LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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clatsqr.f
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1*> \brief \b CLATSQR
2*
3* Definition:
4* ===========
5*
6* SUBROUTINE CLATSQR( 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* COMPLEX A( LDA, * ), T( LDT, * ), WORK( * )
14* ..
15*
16*
17*> \par Purpose:
18* =============
19*>
20*> \verbatim
21*>
22*> CLATSQR computes a blocked Tall-Skinny QR factorization of
23*> a complex M-by-N matrix A for M >= N:
24*>
25*> A = Q * ( R ),
26*> ( 0 )
27*>
28*> where:
29*>
30*> Q is a M-by-M orthogonal matrix, stored on exit in an implicit
31*> form in the elements below the diagonal of the array A and in
32*> the elements of the array T;
33*>
34*> R is an upper-triangular N-by-N matrix, stored on exit in
35*> the elements on and above the diagonal of the array A.
36*>
37*> 0 is a (M-N)-by-N zero matrix, and is not stored.
38*>
39*> \endverbatim
40*
41* Arguments:
42* ==========
43*
44*> \param[in] M
45*> \verbatim
46*> M is INTEGER
47*> The number of rows of the matrix A. M >= 0.
48*> \endverbatim
49*>
50*> \param[in] N
51*> \verbatim
52*> N is INTEGER
53*> The number of columns of the matrix A. M >= N >= 0.
54*> \endverbatim
55*>
56*> \param[in] MB
57*> \verbatim
58*> MB is INTEGER
59*> The row block size to be used in the blocked QR.
60*> MB > N.
61*> \endverbatim
62*>
63*> \param[in] NB
64*> \verbatim
65*> NB is INTEGER
66*> The column block size to be used in the blocked QR.
67*> N >= NB >= 1.
68*> \endverbatim
69*>
70*> \param[in,out] A
71*> \verbatim
72*> A is COMPLEX array, dimension (LDA,N)
73*> On entry, the M-by-N matrix A.
74*> On exit, the elements on and above the diagonal
75*> of the array contain the N-by-N upper triangular matrix R;
76*> the elements below the diagonal represent Q by the columns
77*> of blocked V (see Further Details).
78*> \endverbatim
79*>
80*> \param[in] LDA
81*> \verbatim
82*> LDA is INTEGER
83*> The leading dimension of the array A. LDA >= max(1,M).
84*> \endverbatim
85*>
86*> \param[out] T
87*> \verbatim
88*> T is COMPLEX array,
89*> dimension (LDT, N * Number_of_row_blocks)
90*> where Number_of_row_blocks = CEIL((M-N)/(MB-N))
91*> The blocked upper triangular block reflectors stored in compact form
92*> as a sequence of upper triangular blocks.
93*> See Further Details below.
94*> \endverbatim
95*>
96*> \param[in] LDT
97*> \verbatim
98*> LDT is INTEGER
99*> The leading dimension of the array T. LDT >= NB.
100*> \endverbatim
101*>
102*> \param[out] WORK
103*> \verbatim
104*> (workspace) COMPLEX array, dimension (MAX(1,LWORK))
105*> \endverbatim
106*>
107*> \param[in] LWORK
108*> \verbatim
109*> The dimension of the array WORK. LWORK >= NB*N.
110*> If LWORK = -1, then a workspace query is assumed; the routine
111*> only calculates the optimal size of the WORK array, returns
112*> this value as the first entry of the WORK array, and no error
113*> message related to LWORK is issued by XERBLA.
114*> \endverbatim
115*>
116*> \param[out] INFO
117*> \verbatim
118*> INFO is INTEGER
119*> = 0: successful exit
120*> < 0: if INFO = -i, the i-th argument had an illegal value
121*> \endverbatim
122*
123* Authors:
124* ========
125*
126*> \author Univ. of Tennessee
127*> \author Univ. of California Berkeley
128*> \author Univ. of Colorado Denver
129*> \author NAG Ltd.
130*
131*> \par Further Details:
132* =====================
133*>
134*> \verbatim
135*> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
136*> representing Q as a product of other orthogonal matrices
137*> Q = Q(1) * Q(2) * . . . * Q(k)
138*> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
139*> Q(1) zeros out the subdiagonal entries of rows 1:MB of A
140*> Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
141*> Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
142*> . . .
143*>
144*> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
145*> stored under the diagonal of rows 1:MB of A, and by upper triangular
146*> block reflectors, stored in array T(1:LDT,1:N).
147*> For more information see Further Details in GEQRT.
148*>
149*> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
150*> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
151*> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
152*> The last Q(k) may use fewer rows.
153*> For more information see Further Details in TPQRT.
154*>
155*> For more details of the overall algorithm, see the description of
156*> Sequential TSQR in Section 2.2 of [1].
157*>
158*> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
159*> J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
160*> SIAM J. Sci. Comput, vol. 34, no. 1, 2012
161*> \endverbatim
162*>
163* =====================================================================
164 SUBROUTINE clatsqr( M, N, MB, NB, A, LDA, T, LDT, WORK,
165 \$ LWORK, INFO)
166*
167* -- LAPACK computational routine --
168* -- LAPACK is a software package provided by Univ. of Tennessee, --
169* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
170*
171* .. Scalar Arguments ..
172 INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK
173* ..
174* .. Array Arguments ..
175 COMPLEX A( LDA, * ), WORK( * ), T(LDT, *)
176* ..
177*
178* =====================================================================
179*
180* ..
181* .. Local Scalars ..
182 LOGICAL LQUERY
183 INTEGER I, II, KK, CTR
184* ..
185* .. EXTERNAL FUNCTIONS ..
186 LOGICAL LSAME
187 EXTERNAL lsame
188* .. EXTERNAL SUBROUTINES ..
189 EXTERNAL cgeqrt, ctpqrt, xerbla
190* .. INTRINSIC FUNCTIONS ..
191 INTRINSIC max, min, mod
192* ..
193* .. EXECUTABLE STATEMENTS ..
194*
195* TEST THE INPUT ARGUMENTS
196*
197 info = 0
198*
199 lquery = ( lwork.EQ.-1 )
200*
201 IF( m.LT.0 ) THEN
202 info = -1
203 ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
204 info = -2
205 ELSE IF( mb.LT.1 ) THEN
206 info = -3
207 ELSE IF( nb.LT.1 .OR. ( nb.GT.n .AND. n.GT.0 )) THEN
208 info = -4
209 ELSE IF( lda.LT.max( 1, m ) ) THEN
210 info = -6
211 ELSE IF( ldt.LT.nb ) THEN
212 info = -8
213 ELSE IF( lwork.LT.(n*nb) .AND. (.NOT.lquery) ) THEN
214 info = -10
215 END IF
216 IF( info.EQ.0) THEN
217 work(1) = nb*n
218 END IF
219 IF( info.NE.0 ) THEN
220 CALL xerbla( 'CLATSQR', -info )
221 RETURN
222 ELSE IF (lquery) THEN
223 RETURN
224 END IF
225*
226* Quick return if possible
227*
228 IF( min(m,n).EQ.0 ) THEN
229 RETURN
230 END IF
231*
232* The QR Decomposition
233*
234 IF ((mb.LE.n).OR.(mb.GE.m)) THEN
235 CALL cgeqrt( m, n, nb, a, lda, t, ldt, work, info)
236 RETURN
237 END IF
238 kk = mod((m-n),(mb-n))
239 ii=m-kk+1
240*
241* Compute the QR factorization of the first block A(1:MB,1:N)
242*
243 CALL cgeqrt( mb, n, nb, a(1,1), lda, t, ldt, work, info )
244 ctr = 1
245*
246 DO i = mb+1, ii-mb+n , (mb-n)
247*
248* Compute the QR factorization of the current block A(I:I+MB-N,1:N)
249*
250 CALL ctpqrt( mb-n, n, 0, nb, a(1,1), lda, a( i, 1 ), lda,
251 \$ t(1,ctr * n + 1),
252 \$ ldt, work, info )
253 ctr = ctr + 1
254 END DO
255*
256* Compute the QR factorization of the last block A(II:M,1:N)
257*
258 IF (ii.LE.m) THEN
259 CALL ctpqrt( kk, n, 0, nb, a(1,1), lda, a( ii, 1 ), lda,
260 \$ t(1, ctr * n + 1), ldt,
261 \$ work, info )
262 END IF
263*
264 work( 1 ) = n*nb
265 RETURN
266*
267* End of CLATSQR
268*
269 END
subroutine clatsqr(M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
CLATSQR
Definition: clatsqr.f:166
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgeqrt(M, N, NB, A, LDA, T, LDT, WORK, INFO)
CGEQRT
Definition: cgeqrt.f:141
subroutine ctpqrt(M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
CTPQRT
Definition: ctpqrt.f:189