimension( * ) WORK, integer LWORK, integer INFO )

ZLASWLQ

Purpose:
``` ZLASWLQ computes a blocked Tall-Skinny LQ factorization of
a complexx 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*16 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*16 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*16 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 zlaswlq.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*16 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 zgelqt, ztplqt, xerbla
188* .. INTRINSIC FUNCTIONS ..
189 INTRINSIC max, min, mod
190* ..
191* .. EXECUTABLE STATEMENTS ..
192*
193* TEST THE INPUT ARGUMENTS
194*
195 info = 0
196*
197 lquery = ( lwork.EQ.-1 )
198*
199 IF( m.LT.0 ) THEN
200 info = -1
201 ELSE IF( n.LT.0 .OR. n.LT.m ) THEN
202 info = -2
203 ELSE IF( mb.LT.1 .OR. ( mb.GT.m .AND. m.GT.0 )) THEN
204 info = -3
205 ELSE IF( nb.LE.0 ) THEN
206 info = -4
207 ELSE IF( lda.LT.max( 1, m ) ) THEN
208 info = -6
209 ELSE IF( ldt.LT.mb ) THEN
210 info = -8
211 ELSE IF( ( lwork.LT.m*mb) .AND. (.NOT.lquery) ) THEN
212 info = -10
213 END IF
214 IF( info.EQ.0) THEN
215 work(1) = mb*m
216 END IF
217*
218 IF( info.NE.0 ) THEN
219 CALL xerbla( 'ZLASWLQ', -info )
220 RETURN
221 ELSE IF (lquery) THEN
222 RETURN
223 END IF
224*
225* Quick return if possible
226*
227 IF( min(m,n).EQ.0 ) THEN
228 RETURN
229 END IF
230*
231* The LQ Decomposition
232*
233 IF((m.GE.n).OR.(nb.LE.m).OR.(nb.GE.n)) THEN
234 CALL zgelqt( m, n, mb, a, lda, t, ldt, work, info)
235 RETURN
236 END IF
237*
238 kk = mod((n-m),(nb-m))
239 ii=n-kk+1
240*
241* Compute the LQ factorization of the first block A(1:M,1:NB)
242*
243 CALL zgelqt( m, nb, mb, a(1,1), lda, t, ldt, work, info)
244 ctr = 1
245*
246 DO i = nb+1, ii-nb+m , (nb-m)
247*
248* Compute the QR factorization of the current block A(1:M,I:I+NB-M)
249*
250 CALL ztplqt( m, nb-m, 0, mb, a(1,1), lda, a( 1, i ),
251 \$ lda, t(1, ctr * m + 1),
252 \$ ldt, work, info )
253 ctr = ctr + 1
254 END DO
255*
256* Compute the QR factorization of the last block A(1:M,II:N)
257*
258 IF (ii.LE.n) THEN
259 CALL ztplqt( m, kk, 0, mb, a(1,1), lda, a( 1, ii ),
260 \$ lda, t(1, ctr * m + 1), ldt,
261 \$ work, info )
262 END IF
263*
264 work( 1 ) = m * mb
265 RETURN
266*
267* End of ZLASWLQ
268*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zgelqt(M, N, MB, A, LDA, T, LDT, WORK, INFO)
ZGELQT
Definition: zgelqt.f:139
subroutine ztplqt(M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
ZTPLQT
Definition: ztplqt.f:189
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