 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dlaswlq()

 subroutine dlaswlq ( 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 )

DLASWLQ

Purpose:
``` DLASWLQ computes a blocked Tall-Skinny LQ factorization of
a real 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) DOUBLE PRECISION 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 dlaswlq.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  DOUBLE PRECISION 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 dgelqt, dtplqt, 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.LT.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( 'DLASWLQ', -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 dgelqt( 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 dgelqt( 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 dtplqt( 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 dtplqt( 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 DLASWLQ
268 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dgelqt(M, N, MB, A, LDA, T, LDT, WORK, INFO)
DGELQT
Definition: dgelqt.f:139
subroutine dtplqt(M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
DTPLQT
Definition: dtplqt.f:189
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