LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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dgeqp3.f
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1*> \brief \b DGEQP3
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqp3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, LWORK, M, N
25* ..
26* .. Array Arguments ..
27* INTEGER JPVT( * )
28* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> DGEQP3 computes a QR factorization with column pivoting of a
38*> matrix A: A*P = Q*R using Level 3 BLAS.
39*> \endverbatim
40*
41* Arguments:
42* ==========
43*
44*> \param[in] M
45*> \verbatim
46*> M is INTEGER
47*> The number of rows of the matrix A. M >= 0.
48*> \endverbatim
49*>
50*> \param[in] N
51*> \verbatim
52*> N is INTEGER
53*> The number of columns of the matrix A. N >= 0.
54*> \endverbatim
55*>
56*> \param[in,out] A
57*> \verbatim
58*> A is DOUBLE PRECISION array, dimension (LDA,N)
59*> On entry, the M-by-N matrix A.
60*> On exit, the upper triangle of the array contains the
61*> min(M,N)-by-N upper trapezoidal matrix R; the elements below
62*> the diagonal, together with the array TAU, represent the
63*> orthogonal matrix Q as a product of min(M,N) elementary
64*> reflectors.
65*> \endverbatim
66*>
67*> \param[in] LDA
68*> \verbatim
69*> LDA is INTEGER
70*> The leading dimension of the array A. LDA >= max(1,M).
71*> \endverbatim
72*>
73*> \param[in,out] JPVT
74*> \verbatim
75*> JPVT is INTEGER array, dimension (N)
76*> On entry, if JPVT(J).ne.0, the J-th column of A is permuted
77*> to the front of A*P (a leading column); if JPVT(J)=0,
78*> the J-th column of A is a free column.
79*> On exit, if JPVT(J)=K, then the J-th column of A*P was the
80*> the K-th column of A.
81*> \endverbatim
82*>
83*> \param[out] TAU
84*> \verbatim
85*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
86*> The scalar factors of the elementary reflectors.
87*> \endverbatim
88*>
89*> \param[out] WORK
90*> \verbatim
91*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
92*> On exit, if INFO=0, WORK(1) returns the optimal LWORK.
93*> \endverbatim
94*>
95*> \param[in] LWORK
96*> \verbatim
97*> LWORK is INTEGER
98*> The dimension of the array WORK. LWORK >= 3*N+1.
99*> For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
100*> is the optimal blocksize.
101*>
102*> If LWORK = -1, then a workspace query is assumed; the routine
103*> only calculates the optimal size of the WORK array, returns
104*> this value as the first entry of the WORK array, and no error
105*> message related to LWORK is issued by XERBLA.
106*> \endverbatim
107*>
108*> \param[out] INFO
109*> \verbatim
110*> INFO is INTEGER
111*> = 0: successful exit.
112*> < 0: if INFO = -i, the i-th argument had an illegal value.
113*> \endverbatim
114*
115* Authors:
116* ========
117*
118*> \author Univ. of Tennessee
119*> \author Univ. of California Berkeley
120*> \author Univ. of Colorado Denver
121*> \author NAG Ltd.
122*
123*> \ingroup doubleGEcomputational
124*
125*> \par Further Details:
126* =====================
127*>
128*> \verbatim
129*>
130*> The matrix Q is represented as a product of elementary reflectors
131*>
132*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
133*>
134*> Each H(i) has the form
135*>
136*> H(i) = I - tau * v * v**T
137*>
138*> where tau is a real scalar, and v is a real/complex vector
139*> with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
140*> A(i+1:m,i), and tau in TAU(i).
141*> \endverbatim
142*
143*> \par Contributors:
144* ==================
145*>
146*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
147*> X. Sun, Computer Science Dept., Duke University, USA
148*>
149* =====================================================================
150 SUBROUTINE dgeqp3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
151*
152* -- LAPACK computational routine --
153* -- LAPACK is a software package provided by Univ. of Tennessee, --
154* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155*
156* .. Scalar Arguments ..
157 INTEGER INFO, LDA, LWORK, M, N
158* ..
159* .. Array Arguments ..
160 INTEGER JPVT( * )
161 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
162* ..
163*
164* =====================================================================
165*
166* .. Parameters ..
167 INTEGER INB, INBMIN, IXOVER
168 parameter( inb = 1, inbmin = 2, ixover = 3 )
169* ..
170* .. Local Scalars ..
171 LOGICAL LQUERY
172 INTEGER FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
173 \$ NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
174* ..
175* .. External Subroutines ..
176 EXTERNAL dgeqrf, dlaqp2, dlaqps, dormqr, dswap, xerbla
177* ..
178* .. External Functions ..
179 INTEGER ILAENV
180 DOUBLE PRECISION DNRM2
181 EXTERNAL ilaenv, dnrm2
182* ..
183* .. Intrinsic Functions ..
184 INTRINSIC int, max, min
185* ..
186* .. Executable Statements ..
187*
188* Test input arguments
189* ====================
190*
191 info = 0
192 lquery = ( lwork.EQ.-1 )
193 IF( m.LT.0 ) THEN
194 info = -1
195 ELSE IF( n.LT.0 ) THEN
196 info = -2
197 ELSE IF( lda.LT.max( 1, m ) ) THEN
198 info = -4
199 END IF
200*
201 IF( info.EQ.0 ) THEN
202 minmn = min( m, n )
203 IF( minmn.EQ.0 ) THEN
204 iws = 1
205 lwkopt = 1
206 ELSE
207 iws = 3*n + 1
208 nb = ilaenv( inb, 'DGEQRF', ' ', m, n, -1, -1 )
209 lwkopt = 2*n + ( n + 1 )*nb
210 END IF
211 work( 1 ) = lwkopt
212*
213 IF( ( lwork.LT.iws ) .AND. .NOT.lquery ) THEN
214 info = -8
215 END IF
216 END IF
217*
218 IF( info.NE.0 ) THEN
219 CALL xerbla( 'DGEQP3', -info )
220 RETURN
221 ELSE IF( lquery ) THEN
222 RETURN
223 END IF
224*
225* Move initial columns up front.
226*
227 nfxd = 1
228 DO 10 j = 1, n
229 IF( jpvt( j ).NE.0 ) THEN
230 IF( j.NE.nfxd ) THEN
231 CALL dswap( m, a( 1, j ), 1, a( 1, nfxd ), 1 )
232 jpvt( j ) = jpvt( nfxd )
233 jpvt( nfxd ) = j
234 ELSE
235 jpvt( j ) = j
236 END IF
237 nfxd = nfxd + 1
238 ELSE
239 jpvt( j ) = j
240 END IF
241 10 CONTINUE
242 nfxd = nfxd - 1
243*
244* Factorize fixed columns
245* =======================
246*
247* Compute the QR factorization of fixed columns and update
248* remaining columns.
249*
250 IF( nfxd.GT.0 ) THEN
251 na = min( m, nfxd )
252*CC CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
253 CALL dgeqrf( m, na, a, lda, tau, work, lwork, info )
254 iws = max( iws, int( work( 1 ) ) )
255 IF( na.LT.n ) THEN
256*CC CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
257*CC \$ TAU, A( 1, NA+1 ), LDA, WORK, INFO )
258 CALL dormqr( 'Left', 'Transpose', m, n-na, na, a, lda, tau,
259 \$ a( 1, na+1 ), lda, work, lwork, info )
260 iws = max( iws, int( work( 1 ) ) )
261 END IF
262 END IF
263*
264* Factorize free columns
265* ======================
266*
267 IF( nfxd.LT.minmn ) THEN
268*
269 sm = m - nfxd
270 sn = n - nfxd
271 sminmn = minmn - nfxd
272*
273* Determine the block size.
274*
275 nb = ilaenv( inb, 'DGEQRF', ' ', sm, sn, -1, -1 )
276 nbmin = 2
277 nx = 0
278*
279 IF( ( nb.GT.1 ) .AND. ( nb.LT.sminmn ) ) THEN
280*
281* Determine when to cross over from blocked to unblocked code.
282*
283 nx = max( 0, ilaenv( ixover, 'DGEQRF', ' ', sm, sn, -1,
284 \$ -1 ) )
285*
286*
287 IF( nx.LT.sminmn ) THEN
288*
289* Determine if workspace is large enough for blocked code.
290*
291 minws = 2*sn + ( sn+1 )*nb
292 iws = max( iws, minws )
293 IF( lwork.LT.minws ) THEN
294*
295* Not enough workspace to use optimal NB: Reduce NB and
296* determine the minimum value of NB.
297*
298 nb = ( lwork-2*sn ) / ( sn+1 )
299 nbmin = max( 2, ilaenv( inbmin, 'DGEQRF', ' ', sm, sn,
300 \$ -1, -1 ) )
301*
302*
303 END IF
304 END IF
305 END IF
306*
307* Initialize partial column norms. The first N elements of work
308* store the exact column norms.
309*
310 DO 20 j = nfxd + 1, n
311 work( j ) = dnrm2( sm, a( nfxd+1, j ), 1 )
312 work( n+j ) = work( j )
313 20 CONTINUE
314*
315 IF( ( nb.GE.nbmin ) .AND. ( nb.LT.sminmn ) .AND.
316 \$ ( nx.LT.sminmn ) ) THEN
317*
318* Use blocked code initially.
319*
320 j = nfxd + 1
321*
322* Compute factorization: while loop.
323*
324*
325 topbmn = minmn - nx
326 30 CONTINUE
327 IF( j.LE.topbmn ) THEN
328 jb = min( nb, topbmn-j+1 )
329*
330* Factorize JB columns among columns J:N.
331*
332 CALL dlaqps( m, n-j+1, j-1, jb, fjb, a( 1, j ), lda,
333 \$ jpvt( j ), tau( j ), work( j ), work( n+j ),
334 \$ work( 2*n+1 ), work( 2*n+jb+1 ), n-j+1 )
335*
336 j = j + fjb
337 GO TO 30
338 END IF
339 ELSE
340 j = nfxd + 1
341 END IF
342*
343* Use unblocked code to factor the last or only block.
344*
345*
346 IF( j.LE.minmn )
347 \$ CALL dlaqp2( m, n-j+1, j-1, a( 1, j ), lda, jpvt( j ),
348 \$ tau( j ), work( j ), work( n+j ),
349 \$ work( 2*n+1 ) )
350*
351 END IF
352*
353 work( 1 ) = iws
354 RETURN
355*
356* End of DGEQP3
357*
358 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
DGEQP3
Definition: dgeqp3.f:151
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:146
subroutine dlaqp2(M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
DLAQP2 computes a QR factorization with column pivoting of the matrix block.
Definition: dlaqp2.f:149
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 dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:167