LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqps.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqps.f">
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*> Auxiliary 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*> \ingroup laqps
154*
155*> \par Contributors:
156* ==================
157*>
158*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
159*> X. Sun, Computer Science Dept., Duke University, USA
160*>
161*> \n
162*> Partial column norm updating strategy modified on April 2011
163*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
164*> University of Zagreb, Croatia.
165*
166*> \par References:
167* ================
168*>
169*> LAPACK Working Note 176
170*
171*> \htmlonly
172*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
173*> \endhtmlonly
174*
175* =====================================================================
176 SUBROUTINE slaqps( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
177 \$ VN2, AUXV, F, LDF )
178*
179* -- LAPACK auxiliary routine --
180* -- LAPACK is a software package provided by Univ. of Tennessee, --
181* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
182*
183* .. Scalar Arguments ..
184 INTEGER KB, LDA, LDF, M, N, NB, OFFSET
185* ..
186* .. Array Arguments ..
187 INTEGER JPVT( * )
188 REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
189 \$ vn1( * ), vn2( * )
190* ..
191*
192* =====================================================================
193*
194* .. Parameters ..
195 REAL ZERO, ONE
196 parameter( zero = 0.0e+0, one = 1.0e+0 )
197* ..
198* .. Local Scalars ..
199 INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
200 REAL AKK, TEMP, TEMP2, TOL3Z
201* ..
202* .. External Subroutines ..
203 EXTERNAL sgemm, sgemv, slarfg, sswap
204* ..
205* .. Intrinsic Functions ..
206 INTRINSIC abs, max, min, nint, real, sqrt
207* ..
208* .. External Functions ..
209 INTEGER ISAMAX
210 REAL SLAMCH, SNRM2
211 EXTERNAL isamax, slamch, snrm2
212* ..
213* .. Executable Statements ..
214*
215 lastrk = min( m, n+offset )
216 lsticc = 0
217 k = 0
218 tol3z = sqrt(slamch('Epsilon'))
219*
220* Beginning of while loop.
221*
222 10 CONTINUE
223 IF( ( k.LT.nb ) .AND. ( lsticc.EQ.0 ) ) THEN
224 k = k + 1
225 rk = offset + k
226*
227* Determine ith pivot column and swap if necessary
228*
229 pvt = ( k-1 ) + isamax( n-k+1, vn1( k ), 1 )
230 IF( pvt.NE.k ) THEN
231 CALL sswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
232 CALL sswap( k-1, f( pvt, 1 ), ldf, f( k, 1 ), ldf )
233 itemp = jpvt( pvt )
234 jpvt( pvt ) = jpvt( k )
235 jpvt( k ) = itemp
236 vn1( pvt ) = vn1( k )
237 vn2( pvt ) = vn2( k )
238 END IF
239*
240* Apply previous Householder reflectors to column K:
241* A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
242*
243 IF( k.GT.1 ) THEN
244 CALL sgemv( 'No transpose', m-rk+1, k-1, -one, a( rk, 1 ),
245 \$ lda, f( k, 1 ), ldf, one, a( rk, k ), 1 )
246 END IF
247*
248* Generate elementary reflector H(k).
249*
250 IF( rk.LT.m ) THEN
251 CALL slarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
252 ELSE
253 CALL slarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
254 END IF
255*
256 akk = a( rk, k )
257 a( rk, k ) = one
258*
259* Compute Kth column of F:
260*
261* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
262*
263 IF( k.LT.n ) THEN
264 CALL sgemv( 'Transpose', m-rk+1, n-k, tau( k ),
265 \$ a( rk, k+1 ), lda, a( rk, k ), 1, zero,
266 \$ f( k+1, k ), 1 )
267 END IF
268*
270*
271 DO 20 j = 1, k
272 f( j, k ) = zero
273 20 CONTINUE
274*
275* Incremental updating of F:
276* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
277* *A(RK:M,K).
278*
279 IF( k.GT.1 ) THEN
280 CALL sgemv( 'Transpose', m-rk+1, k-1, -tau( k ), a( rk, 1 ),
281 \$ lda, a( rk, k ), 1, zero, auxv( 1 ), 1 )
282*
283 CALL sgemv( 'No transpose', n, k-1, one, f( 1, 1 ), ldf,
284 \$ auxv( 1 ), 1, one, f( 1, k ), 1 )
285 END IF
286*
287* Update the current row of A:
288* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
289*
290 IF( k.LT.n ) THEN
291 CALL sgemv( 'No transpose', n-k, k, -one, f( k+1, 1 ), ldf,
292 \$ a( rk, 1 ), lda, one, a( rk, k+1 ), lda )
293 END IF
294*
295* Update partial column norms.
296*
297 IF( rk.LT.lastrk ) THEN
298 DO 30 j = k + 1, n
299 IF( vn1( j ).NE.zero ) THEN
300*
301* NOTE: The following 4 lines follow from the analysis in
302* Lapack Working Note 176.
303*
304 temp = abs( a( rk, j ) ) / vn1( j )
305 temp = max( zero, ( one+temp )*( one-temp ) )
306 temp2 = temp*( vn1( j ) / vn2( j ) )**2
307 IF( temp2 .LE. tol3z ) THEN
308 vn2( j ) = real( lsticc )
309 lsticc = j
310 ELSE
311 vn1( j ) = vn1( j )*sqrt( temp )
312 END IF
313 END IF
314 30 CONTINUE
315 END IF
316*
317 a( rk, k ) = akk
318*
319* End of while loop.
320*
321 GO TO 10
322 END IF
323 kb = k
324 rk = offset + kb
325*
326* Apply the block reflector to the rest of the matrix:
327* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
328* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
329*
330 IF( kb.LT.min( n, m-offset ) ) THEN
331 CALL sgemm( 'No transpose', 'Transpose', m-rk, n-kb, kb, -one,
332 \$ a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf, one,
333 \$ a( rk+1, kb+1 ), lda )
334 END IF
335*
336* Recomputation of difficult columns.
337*
338 40 CONTINUE
339 IF( lsticc.GT.0 ) THEN
340 itemp = nint( vn2( lsticc ) )
341 vn1( lsticc ) = snrm2( m-rk, a( rk+1, lsticc ), 1 )
342*
343* NOTE: The computation of VN1( LSTICC ) relies on the fact that
344* SNRM2 does not fail on vectors with norm below the value of
345* SQRT(DLAMCH('S'))
346*
347 vn2( lsticc ) = vn1( lsticc )
348 lsticc = itemp
349 GO TO 40
350 END IF
351*
352 RETURN
353*
354* End of SLAQPS
355*
356 END
subroutine sgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
SGEMM
Definition sgemm.f:188
subroutine sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
SGEMV
Definition sgemv.f:158
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:178
subroutine slarfg(n, alpha, x, incx, tau)
SLARFG generates an elementary reflector (Householder matrix).
Definition slarfg.f:106
subroutine sswap(n, sx, incx, sy, incy)
SSWAP
Definition sswap.f:82