LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgetsls()

 subroutine dgetsls ( character TRANS, integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DGETSLS

Purpose:
``` DGETSLS solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
factorization of A.  It is assumed that A has full rank.

The following options are provided:

1. If TRANS = 'N' and m >= n:  find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.

2. If TRANS = 'N' and m < n:  find the minimum norm solution of
an underdetermined system A * X = B.

3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
an undetermined system A**T * X = B.

4. If TRANS = 'T' and m < n:  find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.

Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.```
Parameters
 [in] TRANS ``` TRANS is CHARACTER*1 = 'N': the linear system involves A; = 'T': the linear system involves A**T.``` [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 matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >=0.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A is overwritten by details of its QR or LQ factorization as returned by DGEQR or DGELQ.``` [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,NRHS) On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'T'. On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors. if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors; if TRANS = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = 'T' and m < n, rows 1 to M of B contain the least squares solution vectors.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= MAX(1,M,N).``` [out] WORK ``` (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If LWORK = -1 or -2, then a workspace query is assumed. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.```

Definition at line 160 of file dgetsls.f.

162 *
163 * -- LAPACK driver routine --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 *
167 * .. Scalar Arguments ..
168  CHARACTER TRANS
169  INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
170 * ..
171 * .. Array Arguments ..
172  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
173 *
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  DOUBLE PRECISION ZERO, ONE
180  parameter( zero = 0.0d0, one = 1.0d0 )
181 * ..
182 * .. Local Scalars ..
183  LOGICAL LQUERY, TRAN
184  INTEGER I, IASCL, IBSCL, J, MAXMN, BROW,
185  \$ SCLLEN, TSZO, TSZM, LWO, LWM, LW1, LW2,
186  \$ WSIZEO, WSIZEM, INFO2
187  DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM, TQ( 5 ), WORKQ( 1 )
188 * ..
189 * .. External Functions ..
190  LOGICAL LSAME
191  DOUBLE PRECISION DLAMCH, DLANGE
192  EXTERNAL lsame, dlabad, dlamch, dlange
193 * ..
194 * .. External Subroutines ..
195  EXTERNAL dgeqr, dgemqr, dlascl, dlaset,
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC dble, max, min, int
200 * ..
201 * .. Executable Statements ..
202 *
203 * Test the input arguments.
204 *
205  info = 0
206  maxmn = max( m, n )
207  tran = lsame( trans, 'T' )
208 *
209  lquery = ( lwork.EQ.-1 .OR. lwork.EQ.-2 )
210  IF( .NOT.( lsame( trans, 'N' ) .OR.
211  \$ lsame( trans, 'T' ) ) ) THEN
212  info = -1
213  ELSE IF( m.LT.0 ) THEN
214  info = -2
215  ELSE IF( n.LT.0 ) THEN
216  info = -3
217  ELSE IF( nrhs.LT.0 ) THEN
218  info = -4
219  ELSE IF( lda.LT.max( 1, m ) ) THEN
220  info = -6
221  ELSE IF( ldb.LT.max( 1, m, n ) ) THEN
222  info = -8
223  END IF
224 *
225  IF( info.EQ.0 ) THEN
226 *
227 * Determine the optimum and minimum LWORK
228 *
229  IF( m.GE.n ) THEN
230  CALL dgeqr( m, n, a, lda, tq, -1, workq, -1, info2 )
231  tszo = int( tq( 1 ) )
232  lwo = int( workq( 1 ) )
233  CALL dgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
234  \$ tszo, b, ldb, workq, -1, info2 )
235  lwo = max( lwo, int( workq( 1 ) ) )
236  CALL dgeqr( m, n, a, lda, tq, -2, workq, -2, info2 )
237  tszm = int( tq( 1 ) )
238  lwm = int( workq( 1 ) )
239  CALL dgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
240  \$ tszm, b, ldb, workq, -1, info2 )
241  lwm = max( lwm, int( workq( 1 ) ) )
242  wsizeo = tszo + lwo
243  wsizem = tszm + lwm
244  ELSE
245  CALL dgelq( m, n, a, lda, tq, -1, workq, -1, info2 )
246  tszo = int( tq( 1 ) )
247  lwo = int( workq( 1 ) )
248  CALL dgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
249  \$ tszo, b, ldb, workq, -1, info2 )
250  lwo = max( lwo, int( workq( 1 ) ) )
251  CALL dgelq( m, n, a, lda, tq, -2, workq, -2, info2 )
252  tszm = int( tq( 1 ) )
253  lwm = int( workq( 1 ) )
254  CALL dgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
255  \$ tszm, b, ldb, workq, -1, info2 )
256  lwm = max( lwm, int( workq( 1 ) ) )
257  wsizeo = tszo + lwo
258  wsizem = tszm + lwm
259  END IF
260 *
261  IF( ( lwork.LT.wsizem ).AND.( .NOT.lquery ) ) THEN
262  info = -10
263  END IF
264 *
265  work( 1 ) = dble( wsizeo )
266 *
267  END IF
268 *
269  IF( info.NE.0 ) THEN
270  CALL xerbla( 'DGETSLS', -info )
271  RETURN
272  END IF
273  IF( lquery ) THEN
274  IF( lwork.EQ.-2 ) work( 1 ) = dble( wsizem )
275  RETURN
276  END IF
277  IF( lwork.LT.wsizeo ) THEN
278  lw1 = tszm
279  lw2 = lwm
280  ELSE
281  lw1 = tszo
282  lw2 = lwo
283  END IF
284 *
285 * Quick return if possible
286 *
287  IF( min( m, n, nrhs ).EQ.0 ) THEN
288  CALL dlaset( 'FULL', max( m, n ), nrhs, zero, zero,
289  \$ b, ldb )
290  RETURN
291  END IF
292 *
293 * Get machine parameters
294 *
295  smlnum = dlamch( 'S' ) / dlamch( 'P' )
296  bignum = one / smlnum
297  CALL dlabad( smlnum, bignum )
298 *
299 * Scale A, B if max element outside range [SMLNUM,BIGNUM]
300 *
301  anrm = dlange( 'M', m, n, a, lda, work )
302  iascl = 0
303  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
304 *
305 * Scale matrix norm up to SMLNUM
306 *
307  CALL dlascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
308  iascl = 1
309  ELSE IF( anrm.GT.bignum ) THEN
310 *
311 * Scale matrix norm down to BIGNUM
312 *
313  CALL dlascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
314  iascl = 2
315  ELSE IF( anrm.EQ.zero ) THEN
316 *
317 * Matrix all zero. Return zero solution.
318 *
319  CALL dlaset( 'F', maxmn, nrhs, zero, zero, b, ldb )
320  GO TO 50
321  END IF
322 *
323  brow = m
324  IF ( tran ) THEN
325  brow = n
326  END IF
327  bnrm = dlange( 'M', brow, nrhs, b, ldb, work )
328  ibscl = 0
329  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
330 *
331 * Scale matrix norm up to SMLNUM
332 *
333  CALL dlascl( 'G', 0, 0, bnrm, smlnum, brow, nrhs, b, ldb,
334  \$ info )
335  ibscl = 1
336  ELSE IF( bnrm.GT.bignum ) THEN
337 *
338 * Scale matrix norm down to BIGNUM
339 *
340  CALL dlascl( 'G', 0, 0, bnrm, bignum, brow, nrhs, b, ldb,
341  \$ info )
342  ibscl = 2
343  END IF
344 *
345  IF ( m.GE.n ) THEN
346 *
347 * compute QR factorization of A
348 *
349  CALL dgeqr( m, n, a, lda, work( lw2+1 ), lw1,
350  \$ work( 1 ), lw2, info )
351  IF ( .NOT.tran ) THEN
352 *
353 * Least-Squares Problem min || A * X - B ||
354 *
355 * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
356 *
357  CALL dgemqr( 'L' , 'T', m, nrhs, n, a, lda,
358  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
359  \$ info )
360 *
361 * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
362 *
363  CALL dtrtrs( 'U', 'N', 'N', n, nrhs,
364  \$ a, lda, b, ldb, info )
365  IF( info.GT.0 ) THEN
366  RETURN
367  END IF
368  scllen = n
369  ELSE
370 *
371 * Overdetermined system of equations A**T * X = B
372 *
373 * B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
374 *
375  CALL dtrtrs( 'U', 'T', 'N', n, nrhs,
376  \$ a, lda, b, ldb, info )
377 *
378  IF( info.GT.0 ) THEN
379  RETURN
380  END IF
381 *
382 * B(N+1:M,1:NRHS) = ZERO
383 *
384  DO 20 j = 1, nrhs
385  DO 10 i = n + 1, m
386  b( i, j ) = zero
387  10 CONTINUE
388  20 CONTINUE
389 *
390 * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
391 *
392  CALL dgemqr( 'L', 'N', m, nrhs, n, a, lda,
393  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
394  \$ info )
395 *
396  scllen = m
397 *
398  END IF
399 *
400  ELSE
401 *
402 * Compute LQ factorization of A
403 *
404  CALL dgelq( m, n, a, lda, work( lw2+1 ), lw1,
405  \$ work( 1 ), lw2, info )
406 *
407 * workspace at least M, optimally M*NB.
408 *
409  IF( .NOT.tran ) THEN
410 *
411 * underdetermined system of equations A * X = B
412 *
413 * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
414 *
415  CALL dtrtrs( 'L', 'N', 'N', m, nrhs,
416  \$ a, lda, b, ldb, info )
417 *
418  IF( info.GT.0 ) THEN
419  RETURN
420  END IF
421 *
422 * B(M+1:N,1:NRHS) = 0
423 *
424  DO 40 j = 1, nrhs
425  DO 30 i = m + 1, n
426  b( i, j ) = zero
427  30 CONTINUE
428  40 CONTINUE
429 *
430 * B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
431 *
432  CALL dgemlq( 'L', 'T', n, nrhs, m, a, lda,
433  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
434  \$ info )
435 *
436 * workspace at least NRHS, optimally NRHS*NB
437 *
438  scllen = n
439 *
440  ELSE
441 *
442 * overdetermined system min || A**T * X - B ||
443 *
444 * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
445 *
446  CALL dgemlq( 'L', 'N', n, nrhs, m, a, lda,
447  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
448  \$ info )
449 *
450 * workspace at least NRHS, optimally NRHS*NB
451 *
452 * B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
453 *
454  CALL dtrtrs( 'Lower', 'Transpose', 'Non-unit', m, nrhs,
455  \$ a, lda, b, ldb, info )
456 *
457  IF( info.GT.0 ) THEN
458  RETURN
459  END IF
460 *
461  scllen = m
462 *
463  END IF
464 *
465  END IF
466 *
467 * Undo scaling
468 *
469  IF( iascl.EQ.1 ) THEN
470  CALL dlascl( 'G', 0, 0, anrm, smlnum, scllen, nrhs, b, ldb,
471  \$ info )
472  ELSE IF( iascl.EQ.2 ) THEN
473  CALL dlascl( 'G', 0, 0, anrm, bignum, scllen, nrhs, b, ldb,
474  \$ info )
475  END IF
476  IF( ibscl.EQ.1 ) THEN
477  CALL dlascl( 'G', 0, 0, smlnum, bnrm, scllen, nrhs, b, ldb,
478  \$ info )
479  ELSE IF( ibscl.EQ.2 ) THEN
480  CALL dlascl( 'G', 0, 0, bignum, bnrm, scllen, nrhs, b, ldb,
481  \$ info )
482  END IF
483 *
484  50 CONTINUE
485  work( 1 ) = dble( tszo + lwo )
486  RETURN
487 *
488 * End of DGETSLS
489 *
subroutine dgelq(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
DGELQ
Definition: dgelq.f:172
subroutine dgemlq(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
DGEMLQ
Definition: dgemlq.f:171
subroutine dgemqr(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
DGEMQR
Definition: dgemqr.f:172
subroutine dgeqr(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
DGEQR
Definition: dgeqr.f:174
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dtrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
DTRTRS
Definition: dtrtrs.f:140
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