LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ sgglse()

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

SGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:
``` SGGLSE 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (N) On exit, X is the solution of the LSE problem.``` [out] WORK ``` WORK is REAL 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 SGEQRF, SGERQF, SORMQR 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 sgglse.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 REAL A( LDA, * ), B( LDB, * ), C( * ), D( * ),
190 \$ WORK( * ), X( * )
191* ..
192*
193* =====================================================================
194*
195* .. Parameters ..
196 REAL ONE
197 parameter( one = 1.0e+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 saxpy, scopy, sgemv, sggrqf, sormqr, sormrq,
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, 'SGEQRF', ' ', m, n, -1, -1 )
242 nb2 = ilaenv( 1, 'SGERQF', ' ', m, n, -1, -1 )
243 nb3 = ilaenv( 1, 'SORMQR', ' ', m, n, p, -1 )
244 nb4 = ilaenv( 1, 'SORMRQ', ' ', 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( 'SGGLSE', -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 sggrqf( 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**T *c = ( c1 ) N-P
282* ( c2 ) M+P-N
283*
284 CALL sormqr( '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 strtrs( '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 scopy( p, d, 1, x( n-p+1 ), 1 )
302*
303* Update c1
304*
305 CALL sgemv( '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 strtrs( '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 scopy( 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 sgemv( '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 strmv( 'Upper', 'No transpose', 'Non unit', nr,
337 \$ a( n-p+1, n-p+1 ), lda, d, 1 )
338 CALL saxpy( nr, -one, d, 1, c( n-p+1 ), 1 )
339 END IF
340*
341* Backward transformation x = Q**T*x
342*
343 CALL sormrq( '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 SGGLSE
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 sormrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMRQ
Definition: sormrq.f:168
subroutine strtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
STRTRS
Definition: strtrs.f:140
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGRQF
Definition: sggrqf.f:214
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine strmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
STRMV
Definition: strmv.f:147
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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