LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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.```
Date
November 2011

Definition at line 182 of file zgglse.f.

182 *
183 * -- LAPACK driver routine (version 3.4.0) --
184 * -- LAPACK is a software package provided by Univ. of Tennessee, --
185 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186 * November 2011
187 *
188 * .. Scalar Arguments ..
189  INTEGER info, lda, ldb, lwork, m, n, p
190 * ..
191 * .. Array Arguments ..
192  COMPLEX*16 a( lda, * ), b( ldb, * ), c( * ), d( * ),
193  \$ work( * ), x( * )
194 * ..
195 *
196 * =====================================================================
197 *
198 * .. Parameters ..
199  COMPLEX*16 cone
200  parameter ( cone = ( 1.0d+0, 0.0d+0 ) )
201 * ..
202 * .. Local Scalars ..
203  LOGICAL lquery
204  INTEGER lopt, lwkmin, lwkopt, mn, nb, nb1, nb2, nb3,
205  \$ nb4, nr
206 * ..
207 * .. External Subroutines ..
208  EXTERNAL xerbla, zaxpy, zcopy, zgemv, zggrqf, ztrmv,
209  \$ ztrtrs, zunmqr, zunmrq
210 * ..
211 * .. External Functions ..
212  INTEGER ilaenv
213  EXTERNAL ilaenv
214 * ..
215 * .. Intrinsic Functions ..
216  INTRINSIC int, max, min
217 * ..
218 * .. Executable Statements ..
219 *
220 * Test the input parameters
221 *
222  info = 0
223  mn = min( m, n )
224  lquery = ( lwork.EQ.-1 )
225  IF( m.LT.0 ) THEN
226  info = -1
227  ELSE IF( n.LT.0 ) THEN
228  info = -2
229  ELSE IF( p.LT.0 .OR. p.GT.n .OR. p.LT.n-m ) THEN
230  info = -3
231  ELSE IF( lda.LT.max( 1, m ) ) THEN
232  info = -5
233  ELSE IF( ldb.LT.max( 1, p ) ) THEN
234  info = -7
235  END IF
236 *
237 * Calculate workspace
238 *
239  IF( info.EQ.0) THEN
240  IF( n.EQ.0 ) THEN
241  lwkmin = 1
242  lwkopt = 1
243  ELSE
244  nb1 = ilaenv( 1, 'ZGEQRF', ' ', m, n, -1, -1 )
245  nb2 = ilaenv( 1, 'ZGERQF', ' ', m, n, -1, -1 )
246  nb3 = ilaenv( 1, 'ZUNMQR', ' ', m, n, p, -1 )
247  nb4 = ilaenv( 1, 'ZUNMRQ', ' ', m, n, p, -1 )
248  nb = max( nb1, nb2, nb3, nb4 )
249  lwkmin = m + n + p
250  lwkopt = p + mn + max( m, n )*nb
251  END IF
252  work( 1 ) = lwkopt
253 *
254  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
255  info = -12
256  END IF
257  END IF
258 *
259  IF( info.NE.0 ) THEN
260  CALL xerbla( 'ZGGLSE', -info )
261  RETURN
262  ELSE IF( lquery ) THEN
263  RETURN
264  END IF
265 *
266 * Quick return if possible
267 *
268  IF( n.EQ.0 )
269  \$ RETURN
270 *
271 * Compute the GRQ factorization of matrices B and A:
272 *
273 * B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P
274 * N-P P ( 0 R22 ) M+P-N
275 * N-P P
276 *
277 * where T12 and R11 are upper triangular, and Q and Z are
278 * unitary.
279 *
280  CALL zggrqf( p, m, n, b, ldb, work, a, lda, work( p+1 ),
281  \$ work( p+mn+1 ), lwork-p-mn, info )
282  lopt = work( p+mn+1 )
283 *
284 * Update c = Z**H *c = ( c1 ) N-P
285 * ( c2 ) M+P-N
286 *
287  CALL zunmqr( 'Left', 'Conjugate Transpose', m, 1, mn, a, lda,
288  \$ work( p+1 ), c, max( 1, m ), work( p+mn+1 ),
289  \$ lwork-p-mn, info )
290  lopt = max( lopt, int( work( p+mn+1 ) ) )
291 *
292 * Solve T12*x2 = d for x2
293 *
294  IF( p.GT.0 ) THEN
295  CALL ztrtrs( 'Upper', 'No transpose', 'Non-unit', p, 1,
296  \$ b( 1, n-p+1 ), ldb, d, p, info )
297 *
298  IF( info.GT.0 ) THEN
299  info = 1
300  RETURN
301  END IF
302 *
303 * Put the solution in X
304 *
305  CALL zcopy( p, d, 1, x( n-p+1 ), 1 )
306 *
307 * Update c1
308 *
309  CALL zgemv( 'No transpose', n-p, p, -cone, a( 1, n-p+1 ), lda,
310  \$ d, 1, cone, c, 1 )
311  END IF
312 *
313 * Solve R11*x1 = c1 for x1
314 *
315  IF( n.GT.p ) THEN
316  CALL ztrtrs( 'Upper', 'No transpose', 'Non-unit', n-p, 1,
317  \$ a, lda, c, n-p, info )
318 *
319  IF( info.GT.0 ) THEN
320  info = 2
321  RETURN
322  END IF
323 *
324 * Put the solutions in X
325 *
326  CALL zcopy( n-p, c, 1, x, 1 )
327  END IF
328 *
329 * Compute the residual vector:
330 *
331  IF( m.LT.n ) THEN
332  nr = m + p - n
333  IF( nr.GT.0 )
334  \$ CALL zgemv( 'No transpose', nr, n-m, -cone, a( n-p+1, m+1 ),
335  \$ lda, d( nr+1 ), 1, cone, c( n-p+1 ), 1 )
336  ELSE
337  nr = p
338  END IF
339  IF( nr.GT.0 ) THEN
340  CALL ztrmv( 'Upper', 'No transpose', 'Non unit', nr,
341  \$ a( n-p+1, n-p+1 ), lda, d, 1 )
342  CALL zaxpy( nr, -cone, d, 1, c( n-p+1 ), 1 )
343  END IF
344 *
345 * Backward transformation x = Q**H*x
346 *
347  CALL zunmrq( 'Left', 'Conjugate Transpose', n, 1, p, b, ldb,
348  \$ work( 1 ), x, n, work( p+mn+1 ), lwork-p-mn, info )
349  work( 1 ) = p + mn + max( lopt, int( work( p+mn+1 ) ) )
350 *
351  RETURN
352 *
353 * End of ZGGLSE
354 *
subroutine zggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
ZGGRQF
Definition: zggrqf.f:216
subroutine ztrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
ZTRTRS
Definition: ztrtrs.f:142
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:52
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:169
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:149
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
subroutine zunmrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMRQ
Definition: zunmrq.f:169
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:53

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