 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgglse()

 subroutine dgglse ( integer M, integer N, integer P, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) C, double precision, dimension( * ) D, double precision, dimension( * ) X, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Download DGGLSE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` DGGLSE 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N) On exit, X is the solution of the LSE problem.``` [out] WORK ``` WORK is DOUBLE PRECISION 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 DGEQRF, SGERQF, DORMQR and SORMRQ. 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 dgglse.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  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ),
190  \$ WORK( * ), X( * )
191 * ..
192 *
193 * =====================================================================
194 *
195 * .. Parameters ..
196  DOUBLE PRECISION ONE
197  parameter( one = 1.0d+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 daxpy, dcopy, dgemv, dggrqf, dormqr, dormrq,
206  \$ dtrmv, dtrtrs, xerbla
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, 'DGEQRF', ' ', m, n, -1, -1 )
242  nb2 = ilaenv( 1, 'DGERQF', ' ', m, n, -1, -1 )
243  nb3 = ilaenv( 1, 'DORMQR', ' ', m, n, p, -1 )
244  nb4 = ilaenv( 1, 'DORMRQ', ' ', 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( 'DGGLSE', -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**T = ( 0 T12 ) P Z**T*A*Q**T = ( 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 * orthogonal.
276 *
277  CALL dggrqf( p, m, n, b, ldb, work, a, lda, work( p+1 ),
278  \$ work( p+mn+1 ), lwork-p-mn, info )
279  lopt = work( p+mn+1 )
280 *
281 * Update c = Z**T *c = ( c1 ) N-P
282 * ( c2 ) M+P-N
283 *
284  CALL dormqr( 'Left', 'Transpose', m, 1, mn, a, lda, work( p+1 ),
285  \$ c, max( 1, m ), work( p+mn+1 ), lwork-p-mn, info )
286  lopt = max( lopt, int( work( p+mn+1 ) ) )
287 *
288 * Solve T12*x2 = d for x2
289 *
290  IF( p.GT.0 ) THEN
291  CALL dtrtrs( 'Upper', 'No transpose', 'Non-unit', p, 1,
292  \$ b( 1, n-p+1 ), ldb, d, p, info )
293 *
294  IF( info.GT.0 ) THEN
295  info = 1
296  RETURN
297  END IF
298 *
299 * Put the solution in X
300 *
301  CALL dcopy( p, d, 1, x( n-p+1 ), 1 )
302 *
303 * Update c1
304 *
305  CALL dgemv( 'No transpose', n-p, p, -one, a( 1, n-p+1 ), lda,
306  \$ d, 1, one, c, 1 )
307  END IF
308 *
309 * Solve R11*x1 = c1 for x1
310 *
311  IF( n.GT.p ) THEN
312  CALL dtrtrs( 'Upper', 'No transpose', 'Non-unit', n-p, 1,
313  \$ a, lda, c, n-p, info )
314 *
315  IF( info.GT.0 ) THEN
316  info = 2
317  RETURN
318  END IF
319 *
320 * Put the solutions in X
321 *
322  CALL dcopy( n-p, c, 1, x, 1 )
323  END IF
324 *
325 * Compute the residual vector:
326 *
327  IF( m.LT.n ) THEN
328  nr = m + p - n
329  IF( nr.GT.0 )
330  \$ CALL dgemv( 'No transpose', nr, n-m, -one, a( n-p+1, m+1 ),
331  \$ lda, d( nr+1 ), 1, one, c( n-p+1 ), 1 )
332  ELSE
333  nr = p
334  END IF
335  IF( nr.GT.0 ) THEN
336  CALL dtrmv( 'Upper', 'No transpose', 'Non unit', nr,
337  \$ a( n-p+1, n-p+1 ), lda, d, 1 )
338  CALL daxpy( nr, -one, d, 1, c( n-p+1 ), 1 )
339  END IF
340 *
341 * Backward transformation x = Q**T*x
342 *
343  CALL dormrq( 'Left', 'Transpose', n, 1, p, b, ldb, work( 1 ), x,
344  \$ n, work( p+mn+1 ), lwork-p-mn, info )
345  work( 1 ) = p + mn + max( lopt, int( work( p+mn+1 ) ) )
346 *
347  RETURN
348 *
349 * End of DGGLSE
350 *
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 dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRMV
Definition: dtrmv.f:147
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:167
subroutine dormrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMRQ
Definition: dormrq.f:167
subroutine dggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGRQF
Definition: dggrqf.f:214
subroutine dtrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
DTRTRS
Definition: dtrtrs.f:140
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