LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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zlaqps.f
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1*> \brief \b ZLAQPS 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/zlaqps.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqps.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqps.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZLAQPS( 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 VN1( * ), VN2( * )
30* COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> ZLAQPS 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*16 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*16 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 COMPLEX*16 array, dimension (NB)
130*> Auxiliary vector.
131*> \endverbatim
132*>
133*> \param[in,out] F
134*> \verbatim
135*> F is COMPLEX*16 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*> \ingroup complex16OTHERauxiliary
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 zlaqps( 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 VN1( * ), VN2( * )
188 COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
189* ..
190*
191* =====================================================================
192*
193* .. Parameters ..
194 DOUBLE PRECISION ZERO, ONE
195 COMPLEX*16 CZERO, CONE
196 parameter( zero = 0.0d+0, one = 1.0d+0,
197 \$ czero = ( 0.0d+0, 0.0d+0 ),
198 \$ cone = ( 1.0d+0, 0.0d+0 ) )
199* ..
200* .. Local Scalars ..
201 INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
202 DOUBLE PRECISION TEMP, TEMP2, TOL3Z
203 COMPLEX*16 AKK
204* ..
205* .. External Subroutines ..
206 EXTERNAL zgemm, zgemv, zlarfg, zswap
207* ..
208* .. Intrinsic Functions ..
209 INTRINSIC abs, dble, dconjg, max, min, nint, sqrt
210* ..
211* .. External Functions ..
212 INTEGER IDAMAX
213 DOUBLE PRECISION DLAMCH, DZNRM2
214 EXTERNAL idamax, dlamch, dznrm2
215* ..
216* .. Executable Statements ..
217*
218 lastrk = min( m, n+offset )
219 lsticc = 0
220 k = 0
221 tol3z = sqrt(dlamch('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 ) + idamax( n-k+1, vn1( k ), 1 )
233 IF( pvt.NE.k ) THEN
234 CALL zswap( m, a( 1, pvt ), 1, a( 1, k ), 1 )
235 CALL zswap( 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)**H.
245*
246 IF( k.GT.1 ) THEN
247 DO 20 j = 1, k - 1
248 f( k, j ) = dconjg( f( k, j ) )
249 20 CONTINUE
250 CALL zgemv( 'No transpose', m-rk+1, k-1, -cone, a( rk, 1 ),
251 \$ lda, f( k, 1 ), ldf, cone, a( rk, k ), 1 )
252 DO 30 j = 1, k - 1
253 f( k, j ) = dconjg( f( k, j ) )
254 30 CONTINUE
255 END IF
256*
257* Generate elementary reflector H(k).
258*
259 IF( rk.LT.m ) THEN
260 CALL zlarfg( m-rk+1, a( rk, k ), a( rk+1, k ), 1, tau( k ) )
261 ELSE
262 CALL zlarfg( 1, a( rk, k ), a( rk, k ), 1, tau( k ) )
263 END IF
264*
265 akk = a( rk, k )
266 a( rk, k ) = cone
267*
268* Compute Kth column of F:
269*
270* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K).
271*
272 IF( k.LT.n ) THEN
273 CALL zgemv( 'Conjugate transpose', m-rk+1, n-k, tau( k ),
274 \$ a( rk, k+1 ), lda, a( rk, k ), 1, czero,
275 \$ f( k+1, k ), 1 )
276 END IF
277*
279*
280 DO 40 j = 1, k
281 f( j, k ) = czero
282 40 CONTINUE
283*
284* Incremental updating of F:
285* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H
286* *A(RK:M,K).
287*
288 IF( k.GT.1 ) THEN
289 CALL zgemv( 'Conjugate transpose', m-rk+1, k-1, -tau( k ),
290 \$ a( rk, 1 ), lda, a( rk, k ), 1, czero,
291 \$ auxv( 1 ), 1 )
292*
293 CALL zgemv( 'No transpose', n, k-1, cone, f( 1, 1 ), ldf,
294 \$ auxv( 1 ), 1, cone, f( 1, k ), 1 )
295 END IF
296*
297* Update the current row of A:
298* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H.
299*
300 IF( k.LT.n ) THEN
301 CALL zgemm( 'No transpose', 'Conjugate transpose', 1, n-k,
302 \$ k, -cone, a( rk, 1 ), lda, f( k+1, 1 ), ldf,
303 \$ cone, a( rk, k+1 ), lda )
304 END IF
305*
306* Update partial column norms.
307*
308 IF( rk.LT.lastrk ) THEN
309 DO 50 j = k + 1, n
310 IF( vn1( j ).NE.zero ) THEN
311*
312* NOTE: The following 4 lines follow from the analysis in
313* Lapack Working Note 176.
314*
315 temp = abs( a( rk, j ) ) / vn1( j )
316 temp = max( zero, ( one+temp )*( one-temp ) )
317 temp2 = temp*( vn1( j ) / vn2( j ) )**2
318 IF( temp2 .LE. tol3z ) THEN
319 vn2( j ) = dble( lsticc )
320 lsticc = j
321 ELSE
322 vn1( j ) = vn1( j )*sqrt( temp )
323 END IF
324 END IF
325 50 CONTINUE
326 END IF
327*
328 a( rk, k ) = akk
329*
330* End of while loop.
331*
332 GO TO 10
333 END IF
334 kb = k
335 rk = offset + kb
336*
337* Apply the block reflector to the rest of the matrix:
338* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
339* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H.
340*
341 IF( kb.LT.min( n, m-offset ) ) THEN
342 CALL zgemm( 'No transpose', 'Conjugate transpose', m-rk, n-kb,
343 \$ kb, -cone, a( rk+1, 1 ), lda, f( kb+1, 1 ), ldf,
344 \$ cone, a( rk+1, kb+1 ), lda )
345 END IF
346*
347* Recomputation of difficult columns.
348*
349 60 CONTINUE
350 IF( lsticc.GT.0 ) THEN
351 itemp = nint( vn2( lsticc ) )
352 vn1( lsticc ) = dznrm2( m-rk, a( rk+1, lsticc ), 1 )
353*
354* NOTE: The computation of VN1( LSTICC ) relies on the fact that
355* SNRM2 does not fail on vectors with norm below the value of
356* SQRT(DLAMCH('S'))
357*
358 vn2( lsticc ) = vn1( lsticc )
359 lsticc = itemp
360 GO TO 60
361 END IF
362*
363 RETURN
364*
365* End of ZLAQPS
366*
367 END
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zlaqps(M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
ZLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BL...
Definition: zlaqps.f:177
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106