LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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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*
8*> \htmlonly
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqps.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqps.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqps.f">
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*> Auxiliary 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*> \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*> \n
161*> Partial column norm updating strategy modified on April 2011
162*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
163*> University of Zagreb, Croatia.
164*
165*> \par References:
166* ================
167*>
168*> LAPACK Working Note 176
169*
170*> \htmlonly
171*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
172*> \endhtmlonly
173*
174* =====================================================================
175 SUBROUTINE dlaqps( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
176 \$ VN2, AUXV, F, LDF )
177*
178* -- LAPACK auxiliary routine --
179* -- LAPACK is a software package provided by Univ. of Tennessee, --
180* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181*
182* .. Scalar Arguments ..
183 INTEGER KB, LDA, LDF, M, N, NB, OFFSET
184* ..
185* .. Array Arguments ..
186 INTEGER JPVT( * )
187 DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
188 \$ vn1( * ), vn2( * )
189* ..
190*
191* =====================================================================
192*
193* .. Parameters ..
194 DOUBLE PRECISION ZERO, ONE
195 parameter( zero = 0.0d+0, one = 1.0d+0 )
196* ..
197* .. Local Scalars ..
198 INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
199 DOUBLE PRECISION AKK, TEMP, TEMP2, TOL3Z
200* ..
201* .. External Subroutines ..
202 EXTERNAL dgemm, dgemv, dlarfg, dswap
203* ..
204* .. Intrinsic Functions ..
205 INTRINSIC abs, dble, max, min, nint, sqrt
206* ..
207* .. External Functions ..
208 INTEGER IDAMAX
209 DOUBLE PRECISION DLAMCH, DNRM2
210 EXTERNAL idamax, dlamch, dnrm2
211* ..
212* .. Executable Statements ..
213*
214 lastrk = min( m, n+offset )
215 lsticc = 0
216 k = 0
217 tol3z = sqrt(dlamch('Epsilon'))
218*
219* Beginning of while loop.
220*
221 10 CONTINUE
222 IF( ( k.LT.nb ) .AND. ( lsticc.EQ.0 ) ) THEN
223 k = k + 1
224 rk = offset + k
225*
226* Determine ith pivot column and swap if necessary
227*
228 pvt = ( k-1 ) + idamax( n-k+1, vn1( k ), 1 )
229 IF( pvt.NE.k ) THEN
230 CALL dswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
231 CALL dswap( k-1, f( pvt, 1 ), ldf, f( k, 1 ), ldf )
232 itemp = jpvt( pvt )
233 jpvt( pvt ) = jpvt( k )
234 jpvt( k ) = itemp
235 vn1( pvt ) = vn1( k )
236 vn2( pvt ) = vn2( k )
237 END IF
238*
239* Apply previous Householder reflectors to column K:
240* A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
241*
242 IF( k.GT.1 ) THEN
243 CALL dgemv( 'No transpose', m-rk+1, k-1, -one, a( rk, 1 ),
244 \$ lda, f( k, 1 ), ldf, one, a( rk, k ), 1 )
245 END IF
246*
247* Generate elementary reflector H(k).
248*
249 IF( rk.LT.m ) THEN
250 CALL dlarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
251 ELSE
252 CALL dlarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
253 END IF
254*
255 akk = a( rk, k )
256 a( rk, k ) = one
257*
258* Compute Kth column of F:
259*
260* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
261*
262 IF( k.LT.n ) THEN
263 CALL dgemv( 'Transpose', m-rk+1, n-k, tau( k ),
264 \$ a( rk, k+1 ), lda, a( rk, k ), 1, zero,
265 \$ f( k+1, k ), 1 )
266 END IF
267*
268* Padding F(1:K,K) with zeros.
269*
270 DO 20 j = 1, k
271 f( j, k ) = zero
272 20 CONTINUE
273*
274* Incremental updating of F:
275* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
276* *A(RK:M,K).
277*
278 IF( k.GT.1 ) THEN
279 CALL dgemv( 'Transpose', m-rk+1, k-1, -tau( k ), a( rk, 1 ),
280 \$ lda, a( rk, k ), 1, zero, auxv( 1 ), 1 )
281*
282 CALL dgemv( 'No transpose', n, k-1, one, f( 1, 1 ), ldf,
283 \$ auxv( 1 ), 1, one, f( 1, k ), 1 )
284 END IF
285*
286* Update the current row of A:
287* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
288*
289 IF( k.LT.n ) THEN
290 CALL dgemv( 'No transpose', n-k, k, -one, f( k+1, 1 ), ldf,
291 \$ a( rk, 1 ), lda, one, a( rk, k+1 ), lda )
292 END IF
293*
294* Update partial column norms.
295*
296 IF( rk.LT.lastrk ) THEN
297 DO 30 j = k + 1, n
298 IF( vn1( j ).NE.zero ) THEN
299*
300* NOTE: The following 4 lines follow from the analysis in
301* Lapack Working Note 176.
302*
303 temp = abs( a( rk, j ) ) / vn1( j )
304 temp = max( zero, ( one+temp )*( one-temp ) )
305 temp2 = temp*( vn1( j ) / vn2( j ) )**2
306 IF( temp2 .LE. tol3z ) THEN
307 vn2( j ) = dble( lsticc )
308 lsticc = j
309 ELSE
310 vn1( j ) = vn1( j )*sqrt( temp )
311 END IF
312 END IF
313 30 CONTINUE
314 END IF
315*
316 a( rk, k ) = akk
317*
318* End of while loop.
319*
320 GO TO 10
321 END IF
322 kb = k
323 rk = offset + kb
324*
325* Apply the block reflector to the rest of the matrix:
326* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
327* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
328*
329 IF( kb.LT.min( n, m-offset ) ) THEN
330 CALL dgemm( 'No transpose', 'Transpose', m-rk, n-kb, kb, -one,
331 \$ a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf, one,
332 \$ a( rk+1, kb+1 ), lda )
333 END IF
334*
335* Recomputation of difficult columns.
336*
337 40 CONTINUE
338 IF( lsticc.GT.0 ) THEN
339 itemp = nint( vn2( lsticc ) )
340 vn1( lsticc ) = dnrm2( m-rk, a( rk+1, lsticc ), 1 )
341*
342* NOTE: The computation of VN1( LSTICC ) relies on the fact that
343* SNRM2 does not fail on vectors with norm below the value of
344* SQRT(DLAMCH('S'))
345*
346 vn2( lsticc ) = vn1( lsticc )
347 lsticc = itemp
348 GO TO 40
349 END IF
350*
351 RETURN
352*
353* End of DLAQPS
354*
355 END
subroutine dgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
DGEMM
Definition dgemm.f:188
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
subroutine dlaqps(m, n, offset, nb, kb, a, lda, jpvt, tau, vn1, vn2, auxv, f, ldf)
DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BL...
Definition dlaqps.f:177
subroutine dlarfg(n, alpha, x, incx, tau)
DLARFG generates an elementary reflector (Householder matrix).
Definition dlarfg.f:106
subroutine dswap(n, dx, incx, dy, incy)
DSWAP
Definition dswap.f:82