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

 subroutine cgelsx ( integer M, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer, dimension( * ) JPVT, real RCOND, integer RANK, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO )

CGELSX solves overdetermined or underdetermined systems for GE matrices

Purpose:
``` This routine is deprecated and has been replaced by routine CGELSY.

CGELSX computes the minimum-norm solution to a complex linear least
squares problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A.  A is an M-by-N
matrix which may be rank-deficient.

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.

The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[  0  R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND.  The order of R11, RANK,
is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated
by unitary transformations from the right, arriving at the
complete orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[  0  0 ]
The minimum-norm solution is then
X = P * Z**H [ inv(T11)*Q1**H*B ]
[        0         ]
where Q1 consists of the first RANK columns of 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 matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of 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 has been overwritten by details of its complete orthogonal factorization.``` [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 M-by-NRHS right hand side matrix B. On exit, the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements N+1:M in that column.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M,N).``` [in,out] JPVT ``` JPVT is INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial column, otherwise it is a free column. Before the QR factorization of A, all initial columns are permuted to the leading positions; only the remaining free columns are moved as a result of column pivoting during the factorization. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A.``` [in] RCOND ``` RCOND is REAL RCOND is used to determine the effective rank of A, which is defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number < 1/RCOND.``` [out] RANK ``` RANK is INTEGER The effective rank of A, i.e., the order of the submatrix R11. This is the same as the order of the submatrix T11 in the complete orthogonal factorization of A.``` [out] WORK ``` WORK is COMPLEX array, dimension (min(M,N) + max( N, 2*min(M,N)+NRHS )),``` [out] RWORK ` RWORK is REAL array, dimension (2*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```

Definition at line 182 of file cgelsx.f.

184*
185* -- LAPACK driver routine --
186* -- LAPACK is a software package provided by Univ. of Tennessee, --
187* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
188*
189* .. Scalar Arguments ..
190 INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
191 REAL RCOND
192* ..
193* .. Array Arguments ..
194 INTEGER JPVT( * )
195 REAL RWORK( * )
196 COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
197* ..
198*
199* =====================================================================
200*
201* .. Parameters ..
202 INTEGER IMAX, IMIN
203 parameter( imax = 1, imin = 2 )
204 REAL ZERO, ONE, DONE, NTDONE
205 parameter( zero = 0.0e+0, one = 1.0e+0, done = zero,
206 \$ ntdone = one )
207 COMPLEX CZERO, CONE
208 parameter( czero = ( 0.0e+0, 0.0e+0 ),
209 \$ cone = ( 1.0e+0, 0.0e+0 ) )
210* ..
211* .. Local Scalars ..
212 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
213 REAL ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
214 \$ SMLNUM
215 COMPLEX C1, C2, S1, S2, T1, T2
216* ..
217* .. External Subroutines ..
218 EXTERNAL cgeqpf, claic1, clascl, claset, clatzm, ctrsm,
220* ..
221* .. External Functions ..
222 REAL CLANGE, SLAMCH
223 EXTERNAL clange, slamch
224* ..
225* .. Intrinsic Functions ..
226 INTRINSIC abs, conjg, max, min
227* ..
228* .. Executable Statements ..
229*
230 mn = min( m, n )
231 ismin = mn + 1
232 ismax = 2*mn + 1
233*
234* Test the input arguments.
235*
236 info = 0
237 IF( m.LT.0 ) THEN
238 info = -1
239 ELSE IF( n.LT.0 ) THEN
240 info = -2
241 ELSE IF( nrhs.LT.0 ) THEN
242 info = -3
243 ELSE IF( lda.LT.max( 1, m ) ) THEN
244 info = -5
245 ELSE IF( ldb.LT.max( 1, m, n ) ) THEN
246 info = -7
247 END IF
248*
249 IF( info.NE.0 ) THEN
250 CALL xerbla( 'CGELSX', -info )
251 RETURN
252 END IF
253*
254* Quick return if possible
255*
256 IF( min( m, n, nrhs ).EQ.0 ) THEN
257 rank = 0
258 RETURN
259 END IF
260*
261* Get machine parameters
262*
263 smlnum = slamch( 'S' ) / slamch( 'P' )
264 bignum = one / smlnum
265 CALL slabad( smlnum, bignum )
266*
267* Scale A, B if max elements outside range [SMLNUM,BIGNUM]
268*
269 anrm = clange( 'M', m, n, a, lda, rwork )
270 iascl = 0
271 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
272*
273* Scale matrix norm up to SMLNUM
274*
275 CALL clascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
276 iascl = 1
277 ELSE IF( anrm.GT.bignum ) THEN
278*
279* Scale matrix norm down to BIGNUM
280*
281 CALL clascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
282 iascl = 2
283 ELSE IF( anrm.EQ.zero ) THEN
284*
285* Matrix all zero. Return zero solution.
286*
287 CALL claset( 'F', max( m, n ), nrhs, czero, czero, b, ldb )
288 rank = 0
289 GO TO 100
290 END IF
291*
292 bnrm = clange( 'M', m, nrhs, b, ldb, rwork )
293 ibscl = 0
294 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
295*
296* Scale matrix norm up to SMLNUM
297*
298 CALL clascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
299 ibscl = 1
300 ELSE IF( bnrm.GT.bignum ) THEN
301*
302* Scale matrix norm down to BIGNUM
303*
304 CALL clascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
305 ibscl = 2
306 END IF
307*
308* Compute QR factorization with column pivoting of A:
309* A * P = Q * R
310*
311 CALL cgeqpf( m, n, a, lda, jpvt, work( 1 ), work( mn+1 ), rwork,
312 \$ info )
313*
314* complex workspace MN+N. Real workspace 2*N. Details of Householder
315* rotations stored in WORK(1:MN).
316*
317* Determine RANK using incremental condition estimation
318*
319 work( ismin ) = cone
320 work( ismax ) = cone
321 smax = abs( a( 1, 1 ) )
322 smin = smax
323 IF( abs( a( 1, 1 ) ).EQ.zero ) THEN
324 rank = 0
325 CALL claset( 'F', max( m, n ), nrhs, czero, czero, b, ldb )
326 GO TO 100
327 ELSE
328 rank = 1
329 END IF
330*
331 10 CONTINUE
332 IF( rank.LT.mn ) THEN
333 i = rank + 1
334 CALL claic1( imin, rank, work( ismin ), smin, a( 1, i ),
335 \$ a( i, i ), sminpr, s1, c1 )
336 CALL claic1( imax, rank, work( ismax ), smax, a( 1, i ),
337 \$ a( i, i ), smaxpr, s2, c2 )
338*
339 IF( smaxpr*rcond.LE.sminpr ) THEN
340 DO 20 i = 1, rank
341 work( ismin+i-1 ) = s1*work( ismin+i-1 )
342 work( ismax+i-1 ) = s2*work( ismax+i-1 )
343 20 CONTINUE
344 work( ismin+rank ) = c1
345 work( ismax+rank ) = c2
346 smin = sminpr
347 smax = smaxpr
348 rank = rank + 1
349 GO TO 10
350 END IF
351 END IF
352*
353* Logically partition R = [ R11 R12 ]
354* [ 0 R22 ]
355* where R11 = R(1:RANK,1:RANK)
356*
357* [R11,R12] = [ T11, 0 ] * Y
358*
359 IF( rank.LT.n )
360 \$ CALL ctzrqf( rank, n, a, lda, work( mn+1 ), info )
361*
362* Details of Householder rotations stored in WORK(MN+1:2*MN)
363*
364* B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
365*
366 CALL cunm2r( 'Left', 'Conjugate transpose', m, nrhs, mn, a, lda,
367 \$ work( 1 ), b, ldb, work( 2*mn+1 ), info )
368*
369* workspace NRHS
370*
371* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
372*
373 CALL ctrsm( 'Left', 'Upper', 'No transpose', 'Non-unit', rank,
374 \$ nrhs, cone, a, lda, b, ldb )
375*
376 DO 40 i = rank + 1, n
377 DO 30 j = 1, nrhs
378 b( i, j ) = czero
379 30 CONTINUE
380 40 CONTINUE
381*
382* B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
383*
384 IF( rank.LT.n ) THEN
385 DO 50 i = 1, rank
386 CALL clatzm( 'Left', n-rank+1, nrhs, a( i, rank+1 ), lda,
387 \$ conjg( work( mn+i ) ), b( i, 1 ),
388 \$ b( rank+1, 1 ), ldb, work( 2*mn+1 ) )
389 50 CONTINUE
390 END IF
391*
392* workspace NRHS
393*
394* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
395*
396 DO 90 j = 1, nrhs
397 DO 60 i = 1, n
398 work( 2*mn+i ) = ntdone
399 60 CONTINUE
400 DO 80 i = 1, n
401 IF( work( 2*mn+i ).EQ.ntdone ) THEN
402 IF( jpvt( i ).NE.i ) THEN
403 k = i
404 t1 = b( k, j )
405 t2 = b( jpvt( k ), j )
406 70 CONTINUE
407 b( jpvt( k ), j ) = t1
408 work( 2*mn+k ) = done
409 t1 = t2
410 k = jpvt( k )
411 t2 = b( jpvt( k ), j )
412 IF( jpvt( k ).NE.i )
413 \$ GO TO 70
414 b( i, j ) = t1
415 work( 2*mn+k ) = done
416 END IF
417 END IF
418 80 CONTINUE
419 90 CONTINUE
420*
421* Undo scaling
422*
423 IF( iascl.EQ.1 ) THEN
424 CALL clascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
425 CALL clascl( 'U', 0, 0, smlnum, anrm, rank, rank, a, lda,
426 \$ info )
427 ELSE IF( iascl.EQ.2 ) THEN
428 CALL clascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
429 CALL clascl( 'U', 0, 0, bignum, anrm, rank, rank, a, lda,
430 \$ info )
431 END IF
432 IF( ibscl.EQ.1 ) THEN
433 CALL clascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
434 ELSE IF( ibscl.EQ.2 ) THEN
435 CALL clascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
436 END IF
437*
438 100 CONTINUE
439*
440 RETURN
441*
442* End of CGELSX
443*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:180
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 cgeqpf(M, N, A, LDA, JPVT, TAU, WORK, RWORK, INFO)
CGEQPF
Definition: cgeqpf.f:148
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 claic1(JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
CLAIC1 applies one step of incremental condition estimation.
Definition: claic1.f:135
subroutine ctzrqf(M, N, A, LDA, TAU, INFO)
CTZRQF
Definition: ctzrqf.f:138
subroutine clatzm(SIDE, M, N, V, INCV, TAU, C1, C2, LDC, WORK)
CLATZM
Definition: clatzm.f:152
subroutine cunm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
CUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf...
Definition: cunm2r.f:159
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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