LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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◆ zgglse()

 subroutine zgglse ( integer M, integer N, integer P, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) C, complex*16, dimension( * ) D, complex*16, dimension( * ) X, complex*16, dimension( * ) WORK, integer LWORK, integer INFO )

ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:
``` ZGGLSE solves the linear equality-constrained least squares (LSE)
problem:

minimize || c - A*x ||_2   subject to   B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

rank(B) = P and  rank( (A) ) = N.
( (B) )

These conditions ensure that the LSE problem has a unique solution,
which is obtained using a generalized RQ factorization of the
matrices (B, A) given by

B = (0 R)*Q,   A = Z*T*Q.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrices A and B. N >= 0.``` [in] P ``` P is INTEGER The number of rows of the matrix B. 0 <= P <= N <= M+P.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix T.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the upper triangle of the subarray B(1:P,N-P+1:N) contains the P-by-P upper triangular matrix R.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).``` [in,out] C ``` C is COMPLEX*16 array, dimension (M) On entry, C contains the right hand side vector for the least squares part of the LSE problem. On exit, the residual sum of squares for the solution is given by the sum of squares of elements N-P+1 to M of vector C.``` [in,out] D ``` D is COMPLEX*16 array, dimension (P) On entry, D contains the right hand side vector for the constrained equation. On exit, D is destroyed.``` [out] X ``` X is COMPLEX*16 array, dimension (N) On exit, X is the solution of the LSE problem.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,M+N+P). For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, where NB is an upper bound for the optimal blocksizes for ZGEQRF, CGERQF, ZUNMQR and CUNMRQ. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. = 1: the upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < P; the least squares solution could not be computed. = 2: the (N-P) by (N-P) part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that rank( (A) ) < N; the least squares solution could not ( (B) ) be computed.```

Definition at line 178 of file zgglse.f.

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*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
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
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