LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ zgetsls()

subroutine zgetsls ( character  TRANS,
integer  M,
integer  N,
integer  NRHS,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( * )  WORK,
integer  LWORK,
integer  INFO 
)

ZGETSLS

Purpose:
 ZGETSLS 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*16 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 ZGEQR or ZGELQ.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
[in,out]B
          B is COMPLEX*16 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*16 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 160 of file zgetsls.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*16 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 COMPLEX*16 CZERO
182 parameter( czero = ( 0.0d+0, 0.0d+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 DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM, DUM( 1 )
190 COMPLEX*16 TQ( 5 ), WORKQ( 1 )
191* ..
192* .. External Functions ..
193 LOGICAL LSAME
194 DOUBLE PRECISION DLAMCH, ZLANGE
195 EXTERNAL lsame, dlabad, dlamch, zlange
196* ..
197* .. External Subroutines ..
198 EXTERNAL zgeqr, zgemqr, zlascl, zlaset,
200* ..
201* .. Intrinsic Functions ..
202 INTRINSIC dble, 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 zgeqr( m, n, a, lda, tq, -1, workq, -1, info2 )
234 tszo = int( tq( 1 ) )
235 lwo = int( workq( 1 ) )
236 CALL zgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
237 $ tszo, b, ldb, workq, -1, info2 )
238 lwo = max( lwo, int( workq( 1 ) ) )
239 CALL zgeqr( m, n, a, lda, tq, -2, workq, -2, info2 )
240 tszm = int( tq( 1 ) )
241 lwm = int( workq( 1 ) )
242 CALL zgemqr( '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 zgelq( m, n, a, lda, tq, -1, workq, -1, info2 )
249 tszo = int( tq( 1 ) )
250 lwo = int( workq( 1 ) )
251 CALL zgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
252 $ tszo, b, ldb, workq, -1, info2 )
253 lwo = max( lwo, int( workq( 1 ) ) )
254 CALL zgelq( m, n, a, lda, tq, -2, workq, -2, info2 )
255 tszm = int( tq( 1 ) )
256 lwm = int( workq( 1 ) )
257 CALL zgemlq( '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 ) = dble( wsizeo )
269*
270 END IF
271*
272 IF( info.NE.0 ) THEN
273 CALL xerbla( 'ZGETSLS', -info )
274 RETURN
275 END IF
276 IF( lquery ) THEN
277 IF( lwork.EQ.-2 ) work( 1 ) = dble( 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 zlaset( 'FULL', max( m, n ), nrhs, czero, czero,
292 $ b, ldb )
293 RETURN
294 END IF
295*
296* Get machine parameters
297*
298 smlnum = dlamch( 'S' ) / dlamch( 'P' )
299 bignum = one / smlnum
300 CALL dlabad( smlnum, bignum )
301*
302* Scale A, B if max element outside range [SMLNUM,BIGNUM]
303*
304 anrm = zlange( '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 zlascl( '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 zlascl( '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 zlaset( '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 = zlange( '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 zlascl( '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 zlascl( '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 zgeqr( 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 zgemqr( '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 ztrtrs( '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 ztrtrs( '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 zgemqr( '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 zgelq( 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 ztrtrs( '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 zgemlq( '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 zgemlq( '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 ztrtrs( '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 zlascl( 'G', 0, 0, anrm, smlnum, scllen, nrhs, b, ldb,
474 $ info )
475 ELSE IF( iascl.EQ.2 ) THEN
476 CALL zlascl( 'G', 0, 0, anrm, bignum, scllen, nrhs, b, ldb,
477 $ info )
478 END IF
479 IF( ibscl.EQ.1 ) THEN
480 CALL zlascl( 'G', 0, 0, smlnum, bnrm, scllen, nrhs, b, ldb,
481 $ info )
482 ELSE IF( ibscl.EQ.2 ) THEN
483 CALL zlascl( 'G', 0, 0, bignum, bnrm, scllen, nrhs, b, ldb,
484 $ info )
485 END IF
486*
487 50 CONTINUE
488 work( 1 ) = dble( tszo + lwo )
489 RETURN
490*
491* End of ZGETSLS
492*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine ztrtrs(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
ZTRTRS
Definition: ztrtrs.f:140
subroutine zgelq(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
ZGELQ
Definition: zgelq.f:172
subroutine zgemlq(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
ZGEMLQ
Definition: zgemlq.f:169
subroutine zgemqr(SIDE, TRANS, M, N, K, A, LDA, T, TSIZE, C, LDC, WORK, LWORK, INFO)
ZGEMQR
Definition: zgemqr.f:172
subroutine zgeqr(M, N, A, LDA, T, TSIZE, WORK, LWORK, INFO)
ZGEQR
Definition: zgeqr.f:174
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