LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
slaqps.f
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1 *> \brief \b SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqps.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
22 * VN2, AUXV, F, LDF )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER KB, LDA, LDF, M, N, NB, OFFSET
26 * ..
27 * .. Array Arguments ..
28 * INTEGER JPVT( * )
29 * REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
30 * \$ VN1( * ), VN2( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> SLAQPS computes a step of QR factorization with column pivoting
40 *> of a real M-by-N matrix A by using Blas-3. It tries to factorize
41 *> NB columns from A starting from the row OFFSET+1, and updates all
42 *> of the matrix with Blas-3 xGEMM.
43 *>
44 *> In some cases, due to catastrophic cancellations, it cannot
45 *> factorize NB columns. Hence, the actual number of factorized
46 *> columns is returned in KB.
47 *>
48 *> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[in] M
55 *> \verbatim
56 *> M is INTEGER
57 *> The number of rows of the matrix A. M >= 0.
58 *> \endverbatim
59 *>
60 *> \param[in] N
61 *> \verbatim
62 *> N is INTEGER
63 *> The number of columns of the matrix A. N >= 0
64 *> \endverbatim
65 *>
66 *> \param[in] OFFSET
67 *> \verbatim
68 *> OFFSET is INTEGER
69 *> The number of rows of A that have been factorized in
70 *> previous steps.
71 *> \endverbatim
72 *>
73 *> \param[in] NB
74 *> \verbatim
75 *> NB is INTEGER
76 *> The number of columns to factorize.
77 *> \endverbatim
78 *>
79 *> \param[out] KB
80 *> \verbatim
81 *> KB is INTEGER
82 *> The number of columns actually factorized.
83 *> \endverbatim
84 *>
85 *> \param[in,out] A
86 *> \verbatim
87 *> A is REAL array, dimension (LDA,N)
88 *> On entry, the M-by-N matrix A.
89 *> On exit, block A(OFFSET+1:M,1:KB) is the triangular
90 *> factor obtained and block A(1:OFFSET,1:N) has been
91 *> accordingly pivoted, but no factorized.
92 *> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
93 *> been updated.
94 *> \endverbatim
95 *>
96 *> \param[in] LDA
97 *> \verbatim
98 *> LDA is INTEGER
99 *> The leading dimension of the array A. LDA >= max(1,M).
100 *> \endverbatim
101 *>
102 *> \param[in,out] JPVT
103 *> \verbatim
104 *> JPVT is INTEGER array, dimension (N)
105 *> JPVT(I) = K <==> Column K of the full matrix A has been
106 *> permuted into position I in AP.
107 *> \endverbatim
108 *>
109 *> \param[out] TAU
110 *> \verbatim
111 *> TAU is REAL array, dimension (KB)
112 *> The scalar factors of the elementary reflectors.
113 *> \endverbatim
114 *>
115 *> \param[in,out] VN1
116 *> \verbatim
117 *> VN1 is REAL array, dimension (N)
118 *> The vector with the partial column norms.
119 *> \endverbatim
120 *>
121 *> \param[in,out] VN2
122 *> \verbatim
123 *> VN2 is REAL array, dimension (N)
124 *> The vector with the exact column norms.
125 *> \endverbatim
126 *>
127 *> \param[in,out] AUXV
128 *> \verbatim
129 *> AUXV is REAL array, dimension (NB)
130 *> Auxiliar vector.
131 *> \endverbatim
132 *>
133 *> \param[in,out] F
134 *> \verbatim
135 *> F is REAL array, dimension (LDF,NB)
136 *> Matrix F**T = L*Y**T*A.
137 *> \endverbatim
138 *>
139 *> \param[in] LDF
140 *> \verbatim
141 *> LDF is INTEGER
142 *> The leading dimension of the array F. LDF >= max(1,N).
143 *> \endverbatim
144 *
145 * Authors:
146 * ========
147 *
148 *> \author Univ. of Tennessee
149 *> \author Univ. of California Berkeley
150 *> \author Univ. of Colorado Denver
151 *> \author NAG Ltd.
152 *
153 *> \date December 2016
154 *
155 *> \ingroup realOTHERauxiliary
156 *
157 *> \par Contributors:
158 * ==================
159 *>
160 *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
161 *> X. Sun, Computer Science Dept., Duke University, USA
162 *>
163 *> \n
164 *> Partial column norm updating strategy modified on April 2011
165 *> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
166 *> University of Zagreb, Croatia.
167 *
168 *> \par References:
169 * ================
170 *>
171 *> LAPACK Working Note 176
172 *
173 *> \htmlonly
174 *> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
175 *> \endhtmlonly
176 *
177 * =====================================================================
178  SUBROUTINE slaqps( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
179  \$ VN2, AUXV, F, LDF )
180 *
181 * -- LAPACK auxiliary routine (version 3.7.0) --
182 * -- LAPACK is a software package provided by Univ. of Tennessee, --
183 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184 * December 2016
185 *
186 * .. Scalar Arguments ..
187  INTEGER KB, LDA, LDF, M, N, NB, OFFSET
188 * ..
189 * .. Array Arguments ..
190  INTEGER JPVT( * )
191  REAL A( lda, * ), AUXV( * ), F( ldf, * ), TAU( * ),
192  \$ vn1( * ), vn2( * )
193 * ..
194 *
195 * =====================================================================
196 *
197 * .. Parameters ..
198  REAL ZERO, ONE
199  parameter( zero = 0.0e+0, one = 1.0e+0 )
200 * ..
201 * .. Local Scalars ..
202  INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
203  REAL AKK, TEMP, TEMP2, TOL3Z
204 * ..
205 * .. External Subroutines ..
206  EXTERNAL sgemm, sgemv, slarfg, sswap
207 * ..
208 * .. Intrinsic Functions ..
209  INTRINSIC abs, max, min, nint, REAL, SQRT
210 * ..
211 * .. External Functions ..
212  INTEGER ISAMAX
213  REAL SLAMCH, SNRM2
214  EXTERNAL isamax, slamch, snrm2
215 * ..
216 * .. Executable Statements ..
217 *
218  lastrk = min( m, n+offset )
219  lsticc = 0
220  k = 0
221  tol3z = sqrt(slamch('Epsilon'))
222 *
223 * Beginning of while loop.
224 *
225  10 CONTINUE
226  IF( ( k.LT.nb ) .AND. ( lsticc.EQ.0 ) ) THEN
227  k = k + 1
228  rk = offset + k
229 *
230 * Determine ith pivot column and swap if necessary
231 *
232  pvt = ( k-1 ) + isamax( n-k+1, vn1( k ), 1 )
233  IF( pvt.NE.k ) THEN
234  CALL sswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
235  CALL sswap( k-1, f( pvt, 1 ), ldf, f( k, 1 ), ldf )
236  itemp = jpvt( pvt )
237  jpvt( pvt ) = jpvt( k )
238  jpvt( k ) = itemp
239  vn1( pvt ) = vn1( k )
240  vn2( pvt ) = vn2( k )
241  END IF
242 *
243 * Apply previous Householder reflectors to column K:
244 * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
245 *
246  IF( k.GT.1 ) THEN
247  CALL sgemv( 'No transpose', m-rk+1, k-1, -one, a( rk, 1 ),
248  \$ lda, f( k, 1 ), ldf, one, a( rk, k ), 1 )
249  END IF
250 *
251 * Generate elementary reflector H(k).
252 *
253  IF( rk.LT.m ) THEN
254  CALL slarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
255  ELSE
256  CALL slarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
257  END IF
258 *
259  akk = a( rk, k )
260  a( rk, k ) = one
261 *
262 * Compute Kth column of F:
263 *
264 * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
265 *
266  IF( k.LT.n ) THEN
267  CALL sgemv( 'Transpose', m-rk+1, n-k, tau( k ),
268  \$ a( rk, k+1 ), lda, a( rk, k ), 1, zero,
269  \$ f( k+1, k ), 1 )
270  END IF
271 *
272 * Padding F(1:K,K) with zeros.
273 *
274  DO 20 j = 1, k
275  f( j, k ) = zero
276  20 CONTINUE
277 *
278 * Incremental updating of F:
279 * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
280 * *A(RK:M,K).
281 *
282  IF( k.GT.1 ) THEN
283  CALL sgemv( 'Transpose', m-rk+1, k-1, -tau( k ), a( rk, 1 ),
284  \$ lda, a( rk, k ), 1, zero, auxv( 1 ), 1 )
285 *
286  CALL sgemv( 'No transpose', n, k-1, one, f( 1, 1 ), ldf,
287  \$ auxv( 1 ), 1, one, f( 1, k ), 1 )
288  END IF
289 *
290 * Update the current row of A:
291 * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
292 *
293  IF( k.LT.n ) THEN
294  CALL sgemv( 'No transpose', n-k, k, -one, f( k+1, 1 ), ldf,
295  \$ a( rk, 1 ), lda, one, a( rk, k+1 ), lda )
296  END IF
297 *
298 * Update partial column norms.
299 *
300  IF( rk.LT.lastrk ) THEN
301  DO 30 j = k + 1, n
302  IF( vn1( j ).NE.zero ) THEN
303 *
304 * NOTE: The following 4 lines follow from the analysis in
305 * Lapack Working Note 176.
306 *
307  temp = abs( a( rk, j ) ) / vn1( j )
308  temp = max( zero, ( one+temp )*( one-temp ) )
309  temp2 = temp*( vn1( j ) / vn2( j ) )**2
310  IF( temp2 .LE. tol3z ) THEN
311  vn2( j ) = REAL( lsticc )
312  lsticc = j
313  ELSE
314  vn1( j ) = vn1( j )*sqrt( temp )
315  END IF
316  END IF
317  30 CONTINUE
318  END IF
319 *
320  a( rk, k ) = akk
321 *
322 * End of while loop.
323 *
324  GO TO 10
325  END IF
326  kb = k
327  rk = offset + kb
328 *
329 * Apply the block reflector to the rest of the matrix:
330 * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
331 * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
332 *
333  IF( kb.LT.min( n, m-offset ) ) THEN
334  CALL sgemm( 'No transpose', 'Transpose', m-rk, n-kb, kb, -one,
335  \$ a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf, one,
336  \$ a( rk+1, kb+1 ), lda )
337  END IF
338 *
339 * Recomputation of difficult columns.
340 *
341  40 CONTINUE
342  IF( lsticc.GT.0 ) THEN
343  itemp = nint( vn2( lsticc ) )
344  vn1( lsticc ) = snrm2( m-rk, a( rk+1, lsticc ), 1 )
345 *
346 * NOTE: The computation of VN1( LSTICC ) relies on the fact that
347 * SNRM2 does not fail on vectors with norm below the value of
348 * SQRT(DLAMCH('S'))
349 *
350  vn2( lsticc ) = vn1( lsticc )
351  lsticc = itemp
352  GO TO 40
353  END IF
354 *
355  RETURN
356 *
357 * End of SLAQPS
358 *
359  END
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:108
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:84
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