1*> \brief <b> DGELSX solves overdetermined or underdetermined systems for GE matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
22*                          WORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
26*       DOUBLE PRECISION   RCOND
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            JPVT( * )
30*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> This routine is deprecated and has been replaced by routine DGELSY.
40*>
41*> DGELSX computes the minimum-norm solution to a real linear least
42*> squares problem:
43*>     minimize || A * X - B ||
44*> using a complete orthogonal factorization of A.  A is an M-by-N
45*> matrix which may be rank-deficient.
46*>
47*> Several right hand side vectors b and solution vectors x can be
48*> handled in a single call; they are stored as the columns of the
49*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
50*> matrix X.
51*>
52*> The routine first computes a QR factorization with column pivoting:
53*>     A * P = Q * [ R11 R12 ]
54*>                 [  0  R22 ]
55*> with R11 defined as the largest leading submatrix whose estimated
56*> condition number is less than 1/RCOND.  The order of R11, RANK,
57*> is the effective rank of A.
58*>
59*> Then, R22 is considered to be negligible, and R12 is annihilated
60*> by orthogonal transformations from the right, arriving at the
61*> complete orthogonal factorization:
62*>    A * P = Q * [ T11 0 ] * Z
63*>                [  0  0 ]
64*> The minimum-norm solution is then
65*>    X = P * Z**T [ inv(T11)*Q1**T*B ]
66*>                 [        0         ]
67*> where Q1 consists of the first RANK columns of Q.
68*> \endverbatim
69*
70*  Arguments:
71*  ==========
72*
73*> \param[in] M
74*> \verbatim
75*>          M is INTEGER
76*>          The number of rows of the matrix A.  M >= 0.
77*> \endverbatim
78*>
79*> \param[in] N
80*> \verbatim
81*>          N is INTEGER
82*>          The number of columns of the matrix A.  N >= 0.
83*> \endverbatim
84*>
85*> \param[in] NRHS
86*> \verbatim
87*>          NRHS is INTEGER
88*>          The number of right hand sides, i.e., the number of
89*>          columns of matrices B and X. NRHS >= 0.
90*> \endverbatim
91*>
92*> \param[in,out] A
93*> \verbatim
94*>          A is DOUBLE PRECISION array, dimension (LDA,N)
95*>          On entry, the M-by-N matrix A.
96*>          On exit, A has been overwritten by details of its
97*>          complete orthogonal factorization.
98*> \endverbatim
99*>
100*> \param[in] LDA
101*> \verbatim
102*>          LDA is INTEGER
103*>          The leading dimension of the array A.  LDA >= max(1,M).
104*> \endverbatim
105*>
106*> \param[in,out] B
107*> \verbatim
108*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
109*>          On entry, the M-by-NRHS right hand side matrix B.
110*>          On exit, the N-by-NRHS solution matrix X.
111*>          If m >= n and RANK = n, the residual sum-of-squares for
112*>          the solution in the i-th column is given by the sum of
113*>          squares of elements N+1:M in that column.
114*> \endverbatim
115*>
116*> \param[in] LDB
117*> \verbatim
118*>          LDB is INTEGER
119*>          The leading dimension of the array B. LDB >= max(1,M,N).
120*> \endverbatim
121*>
122*> \param[in,out] JPVT
123*> \verbatim
124*>          JPVT is INTEGER array, dimension (N)
125*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
126*>          initial column, otherwise it is a free column.  Before
127*>          the QR factorization of A, all initial columns are
128*>          permuted to the leading positions; only the remaining
129*>          free columns are moved as a result of column pivoting
130*>          during the factorization.
131*>          On exit, if JPVT(i) = k, then the i-th column of A*P
132*>          was the k-th column of A.
133*> \endverbatim
134*>
135*> \param[in] RCOND
136*> \verbatim
137*>          RCOND is DOUBLE PRECISION
138*>          RCOND is used to determine the effective rank of A, which
139*>          is defined as the order of the largest leading triangular
140*>          submatrix R11 in the QR factorization with pivoting of A,
141*>          whose estimated condition number < 1/RCOND.
142*> \endverbatim
143*>
144*> \param[out] RANK
145*> \verbatim
146*>          RANK is INTEGER
147*>          The effective rank of A, i.e., the order of the submatrix
148*>          R11.  This is the same as the order of the submatrix T11
149*>          in the complete orthogonal factorization of A.
150*> \endverbatim
151*>
152*> \param[out] WORK
153*> \verbatim
154*>          WORK is DOUBLE PRECISION array, dimension
155*>                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
156*> \endverbatim
157*>
158*> \param[out] INFO
159*> \verbatim
160*>          INFO is INTEGER
161*>          = 0:  successful exit
162*>          < 0:  if INFO = -i, the i-th argument had an illegal value
163*> \endverbatim
164*
165*  Authors:
166*  ========
167*
168*> \author Univ. of Tennessee
169*> \author Univ. of California Berkeley
170*> \author Univ. of Colorado Denver
171*> \author NAG Ltd.
172*
173*> \date November 2011
174*
175*> \ingroup doubleGEsolve
176*
177*  =====================================================================
178      SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
179     $                   WORK, INFO )
180*
181*  -- LAPACK driver routine (version 3.4.0) --
182*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
183*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184*     November 2011
185*
186*     .. Scalar Arguments ..
187      INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
188      DOUBLE PRECISION   RCOND
189*     ..
190*     .. Array Arguments ..
191      INTEGER            JPVT( * )
192      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
193*     ..
194*
195*  =====================================================================
196*
197*     .. Parameters ..
198      INTEGER            IMAX, IMIN
199      PARAMETER          ( IMAX = 1, IMIN = 2 )
200      DOUBLE PRECISION   ZERO, ONE, DONE, NTDONE
201      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO,
202     $                   NTDONE = ONE )
203*     ..
204*     .. Local Scalars ..
205      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
206      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
207     $                   SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
208*     ..
209*     .. External Functions ..
210      DOUBLE PRECISION   DLAMCH, DLANGE
211      EXTERNAL           DLAMCH, DLANGE
212*     ..
213*     .. External Subroutines ..
214      EXTERNAL           DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R,
215     $                   DTRSM, DTZRQF, XERBLA
216*     ..
217*     .. Intrinsic Functions ..
218      INTRINSIC          ABS, MAX, MIN
219*     ..
220*     .. Executable Statements ..
221*
222      MN = MIN( M, N )
223      ISMIN = MN + 1
224      ISMAX = 2*MN + 1
225*
226*     Test the input arguments.
227*
228      INFO = 0
229      IF( M.LT.0 ) THEN
230         INFO = -1
231      ELSE IF( N.LT.0 ) THEN
232         INFO = -2
233      ELSE IF( NRHS.LT.0 ) THEN
234         INFO = -3
235      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
236         INFO = -5
237      ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
238         INFO = -7
239      END IF
240*
241      IF( INFO.NE.0 ) THEN
242         CALL XERBLA( 'DGELSX', -INFO )
243         RETURN
244      END IF
245*
246*     Quick return if possible
247*
248      IF( MIN( M, N, NRHS ).EQ.0 ) THEN
249         RANK = 0
250         RETURN
251      END IF
252*
253*     Get machine parameters
254*
255      SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
256      BIGNUM = ONE / SMLNUM
257      CALL DLABAD( SMLNUM, BIGNUM )
258*
259*     Scale A, B if max elements outside range [SMLNUM,BIGNUM]
260*
261      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
262      IASCL = 0
263      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
264*
265*        Scale matrix norm up to SMLNUM
266*
267         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
268         IASCL = 1
269      ELSE IF( ANRM.GT.BIGNUM ) THEN
270*
271*        Scale matrix norm down to BIGNUM
272*
273         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
274         IASCL = 2
275      ELSE IF( ANRM.EQ.ZERO ) THEN
276*
277*        Matrix all zero. Return zero solution.
278*
279         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
280         RANK = 0
281         GO TO 100
282      END IF
283*
284      BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
285      IBSCL = 0
286      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
287*
288*        Scale matrix norm up to SMLNUM
289*
290         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
291         IBSCL = 1
292      ELSE IF( BNRM.GT.BIGNUM ) THEN
293*
294*        Scale matrix norm down to BIGNUM
295*
296         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
297         IBSCL = 2
298      END IF
299*
300*     Compute QR factorization with column pivoting of A:
301*        A * P = Q * R
302*
303      CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO )
304*
305*     workspace 3*N. Details of Householder rotations stored
306*     in WORK(1:MN).
307*
308*     Determine RANK using incremental condition estimation
309*
310      WORK( ISMIN ) = ONE
311      WORK( ISMAX ) = ONE
312      SMAX = ABS( A( 1, 1 ) )
313      SMIN = SMAX
314      IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
315         RANK = 0
316         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
317         GO TO 100
318      ELSE
319         RANK = 1
320      END IF
321*
322   10 CONTINUE
323      IF( RANK.LT.MN ) THEN
324         I = RANK + 1
325         CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
326     $                A( I, I ), SMINPR, S1, C1 )
327         CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
328     $                A( I, I ), SMAXPR, S2, C2 )
329*
330         IF( SMAXPR*RCOND.LE.SMINPR ) THEN
331            DO 20 I = 1, RANK
332               WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
333               WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
334   20       CONTINUE
335            WORK( ISMIN+RANK ) = C1
336            WORK( ISMAX+RANK ) = C2
337            SMIN = SMINPR
338            SMAX = SMAXPR
339            RANK = RANK + 1
340            GO TO 10
341         END IF
342      END IF
343*
344*     Logically partition R = [ R11 R12 ]
345*                             [  0  R22 ]
346*     where R11 = R(1:RANK,1:RANK)
347*
348*     [R11,R12] = [ T11, 0 ] * Y
349*
350      IF( RANK.LT.N )
351     $   CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
352*
353*     Details of Householder rotations stored in WORK(MN+1:2*MN)
354*
355*     B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
356*
357      CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
358     $             B, LDB, WORK( 2*MN+1 ), INFO )
359*
360*     workspace NRHS
361*
362*     B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
363*
364      CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
365     $            NRHS, ONE, A, LDA, B, LDB )
366*
367      DO 40 I = RANK + 1, N
368         DO 30 J = 1, NRHS
369            B( I, J ) = ZERO
370   30    CONTINUE
371   40 CONTINUE
372*
373*     B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
374*
375      IF( RANK.LT.N ) THEN
376         DO 50 I = 1, RANK
377            CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
378     $                   WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
379     $                   WORK( 2*MN+1 ) )
380   50    CONTINUE
381      END IF
382*
383*     workspace NRHS
384*
385*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
386*
387      DO 90 J = 1, NRHS
388         DO 60 I = 1, N
389            WORK( 2*MN+I ) = NTDONE
390   60    CONTINUE
391         DO 80 I = 1, N
392            IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
393               IF( JPVT( I ).NE.I ) THEN
394                  K = I
395                  T1 = B( K, J )
396                  T2 = B( JPVT( K ), J )
397   70             CONTINUE
398                  B( JPVT( K ), J ) = T1
399                  WORK( 2*MN+K ) = DONE
400                  T1 = T2
401                  K = JPVT( K )
402                  T2 = B( JPVT( K ), J )
403                  IF( JPVT( K ).NE.I )
404     $               GO TO 70
405                  B( I, J ) = T1
406                  WORK( 2*MN+K ) = DONE
407               END IF
408            END IF
409   80    CONTINUE
410   90 CONTINUE
411*
412*     Undo scaling
413*
414      IF( IASCL.EQ.1 ) THEN
415         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
416         CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
417     $                INFO )
418      ELSE IF( IASCL.EQ.2 ) THEN
419         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
420         CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
421     $                INFO )
422      END IF
423      IF( IBSCL.EQ.1 ) THEN
424         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
425      ELSE IF( IBSCL.EQ.2 ) THEN
426         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
427      END IF
428*
429  100 CONTINUE
430*
431      RETURN
432*
433*     End of DGELSX
434*
435      END
436