LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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zgglse.f
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1*> \brief <b> ZGGLSE 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*
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/zgglse.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGGLSE( 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*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
29* \$ WORK( * ), X( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> ZGGLSE 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*16 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*16 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*16 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*16 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*16 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*16 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*> ZGEQRF, CGERQF, ZUNMQR 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 complex16OTHERsolve
176*
177* =====================================================================
178 SUBROUTINE zgglse( 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*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
190 \$ work( * ), x( * )
191* ..
192*
193* =====================================================================
194*
195* .. Parameters ..
196 COMPLEX*16 CONE
197 parameter( cone = ( 1.0d+0, 0.0d+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 xerbla, zaxpy, zcopy, zgemv, zggrqf, ztrmv,
207* ..
208* .. External Functions ..
209 INTEGER ILAENV
210 EXTERNAL ilaenv
211* ..
212* .. Intrinsic Functions ..
213 INTRINSIC int, max, min
214* ..
215* .. Executable Statements ..
216*
217* Test the input parameters
218*
219 info = 0
220 mn = min( m, n )
221 lquery = ( lwork.EQ.-1 )
222 IF( m.LT.0 ) THEN
223 info = -1
224 ELSE IF( n.LT.0 ) THEN
225 info = -2
226 ELSE IF( p.LT.0 .OR. p.GT.n .OR. p.LT.n-m ) THEN
227 info = -3
228 ELSE IF( lda.LT.max( 1, m ) ) THEN
229 info = -5
230 ELSE IF( ldb.LT.max( 1, p ) ) THEN
231 info = -7
232 END IF
233*
234* Calculate workspace
235*
236 IF( info.EQ.0) THEN
237 IF( n.EQ.0 ) THEN
238 lwkmin = 1
239 lwkopt = 1
240 ELSE
241 nb1 = ilaenv( 1, 'ZGEQRF', ' ', m, n, -1, -1 )
242 nb2 = ilaenv( 1, 'ZGERQF', ' ', m, n, -1, -1 )
243 nb3 = ilaenv( 1, 'ZUNMQR', ' ', m, n, p, -1 )
244 nb4 = ilaenv( 1, 'ZUNMRQ', ' ', m, n, p, -1 )
245 nb = max( nb1, nb2, nb3, nb4 )
246 lwkmin = m + n + p
247 lwkopt = p + mn + max( m, n )*nb
248 END IF
249 work( 1 ) = lwkopt
250*
251 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
252 info = -12
253 END IF
254 END IF
255*
256 IF( info.NE.0 ) THEN
257 CALL xerbla( 'ZGGLSE', -info )
258 RETURN
259 ELSE IF( lquery ) THEN
260 RETURN
261 END IF
262*
263* Quick return if possible
264*
265 IF( n.EQ.0 )
266 \$ RETURN
267*
268* Compute the GRQ factorization of matrices B and A:
269*
270* B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P
271* N-P P ( 0 R22 ) M+P-N
272* N-P P
273*
274* where T12 and R11 are upper triangular, and Q and Z are
275* unitary.
276*
277 CALL zggrqf( p, m, n, b, ldb, work, a, lda, work( p+1 ),
278 \$ work( p+mn+1 ), lwork-p-mn, info )
279 lopt = int( work( p+mn+1 ) )
280*
281* Update c = Z**H *c = ( c1 ) N-P
282* ( c2 ) M+P-N
283*
284 CALL zunmqr( 'Left', 'Conjugate Transpose', m, 1, mn, a, lda,
285 \$ work( p+1 ), c, max( 1, m ), work( p+mn+1 ),
286 \$ lwork-p-mn, info )
287 lopt = max( lopt, int( work( p+mn+1 ) ) )
288*
289* Solve T12*x2 = d for x2
290*
291 IF( p.GT.0 ) THEN
292 CALL ztrtrs( 'Upper', 'No transpose', 'Non-unit', p, 1,
293 \$ b( 1, n-p+1 ), ldb, d, p, info )
294*
295 IF( info.GT.0 ) THEN
296 info = 1
297 RETURN
298 END IF
299*
300* Put the solution in X
301*
302 CALL zcopy( p, d, 1, x( n-p+1 ), 1 )
303*
304* Update c1
305*
306 CALL zgemv( 'No transpose', n-p, p, -cone, a( 1, n-p+1 ), lda,
307 \$ d, 1, cone, c, 1 )
308 END IF
309*
310* Solve R11*x1 = c1 for x1
311*
312 IF( n.GT.p ) THEN
313 CALL ztrtrs( 'Upper', 'No transpose', 'Non-unit', n-p, 1,
314 \$ a, lda, c, n-p, info )
315*
316 IF( info.GT.0 ) THEN
317 info = 2
318 RETURN
319 END IF
320*
321* Put the solutions in X
322*
323 CALL zcopy( n-p, c, 1, x, 1 )
324 END IF
325*
326* Compute the residual vector:
327*
328 IF( m.LT.n ) THEN
329 nr = m + p - n
330 IF( nr.GT.0 )
331 \$ CALL zgemv( 'No transpose', nr, n-m, -cone, a( n-p+1, m+1 ),
332 \$ lda, d( nr+1 ), 1, cone, c( n-p+1 ), 1 )
333 ELSE
334 nr = p
335 END IF
336 IF( nr.GT.0 ) THEN
337 CALL ztrmv( 'Upper', 'No transpose', 'Non unit', nr,
338 \$ a( n-p+1, n-p+1 ), lda, d, 1 )
339 CALL zaxpy( nr, -cone, d, 1, c( n-p+1 ), 1 )
340 END IF
341*
342* Backward transformation x = Q**H*x
343*
344 CALL zunmrq( 'Left', 'Conjugate Transpose', n, 1, p, b, ldb,
345 \$ work( 1 ), x, n, work( p+mn+1 ), lwork-p-mn, info )
346 work( 1 ) = p + mn + max( lopt, int( work( p+mn+1 ) ) )
347*
348 RETURN
349*
350* End of ZGGLSE
351*
352 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:147
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine ztrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
ZTRTRS
Definition: ztrtrs.f:140
subroutine zggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
ZGGRQF
Definition: zggrqf.f:214
subroutine zunmrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMRQ
Definition: zunmrq.f:167
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:167
subroutine zgglse(M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO)
ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices
Definition: zgglse.f:180