LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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dlaswlq.f
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1*> \brief \b DLASWLQ
2*
3* Definition:
4* ===========
5*
6* SUBROUTINE DLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK,
7* LWORK, INFO)
8*
9* .. Scalar Arguments ..
10* INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK
11* ..
12* .. Array Arguments ..
13* DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
14* ..
15*
16*
17*> \par Purpose:
18* =============
19*>
20*> \verbatim
21*>
22*> DLASWLQ computes a blocked Tall-Skinny LQ factorization of
23*> a real M-by-N matrix A for M <= N:
24*>
25*> A = ( L 0 ) * Q,
26*>
27*> where:
28*>
29*> Q is a n-by-N orthogonal matrix, stored on exit in an implicit
30*> form in the elements above the diagonal of the array A and in
31*> the elements of the array T;
32*> L is a lower-triangular M-by-M matrix stored on exit in
33*> the elements on and below the diagonal of the array A.
34*> 0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.
35*>
36*> \endverbatim
37*
38* Arguments:
39* ==========
40*
41*> \param[in] M
42*> \verbatim
43*> M is INTEGER
44*> The number of rows of the matrix A. M >= 0.
45*> \endverbatim
46*>
47*> \param[in] N
48*> \verbatim
49*> N is INTEGER
50*> The number of columns of the matrix A. N >= M >= 0.
51*> \endverbatim
52*>
53*> \param[in] MB
54*> \verbatim
55*> MB is INTEGER
56*> The row block size to be used in the blocked QR.
57*> M >= MB >= 1
58*> \endverbatim
59*> \param[in] NB
60*> \verbatim
61*> NB is INTEGER
62*> The column block size to be used in the blocked QR.
63*> NB > 0.
64*> \endverbatim
65*>
66*> \param[in,out] A
67*> \verbatim
68*> A is DOUBLE PRECISION array, dimension (LDA,N)
69*> On entry, the M-by-N matrix A.
70*> On exit, the elements on and below the diagonal
71*> of the array contain the N-by-N lower triangular matrix L;
72*> the elements above the diagonal represent Q by the rows
73*> of blocked V (see Further Details).
74*>
75*> \endverbatim
76*>
77*> \param[in] LDA
78*> \verbatim
79*> LDA is INTEGER
80*> The leading dimension of the array A. LDA >= max(1,M).
81*> \endverbatim
82*>
83*> \param[out] T
84*> \verbatim
85*> T is DOUBLE PRECISION array,
86*> dimension (LDT, N * Number_of_row_blocks)
87*> where Number_of_row_blocks = CEIL((N-M)/(NB-M))
88*> The blocked upper triangular block reflectors stored in compact form
89*> as a sequence of upper triangular blocks.
90*> See Further Details below.
91*> \endverbatim
92*>
93*> \param[in] LDT
94*> \verbatim
95*> LDT is INTEGER
96*> The leading dimension of the array T. LDT >= MB.
97*> \endverbatim
98*>
99*>
100*> \param[out] WORK
101*> \verbatim
102*> (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
103*>
104*> \endverbatim
105*> \param[in] LWORK
106*> \verbatim
107*> The dimension of the array WORK. LWORK >= MB*M.
108*> If LWORK = -1, then a workspace query is assumed; the routine
109*> only calculates the optimal size of the WORK array, returns
110*> this value as the first entry of the WORK array, and no error
111*> message related to LWORK is issued by XERBLA.
112*>
113*> \endverbatim
114*> \param[out] INFO
115*> \verbatim
116*> INFO is INTEGER
117*> = 0: successful exit
118*> < 0: if INFO = -i, the i-th argument had an illegal value
119*> \endverbatim
120*
121* Authors:
122* ========
123*
124*> \author Univ. of Tennessee
125*> \author Univ. of California Berkeley
126*> \author Univ. of Colorado Denver
127*> \author NAG Ltd.
128*
129*> \par Further Details:
130* =====================
131*>
132*> \verbatim
133*> Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
134*> representing Q as a product of other orthogonal matrices
135*> Q = Q(1) * Q(2) * . . . * Q(k)
136*> where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
137*> Q(1) zeros out the upper diagonal entries of rows 1:NB of A
138*> Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
139*> Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
140*> . . .
141*>
142*> Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
143*> stored under the diagonal of rows 1:MB of A, and by upper triangular
144*> block reflectors, stored in array T(1:LDT,1:N).
145*> For more information see Further Details in GELQT.
146*>
147*> Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
148*> stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
149*> block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
150*> The last Q(k) may use fewer rows.
151*> For more information see Further Details in TPQRT.
152*>
153*> For more details of the overall algorithm, see the description of
154*> Sequential TSQR in Section 2.2 of [1].
155*>
156*> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
157*> J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
158*> SIAM J. Sci. Comput, vol. 34, no. 1, 2012
159*> \endverbatim
160*>
161* =====================================================================
162 SUBROUTINE dlaswlq( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
163 $ INFO)
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*
269 END
subroutine dlaswlq(M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
DLASWLQ
Definition: dlaswlq.f:164
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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