LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ cgetsls()

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

CGETSLS

Purpose:
``` CGETSLS solves overdetermined or underdetermined complex 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 = 'C' and m >= n:  find the minimum norm solution of
an undetermined system A**T * X = B.

4. If TRANS = 'C' 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; = 'C': the linear system involves A**H.``` [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 COMPLEX 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 CGEQR or CGELQ.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in,out] B ``` B is COMPLEX 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 = 'C'. 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 = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = 'C' 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) COMPLEX 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 cgetsls.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  COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
173 *
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  REAL ZERO, ONE
180  parameter( zero = 0.0e0, one = 1.0e0 )
181  COMPLEX CZERO
182  parameter( czero = ( 0.0e+0, 0.0e+0 ) )
183 * ..
184 * .. Local Scalars ..
185  LOGICAL LQUERY, TRAN
186  INTEGER I, IASCL, IBSCL, J, MAXMN, BROW,
187  \$ SCLLEN, TSZO, TSZM, LWO, LWM, LW1, LW2,
188  \$ WSIZEO, WSIZEM, INFO2
189  REAL ANRM, BIGNUM, BNRM, SMLNUM, DUM( 1 )
190  COMPLEX TQ( 5 ), WORKQ( 1 )
191 * ..
192 * .. External Functions ..
193  LOGICAL LSAME
194  REAL SLAMCH, CLANGE
195  EXTERNAL lsame, slabad, slamch, clange
196 * ..
197 * .. External Subroutines ..
198  EXTERNAL cgeqr, cgemqr, clascl, claset,
200 * ..
201 * .. Intrinsic Functions ..
202  INTRINSIC real, max, min, int
203 * ..
204 * .. Executable Statements ..
205 *
206 * Test the input arguments.
207 *
208  info = 0
209  maxmn = max( m, n )
210  tran = lsame( trans, 'C' )
211 *
212  lquery = ( lwork.EQ.-1 .OR. lwork.EQ.-2 )
213  IF( .NOT.( lsame( trans, 'N' ) .OR.
214  \$ lsame( trans, 'C' ) ) ) THEN
215  info = -1
216  ELSE IF( m.LT.0 ) THEN
217  info = -2
218  ELSE IF( n.LT.0 ) THEN
219  info = -3
220  ELSE IF( nrhs.LT.0 ) THEN
221  info = -4
222  ELSE IF( lda.LT.max( 1, m ) ) THEN
223  info = -6
224  ELSE IF( ldb.LT.max( 1, m, n ) ) THEN
225  info = -8
226  END IF
227 *
228  IF( info.EQ.0 ) THEN
229 *
230 * Determine the optimum and minimum LWORK
231 *
232  IF( m.GE.n ) THEN
233  CALL cgeqr( m, n, a, lda, tq, -1, workq, -1, info2 )
234  tszo = int( tq( 1 ) )
235  lwo = int( workq( 1 ) )
236  CALL cgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
237  \$ tszo, b, ldb, workq, -1, info2 )
238  lwo = max( lwo, int( workq( 1 ) ) )
239  CALL cgeqr( m, n, a, lda, tq, -2, workq, -2, info2 )
240  tszm = int( tq( 1 ) )
241  lwm = int( workq( 1 ) )
242  CALL cgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
243  \$ tszm, b, ldb, workq, -1, info2 )
244  lwm = max( lwm, int( workq( 1 ) ) )
245  wsizeo = tszo + lwo
246  wsizem = tszm + lwm
247  ELSE
248  CALL cgelq( m, n, a, lda, tq, -1, workq, -1, info2 )
249  tszo = int( tq( 1 ) )
250  lwo = int( workq( 1 ) )
251  CALL cgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
252  \$ tszo, b, ldb, workq, -1, info2 )
253  lwo = max( lwo, int( workq( 1 ) ) )
254  CALL cgelq( m, n, a, lda, tq, -2, workq, -2, info2 )
255  tszm = int( tq( 1 ) )
256  lwm = int( workq( 1 ) )
257  CALL cgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
258  \$ tszm, b, ldb, workq, -1, info2 )
259  lwm = max( lwm, int( workq( 1 ) ) )
260  wsizeo = tszo + lwo
261  wsizem = tszm + lwm
262  END IF
263 *
264  IF( ( lwork.LT.wsizem ).AND.( .NOT.lquery ) ) THEN
265  info = -10
266  END IF
267 *
268  work( 1 ) = real( wsizeo )
269 *
270  END IF
271 *
272  IF( info.NE.0 ) THEN
273  CALL xerbla( 'CGETSLS', -info )
274  RETURN
275  END IF
276  IF( lquery ) THEN
277  IF( lwork.EQ.-2 ) work( 1 ) = real( wsizem )
278  RETURN
279  END IF
280  IF( lwork.LT.wsizeo ) THEN
281  lw1 = tszm
282  lw2 = lwm
283  ELSE
284  lw1 = tszo
285  lw2 = lwo
286  END IF
287 *
288 * Quick return if possible
289 *
290  IF( min( m, n, nrhs ).EQ.0 ) THEN
291  CALL claset( 'FULL', max( m, n ), nrhs, czero, czero,
292  \$ b, ldb )
293  RETURN
294  END IF
295 *
296 * Get machine parameters
297 *
298  smlnum = slamch( 'S' ) / slamch( 'P' )
299  bignum = one / smlnum
300  CALL slabad( smlnum, bignum )
301 *
302 * Scale A, B if max element outside range [SMLNUM,BIGNUM]
303 *
304  anrm = clange( 'M', m, n, a, lda, dum )
305  iascl = 0
306  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
307 *
308 * Scale matrix norm up to SMLNUM
309 *
310  CALL clascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
311  iascl = 1
312  ELSE IF( anrm.GT.bignum ) THEN
313 *
314 * Scale matrix norm down to BIGNUM
315 *
316  CALL clascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
317  iascl = 2
318  ELSE IF( anrm.EQ.zero ) THEN
319 *
320 * Matrix all zero. Return zero solution.
321 *
322  CALL claset( 'F', maxmn, nrhs, czero, czero, b, ldb )
323  GO TO 50
324  END IF
325 *
326  brow = m
327  IF ( tran ) THEN
328  brow = n
329  END IF
330  bnrm = clange( 'M', brow, nrhs, b, ldb, dum )
331  ibscl = 0
332  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
333 *
334 * Scale matrix norm up to SMLNUM
335 *
336  CALL clascl( 'G', 0, 0, bnrm, smlnum, brow, nrhs, b, ldb,
337  \$ info )
338  ibscl = 1
339  ELSE IF( bnrm.GT.bignum ) THEN
340 *
341 * Scale matrix norm down to BIGNUM
342 *
343  CALL clascl( 'G', 0, 0, bnrm, bignum, brow, nrhs, b, ldb,
344  \$ info )
345  ibscl = 2
346  END IF
347 *
348  IF ( m.GE.n ) THEN
349 *
350 * compute QR factorization of A
351 *
352  CALL cgeqr( m, n, a, lda, work( lw2+1 ), lw1,
353  \$ work( 1 ), lw2, info )
354  IF ( .NOT.tran ) THEN
355 *
356 * Least-Squares Problem min || A * X - B ||
357 *
358 * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
359 *
360  CALL cgemqr( 'L' , 'C', m, nrhs, n, a, lda,
361  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
362  \$ info )
363 *
364 * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
365 *
366  CALL ctrtrs( 'U', 'N', 'N', n, nrhs,
367  \$ a, lda, b, ldb, info )
368  IF( info.GT.0 ) THEN
369  RETURN
370  END IF
371  scllen = n
372  ELSE
373 *
374 * Overdetermined system of equations A**T * X = B
375 *
376 * B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
377 *
378  CALL ctrtrs( 'U', 'C', 'N', n, nrhs,
379  \$ a, lda, b, ldb, info )
380 *
381  IF( info.GT.0 ) THEN
382  RETURN
383  END IF
384 *
385 * B(N+1:M,1:NRHS) = CZERO
386 *
387  DO 20 j = 1, nrhs
388  DO 10 i = n + 1, m
389  b( i, j ) = czero
390  10 CONTINUE
391  20 CONTINUE
392 *
393 * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
394 *
395  CALL cgemqr( 'L', 'N', m, nrhs, n, a, lda,
396  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
397  \$ info )
398 *
399  scllen = m
400 *
401  END IF
402 *
403  ELSE
404 *
405 * Compute LQ factorization of A
406 *
407  CALL cgelq( m, n, a, lda, work( lw2+1 ), lw1,
408  \$ work( 1 ), lw2, info )
409 *
410 * workspace at least M, optimally M*NB.
411 *
412  IF( .NOT.tran ) THEN
413 *
414 * underdetermined system of equations A * X = B
415 *
416 * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
417 *
418  CALL ctrtrs( 'L', 'N', 'N', m, nrhs,
419  \$ a, lda, b, ldb, info )
420 *
421  IF( info.GT.0 ) THEN
422  RETURN
423  END IF
424 *
425 * B(M+1:N,1:NRHS) = 0
426 *
427  DO 40 j = 1, nrhs
428  DO 30 i = m + 1, n
429  b( i, j ) = czero
430  30 CONTINUE
431  40 CONTINUE
432 *
433 * B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
434 *
435  CALL cgemlq( 'L', 'C', n, nrhs, m, a, lda,
436  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
437  \$ info )
438 *
439 * workspace at least NRHS, optimally NRHS*NB
440 *
441  scllen = n
442 *
443  ELSE
444 *
445 * overdetermined system min || A**T * X - B ||
446 *
447 * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
448 *
449  CALL cgemlq( 'L', 'N', n, nrhs, m, a, lda,
450  \$ work( lw2+1 ), lw1, b, ldb, work( 1 ), lw2,
451  \$ info )
452 *
453 * workspace at least NRHS, optimally NRHS*NB
454 *
455 * B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
456 *
457  CALL ctrtrs( 'L', 'C', 'N', m, nrhs,
458  \$ a, lda, b, ldb, info )
459 *
460  IF( info.GT.0 ) THEN
461  RETURN
462  END IF
463 *
464  scllen = m
465 *
466  END IF
467 *
468  END IF
469 *
470 * Undo scaling
471 *
472  IF( iascl.EQ.1 ) THEN
473  CALL clascl( 'G', 0, 0, anrm, smlnum, scllen, nrhs, b, ldb,
474  \$ info )
475  ELSE IF( iascl.EQ.2 ) THEN
476  CALL clascl( 'G', 0, 0, anrm, bignum, scllen, nrhs, b, ldb,
477  \$ info )
478  END IF
479  IF( ibscl.EQ.1 ) THEN
480  CALL clascl( 'G', 0, 0, smlnum, bnrm, scllen, nrhs, b, ldb,
481  \$ info )
482  ELSE IF( ibscl.EQ.2 ) THEN
483  CALL clascl( 'G', 0, 0, bignum, bnrm, scllen, nrhs, b, ldb,
484  \$ info )
485  END IF
486 *
487  50 CONTINUE
488  work( 1 ) = real( tszo + lwo )
489  RETURN
490 *
491 * End of CGETSLS
492 *
subroutine cgelq(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
CGELQ
Definition: cgelq.f:172
subroutine cgemlq(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
CGEMLQ
Definition: cgemlq.f:170
subroutine cgemqr(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
CGEMQR
Definition: cgemqr.f:172
subroutine cgeqr(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
CGEQR
Definition: cgeqr.f:174
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function clange(NORM, M, N, A, LDA, WORK)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clange.f:115
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:143
subroutine ctrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
CTRTRS
Definition: ctrtrs.f:140
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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