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sgeqp3.f
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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
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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 *> \date September 2012
124 *
125 *> \ingroup realGEcomputational
126 *
127 *> \par Further Details:
128 * =====================
129 *>
130 *> \verbatim
131 *>
132 *> The matrix Q is represented as a product of elementary reflectors
133 *>
134 *> Q = H(1) H(2) . . . H(k), where k = min(m,n).
135 *>
136 *> Each H(i) has the form
137 *>
138 *> H(i) = I - tau * v * v**T
139 *>
140 *> where tau is a real scalar, and v is a real/complex vector
141 *> with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
142 *> A(i+1:m,i), and tau in TAU(i).
143 *> \endverbatim
144 *
145 *> \par Contributors:
146 * ==================
147 *>
148 *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
149 *> X. Sun, Computer Science Dept., Duke University, USA
150 *>
151 * =====================================================================
152  SUBROUTINE sgeqp3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
153 *
154 * -- LAPACK computational routine (version 3.4.2) --
155 * -- LAPACK is a software package provided by Univ. of Tennessee, --
156 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157 * September 2012
158 *
159 * .. Scalar Arguments ..
160  INTEGER info, lda, lwork, m, n
161 * ..
162 * .. Array Arguments ..
163  INTEGER jpvt( * )
164  REAL a( lda, * ), tau( * ), work( * )
165 * ..
166 *
167 * =====================================================================
168 *
169 * .. Parameters ..
170  INTEGER inb, inbmin, ixover
171  parameter( inb = 1, inbmin = 2, ixover = 3 )
172 * ..
173 * .. Local Scalars ..
174  LOGICAL lquery
175  INTEGER fjb, iws, j, jb, lwkopt, minmn, minws, na, nb,
176  $ nbmin, nfxd, nx, sm, sminmn, sn, topbmn
177 * ..
178 * .. External Subroutines ..
179  EXTERNAL sgeqrf, slaqp2, slaqps, sormqr, sswap, xerbla
180 * ..
181 * .. External Functions ..
182  INTEGER ilaenv
183  REAL snrm2
184  EXTERNAL ilaenv, snrm2
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC int, max, min
188 * ..
189 * .. Executable Statements ..
190 *
191  info = 0
192  lquery = ( lwork.EQ.-1 )
193  IF( m.LT.0 ) THEN
194  info = -1
195  ELSE IF( n.LT.0 ) THEN
196  info = -2
197  ELSE IF( lda.LT.max( 1, m ) ) THEN
198  info = -4
199  END IF
200 *
201  IF( info.EQ.0 ) THEN
202  minmn = min( m, n )
203  IF( minmn.EQ.0 ) THEN
204  iws = 1
205  lwkopt = 1
206  ELSE
207  iws = 3*n + 1
208  nb = ilaenv( inb, 'SGEQRF', ' ', m, n, -1, -1 )
209  lwkopt = 2*n + ( n + 1 )*nb
210  END IF
211  work( 1 ) = lwkopt
212 *
213  IF( ( lwork.LT.iws ) .AND. .NOT.lquery ) THEN
214  info = -8
215  END IF
216  END IF
217 *
218  IF( info.NE.0 ) THEN
219  CALL xerbla( 'SGEQP3', -info )
220  RETURN
221  ELSE IF( lquery ) THEN
222  RETURN
223  END IF
224 *
225 * Quick return if possible.
226 *
227  IF( minmn.EQ.0 ) THEN
228  RETURN
229  END IF
230 *
231 * Move initial columns up front.
232 *
233  nfxd = 1
234  DO 10 j = 1, n
235  IF( jpvt( j ).NE.0 ) THEN
236  IF( j.NE.nfxd ) THEN
237  CALL sswap( m, a( 1, j ), 1, a( 1, nfxd ), 1 )
238  jpvt( j ) = jpvt( nfxd )
239  jpvt( nfxd ) = j
240  ELSE
241  jpvt( j ) = j
242  END IF
243  nfxd = nfxd + 1
244  ELSE
245  jpvt( j ) = j
246  END IF
247  10 CONTINUE
248  nfxd = nfxd - 1
249 *
250 * Factorize fixed columns
251 * =======================
252 *
253 * Compute the QR factorization of fixed columns and update
254 * remaining columns.
255 *
256  IF( nfxd.GT.0 ) THEN
257  na = min( m, nfxd )
258 *CC CALL SGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
259  CALL sgeqrf( m, na, a, lda, tau, work, lwork, info )
260  iws = max( iws, int( work( 1 ) ) )
261  IF( na.LT.n ) THEN
262 *CC CALL SORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
263 *CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
264  CALL sormqr( 'Left', 'Transpose', m, n-na, na, a, lda, tau,
265  $ a( 1, na+1 ), lda, work, lwork, info )
266  iws = max( iws, int( work( 1 ) ) )
267  END IF
268  END IF
269 *
270 * Factorize free columns
271 * ======================
272 *
273  IF( nfxd.LT.minmn ) THEN
274 *
275  sm = m - nfxd
276  sn = n - nfxd
277  sminmn = minmn - nfxd
278 *
279 * Determine the block size.
280 *
281  nb = ilaenv( inb, 'SGEQRF', ' ', sm, sn, -1, -1 )
282  nbmin = 2
283  nx = 0
284 *
285  IF( ( nb.GT.1 ) .AND. ( nb.LT.sminmn ) ) THEN
286 *
287 * Determine when to cross over from blocked to unblocked code.
288 *
289  nx = max( 0, ilaenv( ixover, 'SGEQRF', ' ', sm, sn, -1,
290  $ -1 ) )
291 *
292 *
293  IF( nx.LT.sminmn ) THEN
294 *
295 * Determine if workspace is large enough for blocked code.
296 *
297  minws = 2*sn + ( sn+1 )*nb
298  iws = max( iws, minws )
299  IF( lwork.LT.minws ) THEN
300 *
301 * Not enough workspace to use optimal NB: Reduce NB and
302 * determine the minimum value of NB.
303 *
304  nb = ( lwork-2*sn ) / ( sn+1 )
305  nbmin = max( 2, ilaenv( inbmin, 'SGEQRF', ' ', sm, sn,
306  $ -1, -1 ) )
307 *
308 *
309  END IF
310  END IF
311  END IF
312 *
313 * Initialize partial column norms. The first N elements of work
314 * store the exact column norms.
315 *
316  DO 20 j = nfxd + 1, n
317  work( j ) = snrm2( sm, a( nfxd+1, j ), 1 )
318  work( n+j ) = work( j )
319  20 CONTINUE
320 *
321  IF( ( nb.GE.nbmin ) .AND. ( nb.LT.sminmn ) .AND.
322  $ ( nx.LT.sminmn ) ) THEN
323 *
324 * Use blocked code initially.
325 *
326  j = nfxd + 1
327 *
328 * Compute factorization: while loop.
329 *
330 *
331  topbmn = minmn - nx
332  30 CONTINUE
333  IF( j.LE.topbmn ) THEN
334  jb = min( nb, topbmn-j+1 )
335 *
336 * Factorize JB columns among columns J:N.
337 *
338  CALL slaqps( m, n-j+1, j-1, jb, fjb, a( 1, j ), lda,
339  $ jpvt( j ), tau( j ), work( j ), work( n+j ),
340  $ work( 2*n+1 ), work( 2*n+jb+1 ), n-j+1 )
341 *
342  j = j + fjb
343  go to 30
344  END IF
345  ELSE
346  j = nfxd + 1
347  END IF
348 *
349 * Use unblocked code to factor the last or only block.
350 *
351 *
352  IF( j.LE.minmn )
353  $ CALL slaqp2( m, n-j+1, j-1, a( 1, j ), lda, jpvt( j ),
354  $ tau( j ), work( j ), work( n+j ),
355  $ work( 2*n+1 ) )
356 *
357  END IF
358 *
359  work( 1 ) = iws
360  RETURN
361 *
362 * End of SGEQP3
363 *
364  END