LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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cgglse.f
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1*> \brief <b> CGGLSE solves overdetermined or underdetermined systems for OTHER 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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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgglse.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
22* INFO )
23*
24* .. Scalar Arguments ..
25* INTEGER INFO, LDA, LDB, LWORK, M, N, P
26* ..
27* .. Array Arguments ..
28* COMPLEX A( LDA, * ), B( LDB, * ), C( * ), D( * ),
29* \$ WORK( * ), X( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> CGGLSE solves the linear equality-constrained least squares (LSE)
39*> problem:
40*>
41*> minimize || c - A*x ||_2 subject to B*x = d
42*>
43*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
44*> M-vector, and d is a given P-vector. It is assumed that
45*> P <= N <= M+P, and
46*>
47*> rank(B) = P and rank( (A) ) = N.
48*> ( (B) )
49*>
50*> These conditions ensure that the LSE problem has a unique solution,
51*> which is obtained using a generalized RQ factorization of the
52*> matrices (B, A) given by
53*>
54*> B = (0 R)*Q, A = Z*T*Q.
55*> \endverbatim
56*
57* Arguments:
58* ==========
59*
60*> \param[in] M
61*> \verbatim
62*> M is INTEGER
63*> The number of rows of the matrix A. M >= 0.
64*> \endverbatim
65*>
66*> \param[in] N
67*> \verbatim
68*> N is INTEGER
69*> The number of columns of the matrices A and B. N >= 0.
70*> \endverbatim
71*>
72*> \param[in] P
73*> \verbatim
74*> P is INTEGER
75*> The number of rows of the matrix B. 0 <= P <= N <= M+P.
76*> \endverbatim
77*>
78*> \param[in,out] A
79*> \verbatim
80*> A is COMPLEX array, dimension (LDA,N)
81*> On entry, the M-by-N matrix A.
82*> On exit, the elements on and above the diagonal of the array
83*> contain the min(M,N)-by-N upper trapezoidal matrix T.
84*> \endverbatim
85*>
86*> \param[in] LDA
87*> \verbatim
88*> LDA is INTEGER
89*> The leading dimension of the array A. LDA >= max(1,M).
90*> \endverbatim
91*>
92*> \param[in,out] B
93*> \verbatim
94*> B is COMPLEX array, dimension (LDB,N)
95*> On entry, the P-by-N matrix B.
96*> On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
97*> contains the P-by-P upper triangular matrix R.
98*> \endverbatim
99*>
100*> \param[in] LDB
101*> \verbatim
102*> LDB is INTEGER
103*> The leading dimension of the array B. LDB >= max(1,P).
104*> \endverbatim
105*>
106*> \param[in,out] C
107*> \verbatim
108*> C is COMPLEX array, dimension (M)
109*> On entry, C contains the right hand side vector for the
110*> least squares part of the LSE problem.
111*> On exit, the residual sum of squares for the solution
112*> is given by the sum of squares of elements N-P+1 to M of
113*> vector C.
114*> \endverbatim
115*>
116*> \param[in,out] D
117*> \verbatim
118*> D is COMPLEX array, dimension (P)
119*> On entry, D contains the right hand side vector for the
120*> constrained equation.
121*> On exit, D is destroyed.
122*> \endverbatim
123*>
124*> \param[out] X
125*> \verbatim
126*> X is COMPLEX array, dimension (N)
127*> On exit, X is the solution of the LSE problem.
128*> \endverbatim
129*>
130*> \param[out] WORK
131*> \verbatim
132*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
133*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
134*> \endverbatim
135*>
136*> \param[in] LWORK
137*> \verbatim
138*> LWORK is INTEGER
139*> The dimension of the array WORK. LWORK >= max(1,M+N+P).
140*> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
141*> where NB is an upper bound for the optimal blocksizes for
142*> CGEQRF, CGERQF, CUNMQR and CUNMRQ.
143*>
144*> If LWORK = -1, then a workspace query is assumed; the routine
145*> only calculates the optimal size of the WORK array, returns
146*> this value as the first entry of the WORK array, and no error
147*> message related to LWORK is issued by XERBLA.
148*> \endverbatim
149*>
150*> \param[out] INFO
151*> \verbatim
152*> INFO is INTEGER
153*> = 0: successful exit.
154*> < 0: if INFO = -i, the i-th argument had an illegal value.
155*> = 1: the upper triangular factor R associated with B in the
156*> generalized RQ factorization of the pair (B, A) is
157*> singular, so that rank(B) < P; the least squares
158*> solution could not be computed.
159*> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor
160*> T associated with A in the generalized RQ factorization
161*> of the pair (B, A) is singular, so that
162*> rank( (A) ) < N; the least squares solution could not
163*> ( (B) )
164*> be computed.
165*> \endverbatim
166*
167* Authors:
168* ========
169*
170*> \author Univ. of Tennessee
171*> \author Univ. of California Berkeley
172*> \author Univ. of Colorado Denver
173*> \author NAG Ltd.
174*
175*> \ingroup gglse
176*
177* =====================================================================
178 SUBROUTINE cgglse( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
179 \$ INFO )
180*
181* -- LAPACK driver routine --
182* -- LAPACK is a software package provided by Univ. of Tennessee, --
183* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184*
185* .. Scalar Arguments ..
186 INTEGER INFO, LDA, LDB, LWORK, M, N, P
187* ..
188* .. Array Arguments ..
189 COMPLEX A( LDA, * ), B( LDB, * ), C( * ), D( * ),
190 \$ work( * ), x( * )
191* ..
192*
193* =====================================================================
194*
195* .. Parameters ..
196 COMPLEX CONE
197 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
198* ..
199* .. Local Scalars ..
200 LOGICAL LQUERY
201 INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
202 \$ nb4, nr
203* ..
204* .. External Subroutines ..
205 EXTERNAL caxpy, ccopy, cgemv, cggrqf, ctrmv, ctrtrs,
207* ..
208* .. External Functions ..
209 INTEGER ILAENV
210 REAL SROUNDUP_LWORK
211 EXTERNAL ilaenv, sroundup_lwork
212* ..
213* .. Intrinsic Functions ..
214 INTRINSIC int, max, min
215* ..
216* .. Executable Statements ..
217*
218* Test the input parameters
219*
220 info = 0
221 mn = min( m, n )
222 lquery = ( lwork.EQ.-1 )
223 IF( m.LT.0 ) THEN
224 info = -1
225 ELSE IF( n.LT.0 ) THEN
226 info = -2
227 ELSE IF( p.LT.0 .OR. p.GT.n .OR. p.LT.n-m ) THEN
228 info = -3
229 ELSE IF( lda.LT.max( 1, m ) ) THEN
230 info = -5
231 ELSE IF( ldb.LT.max( 1, p ) ) THEN
232 info = -7
233 END IF
234*
235* Calculate workspace
236*
237 IF( info.EQ.0) THEN
238 IF( n.EQ.0 ) THEN
239 lwkmin = 1
240 lwkopt = 1
241 ELSE
242 nb1 = ilaenv( 1, 'CGEQRF', ' ', m, n, -1, -1 )
243 nb2 = ilaenv( 1, 'CGERQF', ' ', m, n, -1, -1 )
244 nb3 = ilaenv( 1, 'CUNMQR', ' ', m, n, p, -1 )
245 nb4 = ilaenv( 1, 'CUNMRQ', ' ', m, n, p, -1 )
246 nb = max( nb1, nb2, nb3, nb4 )
247 lwkmin = m + n + p
248 lwkopt = p + mn + max( m, n )*nb
249 END IF
250 work( 1 ) = sroundup_lwork(lwkopt)
251*
252 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
253 info = -12
254 END IF
255 END IF
256*
257 IF( info.NE.0 ) THEN
258 CALL xerbla( 'CGGLSE', -info )
259 RETURN
260 ELSE IF( lquery ) THEN
261 RETURN
262 END IF
263*
264* Quick return if possible
265*
266 IF( n.EQ.0 )
267 \$ RETURN
268*
269* Compute the GRQ factorization of matrices B and A:
270*
271* B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P
272* N-P P ( 0 R22 ) M+P-N
273* N-P P
274*
275* where T12 and R11 are upper triangular, and Q and Z are
276* unitary.
277*
278 CALL cggrqf( p, m, n, b, ldb, work, a, lda, work( p+1 ),
279 \$ work( p+mn+1 ), lwork-p-mn, info )
280 lopt = int( work( p+mn+1 ) )
281*
282* Update c = Z**H *c = ( c1 ) N-P
283* ( c2 ) M+P-N
284*
285 CALL cunmqr( 'Left', 'Conjugate Transpose', m, 1, mn, a, lda,
286 \$ work( p+1 ), c, max( 1, m ), work( p+mn+1 ),
287 \$ lwork-p-mn, info )
288 lopt = max( lopt, int( work( p+mn+1 ) ) )
289*
290* Solve T12*x2 = d for x2
291*
292 IF( p.GT.0 ) THEN
293 CALL ctrtrs( 'Upper', 'No transpose', 'Non-unit', p, 1,
294 \$ b( 1, n-p+1 ), ldb, d, p, info )
295*
296 IF( info.GT.0 ) THEN
297 info = 1
298 RETURN
299 END IF
300*
301* Put the solution in X
302*
303 CALL ccopy( p, d, 1, x( n-p+1 ), 1 )
304*
305* Update c1
306*
307 CALL cgemv( 'No transpose', n-p, p, -cone, a( 1, n-p+1 ), lda,
308 \$ d, 1, cone, c, 1 )
309 END IF
310*
311* Solve R11*x1 = c1 for x1
312*
313 IF( n.GT.p ) THEN
314 CALL ctrtrs( 'Upper', 'No transpose', 'Non-unit', n-p, 1,
315 \$ a, lda, c, n-p, info )
316*
317 IF( info.GT.0 ) THEN
318 info = 2
319 RETURN
320 END IF
321*
322* Put the solutions in X
323*
324 CALL ccopy( n-p, c, 1, x, 1 )
325 END IF
326*
327* Compute the residual vector:
328*
329 IF( m.LT.n ) THEN
330 nr = m + p - n
331 IF( nr.GT.0 )
332 \$ CALL cgemv( 'No transpose', nr, n-m, -cone, a( n-p+1, m+1 ),
333 \$ lda, d( nr+1 ), 1, cone, c( n-p+1 ), 1 )
334 ELSE
335 nr = p
336 END IF
337 IF( nr.GT.0 ) THEN
338 CALL ctrmv( 'Upper', 'No transpose', 'Non unit', nr,
339 \$ a( n-p+1, n-p+1 ), lda, d, 1 )
340 CALL caxpy( nr, -cone, d, 1, c( n-p+1 ), 1 )
341 END IF
342*
343* Backward transformation x = Q**H*x
344*
345 CALL cunmrq( 'Left', 'Conjugate Transpose', n, 1, p, b, ldb,
346 \$ work( 1 ), x, n, work( p+mn+1 ), lwork-p-mn, info )
347 work( 1 ) = p + mn + max( lopt, int( work( p+mn+1 ) ) )
348*
349 RETURN
350*
351* End of CGGLSE
352*
353 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
subroutine cgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
CGGLSE solves overdetermined or underdetermined systems for OTHER matrices
Definition cgglse.f:180
subroutine cggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
CGGRQF
Definition cggrqf.f:214
subroutine ctrmv(uplo, trans, diag, n, a, lda, x, incx)
CTRMV
Definition ctrmv.f:147
subroutine ctrtrs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, info)
CTRTRS
Definition ctrtrs.f:140
subroutine cunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMQR
Definition cunmqr.f:168
subroutine cunmrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMRQ
Definition cunmrq.f:168