LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sgeqp3.f
Go to the documentation of this file.
1 *> \brief \b SGEQP3
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SGEQP3 + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgeqp3.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgeqp3.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgeqp3.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, LWORK, M, N
25 * ..
26 * .. Array Arguments ..
27 * INTEGER JPVT( * )
28 * REAL A( LDA, * ), TAU( * ), WORK( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SGEQP3 computes a QR factorization with column pivoting of a
38 *> matrix A: A*P = Q*R using Level 3 BLAS.
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. N >= 0.
54 *> \endverbatim
55 *>
56 *> \param[in,out] A
57 *> \verbatim
58 *> A is REAL array, dimension (LDA,N)
59 *> On entry, the M-by-N matrix A.
60 *> On exit, the upper triangle of the array contains the
61 *> min(M,N)-by-N upper trapezoidal matrix R; the elements below
62 *> the diagonal, together with the array TAU, represent the
63 *> orthogonal matrix Q as a product of min(M,N) elementary
64 *> reflectors.
65 *> \endverbatim
66 *>
67 *> \param[in] LDA
68 *> \verbatim
69 *> LDA is INTEGER
70 *> The leading dimension of the array A. LDA >= max(1,M).
71 *> \endverbatim
72 *>
73 *> \param[in,out] JPVT
74 *> \verbatim
75 *> JPVT is INTEGER array, dimension (N)
76 *> On entry, if JPVT(J).ne.0, the J-th column of A is permuted
77 *> to the front of A*P (a leading column); if JPVT(J)=0,
78 *> the J-th column of A is a free column.
79 *> On exit, if JPVT(J)=K, then the J-th column of A*P was the
80 *> the K-th column of A.
81 *> \endverbatim
82 *>
83 *> \param[out] TAU
84 *> \verbatim
85 *> TAU is REAL array, dimension (min(M,N))
86 *> The scalar factors of the elementary reflectors.
87 *> \endverbatim
88 *>
89 *> \param[out] WORK
90 *> \verbatim
91 *> WORK is REAL array, dimension (MAX(1,LWORK))
92 *> On exit, if INFO=0, WORK(1) returns the optimal LWORK.
93 *> \endverbatim
94 *>
95 *> \param[in] LWORK
96 *> \verbatim
97 *> LWORK is INTEGER
98 *> The dimension of the array WORK. LWORK >= 3*N+1.
99 *> For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
100 *> is the optimal blocksize.
101 *>
102 *> If LWORK = -1, then a workspace query is assumed; the routine
103 *> only calculates the optimal size of the WORK array, returns
104 *> this value as the first entry of the WORK array, and no error
105 *> message related to LWORK is issued by XERBLA.
106 *> \endverbatim
107 *>
108 *> \param[out] INFO
109 *> \verbatim
110 *> INFO is INTEGER
111 *> = 0: successful exit.
112 *> < 0: if INFO = -i, the i-th argument had an illegal value.
113 *> \endverbatim
114 *
115 * Authors:
116 * ========
117 *
118 *> \author Univ. of Tennessee
119 *> \author Univ. of California Berkeley
120 *> \author Univ. of Colorado Denver
121 *> \author NAG Ltd.
122 *
123 *> \ingroup realGEcomputational
124 *
125 *> \par Further Details:
126 * =====================
127 *>
128 *> \verbatim
129 *>
130 *> The matrix Q is represented as a product of elementary reflectors
131 *>
132 *> Q = H(1) H(2) . . . H(k), where k = min(m,n).
133 *>
134 *> Each H(i) has the form
135 *>
136 *> H(i) = I - tau * v * v**T
137 *>
138 *> where tau is a real scalar, and v is a real/complex vector
139 *> with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
140 *> A(i+1:m,i), and tau in TAU(i).
141 *> \endverbatim
142 *
143 *> \par Contributors:
144 * ==================
145 *>
146 *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
147 *> X. Sun, Computer Science Dept., Duke University, USA
148 *>
149 * =====================================================================
150  SUBROUTINE sgeqp3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
151 *
152 * -- LAPACK computational routine --
153 * -- LAPACK is a software package provided by Univ. of Tennessee, --
154 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155 *
156 * .. Scalar Arguments ..
157  INTEGER INFO, LDA, LWORK, M, N
158 * ..
159 * .. Array Arguments ..
160  INTEGER JPVT( * )
161  REAL A( LDA, * ), TAU( * ), WORK( * )
162 * ..
163 *
164 * =====================================================================
165 *
166 * .. Parameters ..
167  INTEGER INB, INBMIN, IXOVER
168  parameter( inb = 1, inbmin = 2, ixover = 3 )
169 * ..
170 * .. Local Scalars ..
171  LOGICAL LQUERY
172  INTEGER FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
173  $ NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
174 * ..
175 * .. External Subroutines ..
176  EXTERNAL sgeqrf, slaqp2, slaqps, sormqr, sswap, xerbla
177 * ..
178 * .. External Functions ..
179  INTEGER ILAENV
180  REAL SNRM2
181  EXTERNAL ilaenv, snrm2
182 * ..
183 * .. Intrinsic Functions ..
184  INTRINSIC int, max, min
185 * Test input arguments
186 * ====================
187 *
188  info = 0
189  lquery = ( lwork.EQ.-1 )
190  IF( m.LT.0 ) THEN
191  info = -1
192  ELSE IF( n.LT.0 ) THEN
193  info = -2
194  ELSE IF( lda.LT.max( 1, m ) ) THEN
195  info = -4
196  END IF
197 *
198  IF( info.EQ.0 ) THEN
199  minmn = min( m, n )
200  IF( minmn.EQ.0 ) THEN
201  iws = 1
202  lwkopt = 1
203  ELSE
204  iws = 3*n + 1
205  nb = ilaenv( inb, 'SGEQRF', ' ', m, n, -1, -1 )
206  lwkopt = 2*n + ( n + 1 )*nb
207  END IF
208  work( 1 ) = lwkopt
209 *
210  IF( ( lwork.LT.iws ) .AND. .NOT.lquery ) THEN
211  info = -8
212  END IF
213  END IF
214 *
215  IF( info.NE.0 ) THEN
216  CALL xerbla( 'SGEQP3', -info )
217  RETURN
218  ELSE IF( lquery ) THEN
219  RETURN
220  END IF
221 *
222 * Move initial columns up front.
223 *
224  nfxd = 1
225  DO 10 j = 1, n
226  IF( jpvt( j ).NE.0 ) THEN
227  IF( j.NE.nfxd ) THEN
228  CALL sswap( m, a( 1, j ), 1, a( 1, nfxd ), 1 )
229  jpvt( j ) = jpvt( nfxd )
230  jpvt( nfxd ) = j
231  ELSE
232  jpvt( j ) = j
233  END IF
234  nfxd = nfxd + 1
235  ELSE
236  jpvt( j ) = j
237  END IF
238  10 CONTINUE
239  nfxd = nfxd - 1
240 *
241 * Factorize fixed columns
242 * =======================
243 *
244 * Compute the QR factorization of fixed columns and update
245 * remaining columns.
246 *
247  IF( nfxd.GT.0 ) THEN
248  na = min( m, nfxd )
249 *CC CALL SGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
250  CALL sgeqrf( m, na, a, lda, tau, work, lwork, info )
251  iws = max( iws, int( work( 1 ) ) )
252  IF( na.LT.n ) THEN
253 *CC CALL SORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
254 *CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
255  CALL sormqr( 'Left', 'Transpose', m, n-na, na, a, lda, tau,
256  $ a( 1, na+1 ), lda, work, lwork, info )
257  iws = max( iws, int( work( 1 ) ) )
258  END IF
259  END IF
260 *
261 * Factorize free columns
262 * ======================
263 *
264  IF( nfxd.LT.minmn ) THEN
265 *
266  sm = m - nfxd
267  sn = n - nfxd
268  sminmn = minmn - nfxd
269 *
270 * Determine the block size.
271 *
272  nb = ilaenv( inb, 'SGEQRF', ' ', sm, sn, -1, -1 )
273  nbmin = 2
274  nx = 0
275 *
276  IF( ( nb.GT.1 ) .AND. ( nb.LT.sminmn ) ) THEN
277 *
278 * Determine when to cross over from blocked to unblocked code.
279 *
280  nx = max( 0, ilaenv( ixover, 'SGEQRF', ' ', sm, sn, -1,
281  $ -1 ) )
282 *
283 *
284  IF( nx.LT.sminmn ) THEN
285 *
286 * Determine if workspace is large enough for blocked code.
287 *
288  minws = 2*sn + ( sn+1 )*nb
289  iws = max( iws, minws )
290  IF( lwork.LT.minws ) THEN
291 *
292 * Not enough workspace to use optimal NB: Reduce NB and
293 * determine the minimum value of NB.
294 *
295  nb = ( lwork-2*sn ) / ( sn+1 )
296  nbmin = max( 2, ilaenv( inbmin, 'SGEQRF', ' ', sm, sn,
297  $ -1, -1 ) )
298 *
299 *
300  END IF
301  END IF
302  END IF
303 *
304 * Initialize partial column norms. The first N elements of work
305 * store the exact column norms.
306 *
307  DO 20 j = nfxd + 1, n
308  work( j ) = snrm2( sm, a( nfxd+1, j ), 1 )
309  work( n+j ) = work( j )
310  20 CONTINUE
311 *
312  IF( ( nb.GE.nbmin ) .AND. ( nb.LT.sminmn ) .AND.
313  $ ( nx.LT.sminmn ) ) THEN
314 *
315 * Use blocked code initially.
316 *
317  j = nfxd + 1
318 *
319 * Compute factorization: while loop.
320 *
321 *
322  topbmn = minmn - nx
323  30 CONTINUE
324  IF( j.LE.topbmn ) THEN
325  jb = min( nb, topbmn-j+1 )
326 *
327 * Factorize JB columns among columns J:N.
328 *
329  CALL slaqps( m, n-j+1, j-1, jb, fjb, a( 1, j ), lda,
330  $ jpvt( j ), tau( j ), work( j ), work( n+j ),
331  $ work( 2*n+1 ), work( 2*n+jb+1 ), n-j+1 )
332 *
333  j = j + fjb
334  GO TO 30
335  END IF
336  ELSE
337  j = nfxd + 1
338  END IF
339 *
340 * Use unblocked code to factor the last or only block.
341 *
342 *
343  IF( j.LE.minmn )
344  $ CALL slaqp2( m, n-j+1, j-1, a( 1, j ), lda, jpvt( j ),
345  $ tau( j ), work( j ), work( n+j ),
346  $ work( 2*n+1 ) )
347 *
348  END IF
349 *
350  work( 1 ) = iws
351  RETURN
352 *
353 * End of SGEQP3
354 *
355  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
SGEQP3
Definition: sgeqp3.f:151
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:145
subroutine slaqp2(M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
SLAQP2 computes a QR factorization with column pivoting of the matrix block.
Definition: slaqp2.f:149
subroutine slaqps(M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BL...
Definition: slaqps.f:178
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82