1*> \brief \b DGEQP3
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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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*> \date November 2015
124*
125*> \ingroup doubleGEcomputational
126*
127*> \par Further Details:
128*  =====================
129*>
130*> \verbatim
131*>
132*>  The matrix Q is represented as a product of elementary reflectors
133*>
134*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
135*>
136*>  Each H(i) has the form
137*>
138*>     H(i) = I - tau * v * v**T
139*>
140*>  where tau is a real scalar, and v is a real/complex vector
141*>  with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
142*>  A(i+1:m,i), and tau in TAU(i).
143*> \endverbatim
144*
145*> \par Contributors:
146*  ==================
147*>
148*>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
149*>    X. Sun, Computer Science Dept., Duke University, USA
150*>
151*  =====================================================================
152      SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
153*
154*  -- LAPACK computational routine (version 3.6.0) --
155*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
156*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157*     November 2015
158*
159*     .. Scalar Arguments ..
160      INTEGER            INFO, LDA, LWORK, M, N
161*     ..
162*     .. Array Arguments ..
163      INTEGER            JPVT( * )
164      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
165*     ..
166*
167*  =====================================================================
168*
169*     .. Parameters ..
170      INTEGER            INB, INBMIN, IXOVER
171      PARAMETER          ( INB = 1, INBMIN = 2, IXOVER = 3 )
172*     ..
173*     .. Local Scalars ..
174      LOGICAL            LQUERY
175      INTEGER            FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
176     $                   NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
177*     ..
178*     .. External Subroutines ..
179      EXTERNAL           DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP, XERBLA
180*     ..
181*     .. External Functions ..
182      INTEGER            ILAENV
183      DOUBLE PRECISION   DNRM2
184      EXTERNAL           ILAENV, DNRM2
185*     ..
186*     .. Intrinsic Functions ..
187      INTRINSIC          INT, MAX, MIN
188*     ..
189*     .. Executable Statements ..
190*
191*     Test input arguments
192*  ====================
193*
194      INFO = 0
195      LQUERY = ( LWORK.EQ.-1 )
196      IF( M.LT.0 ) THEN
197         INFO = -1
198      ELSE IF( N.LT.0 ) THEN
199         INFO = -2
200      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
201         INFO = -4
202      END IF
203*
204      IF( INFO.EQ.0 ) THEN
205         MINMN = MIN( M, N )
206         IF( MINMN.EQ.0 ) THEN
207            IWS = 1
208            LWKOPT = 1
209         ELSE
210            IWS = 3*N + 1
211            NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 )
212            LWKOPT = 2*N + ( N + 1 )*NB
213         END IF
214         WORK( 1 ) = LWKOPT
215*
216         IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN
217            INFO = -8
218         END IF
219      END IF
220*
221      IF( INFO.NE.0 ) THEN
222         CALL XERBLA( 'DGEQP3', -INFO )
223         RETURN
224      ELSE IF( LQUERY ) THEN
225         RETURN
226      END IF
227*
228*     Move initial columns up front.
229*
230      NFXD = 1
231      DO 10 J = 1, N
232         IF( JPVT( J ).NE.0 ) THEN
233            IF( J.NE.NFXD ) THEN
234               CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 )
235               JPVT( J ) = JPVT( NFXD )
236               JPVT( NFXD ) = J
237            ELSE
238               JPVT( J ) = J
239            END IF
240            NFXD = NFXD + 1
241         ELSE
242            JPVT( J ) = J
243         END IF
244   10 CONTINUE
245      NFXD = NFXD - 1
246*
247*     Factorize fixed columns
248*  =======================
249*
250*     Compute the QR factorization of fixed columns and update
251*     remaining columns.
252*
253      IF( NFXD.GT.0 ) THEN
254         NA = MIN( M, NFXD )
255*CC      CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
256         CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO )
257         IWS = MAX( IWS, INT( WORK( 1 ) ) )
258         IF( NA.LT.N ) THEN
259*CC         CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
260*CC  $                   TAU, A( 1, NA+1 ), LDA, WORK, INFO )
261            CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA, TAU,
262     $                   A( 1, NA+1 ), LDA, WORK, LWORK, INFO )
263            IWS = MAX( IWS, INT( WORK( 1 ) ) )
264         END IF
265      END IF
266*
267*     Factorize free columns
268*  ======================
269*
270      IF( NFXD.LT.MINMN ) THEN
271*
272         SM = M - NFXD
273         SN = N - NFXD
274         SMINMN = MINMN - NFXD
275*
276*        Determine the block size.
277*
278         NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 )
279         NBMIN = 2
280         NX = 0
281*
282         IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN
283*
284*           Determine when to cross over from blocked to unblocked code.
285*
286            NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1,
287     $           -1 ) )
288*
289*
290            IF( NX.LT.SMINMN ) THEN
291*
292*              Determine if workspace is large enough for blocked code.
293*
294               MINWS = 2*SN + ( SN+1 )*NB
295               IWS = MAX( IWS, MINWS )
296               IF( LWORK.LT.MINWS ) THEN
297*
298*                 Not enough workspace to use optimal NB: Reduce NB and
299*                 determine the minimum value of NB.
300*
301                  NB = ( LWORK-2*SN ) / ( SN+1 )
302                  NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM, SN,
303     $                    -1, -1 ) )
304*
305*
306               END IF
307            END IF
308         END IF
309*
310*        Initialize partial column norms. The first N elements of work
311*        store the exact column norms.
312*
313         DO 20 J = NFXD + 1, N
314            WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 )
315            WORK( N+J ) = WORK( J )
316   20    CONTINUE
317*
318         IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND.
319     $       ( NX.LT.SMINMN ) ) THEN
320*
321*           Use blocked code initially.
322*
323            J = NFXD + 1
324*
325*           Compute factorization: while loop.
326*
327*
328            TOPBMN = MINMN - NX
329   30       CONTINUE
330            IF( J.LE.TOPBMN ) THEN
331               JB = MIN( NB, TOPBMN-J+1 )
332*
333*              Factorize JB columns among columns J:N.
334*
335               CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA,
336     $                      JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ),
337     $                      WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 )
338*
339               J = J + FJB
340               GO TO 30
341            END IF
342         ELSE
343            J = NFXD + 1
344         END IF
345*
346*        Use unblocked code to factor the last or only block.
347*
348*
349         IF( J.LE.MINMN )
350     $      CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ),
351     $                   TAU( J ), WORK( J ), WORK( N+J ),
352     $                   WORK( 2*N+1 ) )
353*
354      END IF
355*
356      WORK( 1 ) = IWS
357      RETURN
358*
359*     End of DGEQP3
360*
361      END
362