LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dlaqps.f
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1 *> \brief \b DLAQPS 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 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAQPS( 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 * DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
30 * \$ VN1( * ), VN2( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DLAQPS 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (KB)
112 *> The scalar factors of the elementary reflectors.
113 *> \endverbatim
114 *>
115 *> \param[in,out] VN1
116 *> \verbatim
117 *> VN1 is DOUBLE PRECISION array, dimension (N)
118 *> The vector with the partial column norms.
119 *> \endverbatim
120 *>
121 *> \param[in,out] VN2
122 *> \verbatim
123 *> VN2 is DOUBLE PRECISION array, dimension (N)
124 *> The vector with the exact column norms.
125 *> \endverbatim
126 *>
127 *> \param[in,out] AUXV
128 *> \verbatim
129 *> AUXV is DOUBLE PRECISION array, dimension (NB)
130 *> Auxiliar vector.
131 *> \endverbatim
132 *>
133 *> \param[in,out] F
134 *> \verbatim
135 *> F is DOUBLE PRECISION 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 September 2012
154 *
155 *> \ingroup doubleOTHERauxiliary
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 *> \n
163 *> Partial column norm updating strategy modified on April 2011
164 *> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
165 *> University of Zagreb, Croatia.
166 *
167 *> \par References:
168 * ================
169 *>
170 *> LAPACK Working Note 176
171 *
172 *> \htmlonly
173 *> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
174 *> \endhtmlonly
175 *
176 * =====================================================================
177  SUBROUTINE dlaqps( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
178  \$ vn2, auxv, f, ldf )
179 *
180 * -- LAPACK auxiliary routine (version 3.4.2) --
181 * -- LAPACK is a software package provided by Univ. of Tennessee, --
182 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183 * September 2012
184 *
185 * .. Scalar Arguments ..
186  INTEGER kb, lda, ldf, m, n, nb, offset
187 * ..
188 * .. Array Arguments ..
189  INTEGER jpvt( * )
190  DOUBLE PRECISION a( lda, * ), auxv( * ), f( ldf, * ), tau( * ),
191  \$ vn1( * ), vn2( * )
192 * ..
193 *
194 * =====================================================================
195 *
196 * .. Parameters ..
197  DOUBLE PRECISION zero, one
198  parameter( zero = 0.0d+0, one = 1.0d+0 )
199 * ..
200 * .. Local Scalars ..
201  INTEGER itemp, j, k, lastrk, lsticc, pvt, rk
202  DOUBLE PRECISION akk, temp, temp2, tol3z
203 * ..
204 * .. External Subroutines ..
205  EXTERNAL dgemm, dgemv, dlarfg, dswap
206 * ..
207 * .. Intrinsic Functions ..
208  INTRINSIC abs, dble, max, min, nint, sqrt
209 * ..
210 * .. External Functions ..
211  INTEGER idamax
212  DOUBLE PRECISION dlamch, dnrm2
213  EXTERNAL idamax, dlamch, dnrm2
214 * ..
215 * .. Executable Statements ..
216 *
217  lastrk = min( m, n+offset )
218  lsticc = 0
219  k = 0
220  tol3z = sqrt(dlamch('Epsilon'))
221 *
222 * Beginning of while loop.
223 *
224  10 continue
225  IF( ( k.LT.nb ) .AND. ( lsticc.EQ.0 ) ) THEN
226  k = k + 1
227  rk = offset + k
228 *
229 * Determine ith pivot column and swap if necessary
230 *
231  pvt = ( k-1 ) + idamax( n-k+1, vn1( k ), 1 )
232  IF( pvt.NE.k ) THEN
233  CALL dswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
234  CALL dswap( k-1, f( pvt, 1 ), ldf, f( k, 1 ), ldf )
235  itemp = jpvt( pvt )
236  jpvt( pvt ) = jpvt( k )
237  jpvt( k ) = itemp
238  vn1( pvt ) = vn1( k )
239  vn2( pvt ) = vn2( k )
240  END IF
241 *
242 * Apply previous Householder reflectors to column K:
243 * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
244 *
245  IF( k.GT.1 ) THEN
246  CALL dgemv( 'No transpose', m-rk+1, k-1, -one, a( rk, 1 ),
247  \$ lda, f( k, 1 ), ldf, one, a( rk, k ), 1 )
248  END IF
249 *
250 * Generate elementary reflector H(k).
251 *
252  IF( rk.LT.m ) THEN
253  CALL dlarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
254  ELSE
255  CALL dlarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
256  END IF
257 *
258  akk = a( rk, k )
259  a( rk, k ) = one
260 *
261 * Compute Kth column of F:
262 *
263 * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
264 *
265  IF( k.LT.n ) THEN
266  CALL dgemv( 'Transpose', m-rk+1, n-k, tau( k ),
267  \$ a( rk, k+1 ), lda, a( rk, k ), 1, zero,
268  \$ f( k+1, k ), 1 )
269  END IF
270 *
271 * Padding F(1:K,K) with zeros.
272 *
273  DO 20 j = 1, k
274  f( j, k ) = zero
275  20 continue
276 *
277 * Incremental updating of F:
278 * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
279 * *A(RK:M,K).
280 *
281  IF( k.GT.1 ) THEN
282  CALL dgemv( 'Transpose', m-rk+1, k-1, -tau( k ), a( rk, 1 ),
283  \$ lda, a( rk, k ), 1, zero, auxv( 1 ), 1 )
284 *
285  CALL dgemv( 'No transpose', n, k-1, one, f( 1, 1 ), ldf,
286  \$ auxv( 1 ), 1, one, f( 1, k ), 1 )
287  END IF
288 *
289 * Update the current row of A:
290 * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
291 *
292  IF( k.LT.n ) THEN
293  CALL dgemv( 'No transpose', n-k, k, -one, f( k+1, 1 ), ldf,
294  \$ a( rk, 1 ), lda, one, a( rk, k+1 ), lda )
295  END IF
296 *
297 * Update partial column norms.
298 *
299  IF( rk.LT.lastrk ) THEN
300  DO 30 j = k + 1, n
301  IF( vn1( j ).NE.zero ) THEN
302 *
303 * NOTE: The following 4 lines follow from the analysis in
304 * Lapack Working Note 176.
305 *
306  temp = abs( a( rk, j ) ) / vn1( j )
307  temp = max( zero, ( one+temp )*( one-temp ) )
308  temp2 = temp*( vn1( j ) / vn2( j ) )**2
309  IF( temp2 .LE. tol3z ) THEN
310  vn2( j ) = dble( lsticc )
311  lsticc = j
312  ELSE
313  vn1( j ) = vn1( j )*sqrt( temp )
314  END IF
315  END IF
316  30 continue
317  END IF
318 *
319  a( rk, k ) = akk
320 *
321 * End of while loop.
322 *
323  go to 10
324  END IF
325  kb = k
326  rk = offset + kb
327 *
328 * Apply the block reflector to the rest of the matrix:
329 * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
330 * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
331 *
332  IF( kb.LT.min( n, m-offset ) ) THEN
333  CALL dgemm( 'No transpose', 'Transpose', m-rk, n-kb, kb, -one,
334  \$ a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf, one,
335  \$ a( rk+1, kb+1 ), lda )
336  END IF
337 *
338 * Recomputation of difficult columns.
339 *
340  40 continue
341  IF( lsticc.GT.0 ) THEN
342  itemp = nint( vn2( lsticc ) )
343  vn1( lsticc ) = dnrm2( m-rk, a( rk+1, lsticc ), 1 )
344 *
345 * NOTE: The computation of VN1( LSTICC ) relies on the fact that
346 * SNRM2 does not fail on vectors with norm below the value of
347 * SQRT(DLAMCH('S'))
348 *
349  vn2( lsticc ) = vn1( lsticc )
350  lsticc = itemp
351  go to 40
352  END IF
353 *
354  return
355 *
356 * End of DLAQPS
357 *
358  END