LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
claqps.f
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1 *> \brief \b CLAQPS 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 CLAQPS( 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 VN1( * ), VN2( * )
30 * COMPLEX A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CLAQPS computes a step of QR factorization with column pivoting
40 *> of a complex 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (NB)
130 *> Auxiliar vector.
131 *> \endverbatim
132 *>
133 *> \param[in,out] F
134 *> \verbatim
135 *> F is COMPLEX array, dimension (LDF,NB)
136 *> Matrix F**H = L * Y**H * 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 complexOTHERauxiliary
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 claqps( 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 VN1( * ), VN2( * )
192  COMPLEX A( lda, * ), AUXV( * ), F( ldf, * ), TAU( * )
193 * ..
194 *
195 * =====================================================================
196 *
197 * .. Parameters ..
198  REAL ZERO, ONE
199  COMPLEX CZERO, CONE
200  parameter( zero = 0.0e+0, one = 1.0e+0,
201  \$ czero = ( 0.0e+0, 0.0e+0 ),
202  \$ cone = ( 1.0e+0, 0.0e+0 ) )
203 * ..
204 * .. Local Scalars ..
205  INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
206  REAL TEMP, TEMP2, TOL3Z
207  COMPLEX AKK
208 * ..
209 * .. External Subroutines ..
210  EXTERNAL cgemm, cgemv, clarfg, cswap
211 * ..
212 * .. Intrinsic Functions ..
213  INTRINSIC abs, conjg, max, min, nint, REAL, SQRT
214 * ..
215 * .. External Functions ..
216  INTEGER ISAMAX
217  REAL SCNRM2, SLAMCH
218  EXTERNAL isamax, scnrm2, slamch
219 * ..
220 * .. Executable Statements ..
221 *
222  lastrk = min( m, n+offset )
223  lsticc = 0
224  k = 0
225  tol3z = sqrt(slamch('Epsilon'))
226 *
227 * Beginning of while loop.
228 *
229  10 CONTINUE
230  IF( ( k.LT.nb ) .AND. ( lsticc.EQ.0 ) ) THEN
231  k = k + 1
232  rk = offset + k
233 *
234 * Determine ith pivot column and swap if necessary
235 *
236  pvt = ( k-1 ) + isamax( n-k+1, vn1( k ), 1 )
237  IF( pvt.NE.k ) THEN
238  CALL cswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
239  CALL cswap( k-1, f( pvt, 1 ), ldf, f( k, 1 ), ldf )
240  itemp = jpvt( pvt )
241  jpvt( pvt ) = jpvt( k )
242  jpvt( k ) = itemp
243  vn1( pvt ) = vn1( k )
244  vn2( pvt ) = vn2( k )
245  END IF
246 *
247 * Apply previous Householder reflectors to column K:
248 * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H.
249 *
250  IF( k.GT.1 ) THEN
251  DO 20 j = 1, k - 1
252  f( k, j ) = conjg( f( k, j ) )
253  20 CONTINUE
254  CALL cgemv( 'No transpose', m-rk+1, k-1, -cone, a( rk, 1 ),
255  \$ lda, f( k, 1 ), ldf, cone, a( rk, k ), 1 )
256  DO 30 j = 1, k - 1
257  f( k, j ) = conjg( f( k, j ) )
258  30 CONTINUE
259  END IF
260 *
261 * Generate elementary reflector H(k).
262 *
263  IF( rk.LT.m ) THEN
264  CALL clarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
265  ELSE
266  CALL clarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
267  END IF
268 *
269  akk = a( rk, k )
270  a( rk, k ) = cone
271 *
272 * Compute Kth column of F:
273 *
274 * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K).
275 *
276  IF( k.LT.n ) THEN
277  CALL cgemv( 'Conjugate transpose', m-rk+1, n-k, tau( k ),
278  \$ a( rk, k+1 ), lda, a( rk, k ), 1, czero,
279  \$ f( k+1, k ), 1 )
280  END IF
281 *
282 * Padding F(1:K,K) with zeros.
283 *
284  DO 40 j = 1, k
285  f( j, k ) = czero
286  40 CONTINUE
287 *
288 * Incremental updating of F:
289 * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H
290 * *A(RK:M,K).
291 *
292  IF( k.GT.1 ) THEN
293  CALL cgemv( 'Conjugate transpose', m-rk+1, k-1, -tau( k ),
294  \$ a( rk, 1 ), lda, a( rk, k ), 1, czero,
295  \$ auxv( 1 ), 1 )
296 *
297  CALL cgemv( 'No transpose', n, k-1, cone, f( 1, 1 ), ldf,
298  \$ auxv( 1 ), 1, cone, f( 1, k ), 1 )
299  END IF
300 *
301 * Update the current row of A:
302 * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H.
303 *
304  IF( k.LT.n ) THEN
305  CALL cgemm( 'No transpose', 'Conjugate transpose', 1, n-k,
306  \$ k, -cone, a( rk, 1 ), lda, f( k+1, 1 ), ldf,
307  \$ cone, a( rk, k+1 ), lda )
308  END IF
309 *
310 * Update partial column norms.
311 *
312  IF( rk.LT.lastrk ) THEN
313  DO 50 j = k + 1, n
314  IF( vn1( j ).NE.zero ) THEN
315 *
316 * NOTE: The following 4 lines follow from the analysis in
317 * Lapack Working Note 176.
318 *
319  temp = abs( a( rk, j ) ) / vn1( j )
320  temp = max( zero, ( one+temp )*( one-temp ) )
321  temp2 = temp*( vn1( j ) / vn2( j ) )**2
322  IF( temp2 .LE. tol3z ) THEN
323  vn2( j ) = REAL( lsticc )
324  lsticc = j
325  ELSE
326  vn1( j ) = vn1( j )*sqrt( temp )
327  END IF
328  END IF
329  50 CONTINUE
330  END IF
331 *
332  a( rk, k ) = akk
333 *
334 * End of while loop.
335 *
336  GO TO 10
337  END IF
338  kb = k
339  rk = offset + kb
340 *
341 * Apply the block reflector to the rest of the matrix:
342 * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
343 * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H.
344 *
345  IF( kb.LT.min( n, m-offset ) ) THEN
346  CALL cgemm( 'No transpose', 'Conjugate transpose', m-rk, n-kb,
347  \$ kb, -cone, a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf,
348  \$ cone, a( rk+1, kb+1 ), lda )
349  END IF
350 *
351 * Recomputation of difficult columns.
352 *
353  60 CONTINUE
354  IF( lsticc.GT.0 ) THEN
355  itemp = nint( vn2( lsticc ) )
356  vn1( lsticc ) = scnrm2( m-rk, a( rk+1, lsticc ), 1 )
357 *
358 * NOTE: The computation of VN1( LSTICC ) relies on the fact that
359 * SNRM2 does not fail on vectors with norm below the value of
360 * SQRT(DLAMCH('S'))
361 *
362  vn2( lsticc ) = vn1( lsticc )
363  lsticc = itemp
364  GO TO 60
365  END IF
366 *
367  RETURN
368 *
369 * End of CLAQPS
370 *
371  END
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:108
subroutine claqps(M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BL...
Definition: claqps.f:180
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:83
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:189