LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
zlatsqr.f
Go to the documentation of this file.
1 *> \brief \b ZLATSQR
2 *
3 * Definition:
4 * ===========
5 *
6 * SUBROUTINE ZLATSQR( 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*16 A( LDA, * ), T( LDT, * ), WORK( * )
14 * ..
15 *
16 *
17 *> \par Purpose:
18 * =============
19 *>
20 *> \verbatim
21 *>
22 *> ZLATSQR 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*16 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*16 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*16 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).
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.
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 zlatsqr( 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*16 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 zgeqrt, ztpqrt, 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.LE.n ) 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 = -5
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( 'ZLATSQR', -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 zgeqrt( 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 zgeqrt( 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 ztpqrt( 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 ztpqrt( 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 ZLATSQR
268 *
269  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgeqrt(M, N, NB, A, LDA, T, LDT, WORK, INFO)
ZGEQRT
Definition: zgeqrt.f:141
subroutine ztpqrt(M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
ZTPQRT
Definition: ztpqrt.f:189
subroutine zlatsqr(M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
ZLATSQR
Definition: zlatsqr.f:166