LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sggglm()

 subroutine sggglm ( integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) d, real, dimension( * ) x, real, dimension( * ) y, real, dimension( * ) work, integer lwork, integer info )

SGGGLM

Purpose:
``` SGGGLM solves a general Gauss-Markov linear model (GLM) problem:

minimize || y ||_2   subject to   d = A*x + B*y
x

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

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

Under these assumptions, the constrained equation is always
consistent, and there is a unique solution x and a minimal 2-norm
solution y, which is obtained using a generalized QR factorization
of the matrices (A, B) given by

A = Q*(R),   B = Q*T*Z.
(0)

In particular, if matrix B is square nonsingular, then the problem
GLM is equivalent to the following weighted linear least squares
problem

minimize || inv(B)*(d-A*x) ||_2
x

where inv(B) denotes the inverse of B.```
Parameters
 [in] N ``` N is INTEGER The number of rows of the matrices A and B. N >= 0.``` [in] M ``` M is INTEGER The number of columns of the matrix A. 0 <= M <= N.``` [in] P ``` P is INTEGER The number of columns of the matrix B. P >= N-M.``` [in,out] A ``` A is REAL array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is REAL array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] D ``` D is REAL array, dimension (N) On entry, D is the left hand side of the GLM equation. On exit, D is destroyed.``` [out] X ` X is REAL array, dimension (M)` [out] Y ``` Y is REAL array, dimension (P) On exit, X and Y are the solutions of the GLM 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,N+M+P). For optimum performance, LWORK >= M+min(N,P)+max(N,P)*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 A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed.```

Definition at line 183 of file sggglm.f.

185*
186* -- LAPACK driver routine --
187* -- LAPACK is a software package provided by Univ. of Tennessee, --
188* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189*
190* .. Scalar Arguments ..
191 INTEGER INFO, LDA, LDB, LWORK, M, N, P
192* ..
193* .. Array Arguments ..
194 REAL A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
195 \$ X( * ), Y( * )
196* ..
197*
198* ===================================================================
199*
200* .. Parameters ..
201 REAL ZERO, ONE
202 parameter( zero = 0.0e+0, one = 1.0e+0 )
203* ..
204* .. Local Scalars ..
205 LOGICAL LQUERY
206 INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
207 \$ NB4, NP
208* ..
209* .. External Subroutines ..
210 EXTERNAL scopy, sgemv, sggqrf, sormqr, sormrq, strtrs,
211 \$ xerbla
212* ..
213* .. External Functions ..
214 INTEGER ILAENV
215 REAL SROUNDUP_LWORK
216 EXTERNAL ilaenv, sroundup_lwork
217* ..
218* .. Intrinsic Functions ..
219 INTRINSIC int, max, min
220* ..
221* .. Executable Statements ..
222*
223* Test the input parameters
224*
225 info = 0
226 np = min( n, p )
227 lquery = ( lwork.EQ.-1 )
228 IF( n.LT.0 ) THEN
229 info = -1
230 ELSE IF( m.LT.0 .OR. m.GT.n ) THEN
231 info = -2
232 ELSE IF( p.LT.0 .OR. p.LT.n-m ) THEN
233 info = -3
234 ELSE IF( lda.LT.max( 1, n ) ) THEN
235 info = -5
236 ELSE IF( ldb.LT.max( 1, n ) ) THEN
237 info = -7
238 END IF
239*
240* Calculate workspace
241*
242 IF( info.EQ.0) THEN
243 IF( n.EQ.0 ) THEN
244 lwkmin = 1
245 lwkopt = 1
246 ELSE
247 nb1 = ilaenv( 1, 'SGEQRF', ' ', n, m, -1, -1 )
248 nb2 = ilaenv( 1, 'SGERQF', ' ', n, m, -1, -1 )
249 nb3 = ilaenv( 1, 'SORMQR', ' ', n, m, p, -1 )
250 nb4 = ilaenv( 1, 'SORMRQ', ' ', n, m, p, -1 )
251 nb = max( nb1, nb2, nb3, nb4 )
252 lwkmin = m + n + p
253 lwkopt = m + np + max( n, p )*nb
254 END IF
255 work( 1 ) = sroundup_lwork(lwkopt)
256*
257 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
258 info = -12
259 END IF
260 END IF
261*
262 IF( info.NE.0 ) THEN
263 CALL xerbla( 'SGGGLM', -info )
264 RETURN
265 ELSE IF( lquery ) THEN
266 RETURN
267 END IF
268*
269* Quick return if possible
270*
271 IF( n.EQ.0 ) THEN
272 DO i = 1, m
273 x(i) = zero
274 END DO
275 DO i = 1, p
276 y(i) = zero
277 END DO
278 RETURN
279 END IF
280*
281* Compute the GQR factorization of matrices A and B:
282*
283* Q**T*A = ( R11 ) M, Q**T*B*Z**T = ( T11 T12 ) M
284* ( 0 ) N-M ( 0 T22 ) N-M
285* M M+P-N N-M
286*
287* where R11 and T22 are upper triangular, and Q and Z are
288* orthogonal.
289*
290 CALL sggqrf( n, m, p, a, lda, work, b, ldb, work( m+1 ),
291 \$ work( m+np+1 ), lwork-m-np, info )
292 lopt = int( work( m+np+1 ) )
293*
294* Update left-hand-side vector d = Q**T*d = ( d1 ) M
295* ( d2 ) N-M
296*
297 CALL sormqr( 'Left', 'Transpose', n, 1, m, a, lda, work, d,
298 \$ max( 1, n ), work( m+np+1 ), lwork-m-np, info )
299 lopt = max( lopt, int( work( m+np+1 ) ) )
300*
301* Solve T22*y2 = d2 for y2
302*
303 IF( n.GT.m ) THEN
304 CALL strtrs( 'Upper', 'No transpose', 'Non unit', n-m, 1,
305 \$ b( m+1, m+p-n+1 ), ldb, d( m+1 ), n-m, info )
306*
307 IF( info.GT.0 ) THEN
308 info = 1
309 RETURN
310 END IF
311*
312 CALL scopy( n-m, d( m+1 ), 1, y( m+p-n+1 ), 1 )
313 END IF
314*
315* Set y1 = 0
316*
317 DO 10 i = 1, m + p - n
318 y( i ) = zero
319 10 CONTINUE
320*
321* Update d1 = d1 - T12*y2
322*
323 CALL sgemv( 'No transpose', m, n-m, -one, b( 1, m+p-n+1 ), ldb,
324 \$ y( m+p-n+1 ), 1, one, d, 1 )
325*
326* Solve triangular system: R11*x = d1
327*
328 IF( m.GT.0 ) THEN
329 CALL strtrs( 'Upper', 'No Transpose', 'Non unit', m, 1, a, lda,
330 \$ d, m, info )
331*
332 IF( info.GT.0 ) THEN
333 info = 2
334 RETURN
335 END IF
336*
337* Copy D to X
338*
339 CALL scopy( m, d, 1, x, 1 )
340 END IF
341*
342* Backward transformation y = Z**T *y
343*
344 CALL sormrq( 'Left', 'Transpose', p, 1, np,
345 \$ b( max( 1, n-p+1 ), 1 ), ldb, work( m+1 ), y,
346 \$ max( 1, p ), work( m+np+1 ), lwork-m-np, info )
347 work( 1 ) = m + np + max( lopt, int( work( m+np+1 ) ) )
348*
349 RETURN
350*
351* End of SGGGLM
352*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
SGEMV
Definition sgemv.f:158
subroutine sggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGQRF
Definition sggqrf.f:215
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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 sormrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMRQ
Definition sormrq.f:168
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