LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
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 *
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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 November 2015
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 3.6.0) --
155 * -- LAPACK is a software package provided by Univ. of Tennessee, --
156 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157 * November 2015
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 * Test input arguments
189 * ====================
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 * Move initial columns up front.
226 *
227  nfxd = 1
228  DO 10 j = 1, n
229  IF( jpvt( j ).NE.0 ) THEN
230  IF( j.NE.nfxd ) THEN
231  CALL sswap( m, a( 1, j ), 1, a( 1, nfxd ), 1 )
232  jpvt( j ) = jpvt( nfxd )
233  jpvt( nfxd ) = j
234  ELSE
235  jpvt( j ) = j
236  END IF
237  nfxd = nfxd + 1
238  ELSE
239  jpvt( j ) = j
240  END IF
241  10 CONTINUE
242  nfxd = nfxd - 1
243 *
244 * Factorize fixed columns
245 * =======================
246 *
247 * Compute the QR factorization of fixed columns and update
248 * remaining columns.
249 *
250  IF( nfxd.GT.0 ) THEN
251  na = min( m, nfxd )
252 *CC CALL SGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
253  CALL sgeqrf( m, na, a, lda, tau, work, lwork, info )
254  iws = max( iws, int( work( 1 ) ) )
255  IF( na.LT.n ) THEN
256 *CC CALL SORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
257 *CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
258  CALL sormqr( 'Left', 'Transpose', m, n-na, na, a, lda, tau,
259  $ a( 1, na+1 ), lda, work, lwork, info )
260  iws = max( iws, int( work( 1 ) ) )
261  END IF
262  END IF
263 *
264 * Factorize free columns
265 * ======================
266 *
267  IF( nfxd.LT.minmn ) THEN
268 *
269  sm = m - nfxd
270  sn = n - nfxd
271  sminmn = minmn - nfxd
272 *
273 * Determine the block size.
274 *
275  nb = ilaenv( inb, 'SGEQRF', ' ', sm, sn, -1, -1 )
276  nbmin = 2
277  nx = 0
278 *
279  IF( ( nb.GT.1 ) .AND. ( nb.LT.sminmn ) ) THEN
280 *
281 * Determine when to cross over from blocked to unblocked code.
282 *
283  nx = max( 0, ilaenv( ixover, 'SGEQRF', ' ', sm, sn, -1,
284  $ -1 ) )
285 *
286 *
287  IF( nx.LT.sminmn ) THEN
288 *
289 * Determine if workspace is large enough for blocked code.
290 *
291  minws = 2*sn + ( sn+1 )*nb
292  iws = max( iws, minws )
293  IF( lwork.LT.minws ) THEN
294 *
295 * Not enough workspace to use optimal NB: Reduce NB and
296 * determine the minimum value of NB.
297 *
298  nb = ( lwork-2*sn ) / ( sn+1 )
299  nbmin = max( 2, ilaenv( inbmin, 'SGEQRF', ' ', sm, sn,
300  $ -1, -1 ) )
301 *
302 *
303  END IF
304  END IF
305  END IF
306 *
307 * Initialize partial column norms. The first N elements of work
308 * store the exact column norms.
309 *
310  DO 20 j = nfxd + 1, n
311  work( j ) = snrm2( sm, a( nfxd+1, j ), 1 )
312  work( n+j ) = work( j )
313  20 CONTINUE
314 *
315  IF( ( nb.GE.nbmin ) .AND. ( nb.LT.sminmn ) .AND.
316  $ ( nx.LT.sminmn ) ) THEN
317 *
318 * Use blocked code initially.
319 *
320  j = nfxd + 1
321 *
322 * Compute factorization: while loop.
323 *
324 *
325  topbmn = minmn - nx
326  30 CONTINUE
327  IF( j.LE.topbmn ) THEN
328  jb = min( nb, topbmn-j+1 )
329 *
330 * Factorize JB columns among columns J:N.
331 *
332  CALL slaqps( m, n-j+1, j-1, jb, fjb, a( 1, j ), lda,
333  $ jpvt( j ), tau( j ), work( j ), work( n+j ),
334  $ work( 2*n+1 ), work( 2*n+jb+1 ), n-j+1 )
335 *
336  j = j + fjb
337  GO TO 30
338  END IF
339  ELSE
340  j = nfxd + 1
341  END IF
342 *
343 * Use unblocked code to factor the last or only block.
344 *
345 *
346  IF( j.LE.minmn )
347  $ CALL slaqp2( m, n-j+1, j-1, a( 1, j ), lda, jpvt( j ),
348  $ tau( j ), work( j ), work( n+j ),
349  $ work( 2*n+1 ) )
350 *
351  END IF
352 *
353  work( 1 ) = iws
354  RETURN
355 *
356 * End of SGEQP3
357 *
358  END
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:170
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
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:151
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:180
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:53
subroutine sgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
SGEQP3
Definition: sgeqp3.f:153