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

 subroutine dgelsy ( integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer, dimension( * ) JPVT, double precision RCOND, integer RANK, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DGELSY solves overdetermined or underdetermined systems for GE matrices

Purpose:
``` DGELSY computes the minimum-norm solution to a real 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 orthogonal 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**T [ inv(T11)*Q1**T*B ]
[        0         ]
where Q1 consists of the first RANK columns of Q.

This routine is basically identical to the original xGELSX except
three differences:
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas-3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas-3.
o The permutation of matrix B (the right hand side) is faster and
more simple.```
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B. On exit, the N-by-NRHS solution matrix X.``` [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 permuted to the front of AP, otherwise column i is a free column. On exit, if JPVT(i) = k, then the i-th column of AP was the k-th column of A.``` [in] RCOND ``` RCOND is DOUBLE PRECISION 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 DOUBLE PRECISION 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. The unblocked strategy requires that: LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN = min( M, N ). The block algorithm requires that: LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper bound on the blocksize returned by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, and DORMRZ. 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.```
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

Definition at line 202 of file dgelsy.f.

204*
205* -- LAPACK driver routine --
206* -- LAPACK is a software package provided by Univ. of Tennessee, --
207* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208*
209* .. Scalar Arguments ..
210 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
211 DOUBLE PRECISION RCOND
212* ..
213* .. Array Arguments ..
214 INTEGER JPVT( * )
215 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
216* ..
217*
218* =====================================================================
219*
220* .. Parameters ..
221 INTEGER IMAX, IMIN
222 parameter( imax = 1, imin = 2 )
223 DOUBLE PRECISION ZERO, ONE
224 parameter( zero = 0.0d+0, one = 1.0d+0 )
225* ..
226* .. Local Scalars ..
227 LOGICAL LQUERY
228 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
229 \$ LWKOPT, MN, NB, NB1, NB2, NB3, NB4
230 DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
231 \$ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
232* ..
233* .. External Functions ..
234 INTEGER ILAENV
235 DOUBLE PRECISION DLAMCH, DLANGE
236 EXTERNAL ilaenv, dlamch, dlange
237* ..
238* .. External Subroutines ..
239 EXTERNAL dcopy, dgeqp3, dlabad, dlaic1, dlascl, dlaset,
241* ..
242* .. Intrinsic Functions ..
243 INTRINSIC abs, max, min
244* ..
245* .. Executable Statements ..
246*
247 mn = min( m, n )
248 ismin = mn + 1
249 ismax = 2*mn + 1
250*
251* Test the input arguments.
252*
253 info = 0
254 lquery = ( lwork.EQ.-1 )
255 IF( m.LT.0 ) THEN
256 info = -1
257 ELSE IF( n.LT.0 ) THEN
258 info = -2
259 ELSE IF( nrhs.LT.0 ) THEN
260 info = -3
261 ELSE IF( lda.LT.max( 1, m ) ) THEN
262 info = -5
263 ELSE IF( ldb.LT.max( 1, m, n ) ) THEN
264 info = -7
265 END IF
266*
267* Figure out optimal block size
268*
269 IF( info.EQ.0 ) THEN
270 IF( mn.EQ.0 .OR. nrhs.EQ.0 ) THEN
271 lwkmin = 1
272 lwkopt = 1
273 ELSE
274 nb1 = ilaenv( 1, 'DGEQRF', ' ', m, n, -1, -1 )
275 nb2 = ilaenv( 1, 'DGERQF', ' ', m, n, -1, -1 )
276 nb3 = ilaenv( 1, 'DORMQR', ' ', m, n, nrhs, -1 )
277 nb4 = ilaenv( 1, 'DORMRQ', ' ', m, n, nrhs, -1 )
278 nb = max( nb1, nb2, nb3, nb4 )
279 lwkmin = mn + max( 2*mn, n + 1, mn + nrhs )
280 lwkopt = max( lwkmin,
281 \$ mn + 2*n + nb*( n + 1 ), 2*mn + nb*nrhs )
282 END IF
283 work( 1 ) = lwkopt
284*
285 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
286 info = -12
287 END IF
288 END IF
289*
290 IF( info.NE.0 ) THEN
291 CALL xerbla( 'DGELSY', -info )
292 RETURN
293 ELSE IF( lquery ) THEN
294 RETURN
295 END IF
296*
297* Quick return if possible
298*
299 IF( mn.EQ.0 .OR. nrhs.EQ.0 ) THEN
300 rank = 0
301 RETURN
302 END IF
303*
304* Get machine parameters
305*
306 smlnum = dlamch( 'S' ) / dlamch( 'P' )
307 bignum = one / smlnum
308 CALL dlabad( smlnum, bignum )
309*
310* Scale A, B if max entries outside range [SMLNUM,BIGNUM]
311*
312 anrm = dlange( 'M', m, n, a, lda, work )
313 iascl = 0
314 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
315*
316* Scale matrix norm up to SMLNUM
317*
318 CALL dlascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
319 iascl = 1
320 ELSE IF( anrm.GT.bignum ) THEN
321*
322* Scale matrix norm down to BIGNUM
323*
324 CALL dlascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
325 iascl = 2
326 ELSE IF( anrm.EQ.zero ) THEN
327*
328* Matrix all zero. Return zero solution.
329*
330 CALL dlaset( 'F', max( m, n ), nrhs, zero, zero, b, ldb )
331 rank = 0
332 GO TO 70
333 END IF
334*
335 bnrm = dlange( 'M', m, nrhs, b, ldb, work )
336 ibscl = 0
337 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
338*
339* Scale matrix norm up to SMLNUM
340*
341 CALL dlascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
342 ibscl = 1
343 ELSE IF( bnrm.GT.bignum ) THEN
344*
345* Scale matrix norm down to BIGNUM
346*
347 CALL dlascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
348 ibscl = 2
349 END IF
350*
351* Compute QR factorization with column pivoting of A:
352* A * P = Q * R
353*
354 CALL dgeqp3( m, n, a, lda, jpvt, work( 1 ), work( mn+1 ),
355 \$ lwork-mn, info )
356 wsize = mn + work( mn+1 )
357*
358* workspace: MN+2*N+NB*(N+1).
359* Details of Householder rotations stored in WORK(1:MN).
360*
361* Determine RANK using incremental condition estimation
362*
363 work( ismin ) = one
364 work( ismax ) = one
365 smax = abs( a( 1, 1 ) )
366 smin = smax
367 IF( abs( a( 1, 1 ) ).EQ.zero ) THEN
368 rank = 0
369 CALL dlaset( 'F', max( m, n ), nrhs, zero, zero, b, ldb )
370 GO TO 70
371 ELSE
372 rank = 1
373 END IF
374*
375 10 CONTINUE
376 IF( rank.LT.mn ) THEN
377 i = rank + 1
378 CALL dlaic1( imin, rank, work( ismin ), smin, a( 1, i ),
379 \$ a( i, i ), sminpr, s1, c1 )
380 CALL dlaic1( imax, rank, work( ismax ), smax, a( 1, i ),
381 \$ a( i, i ), smaxpr, s2, c2 )
382*
383 IF( smaxpr*rcond.LE.sminpr ) THEN
384 DO 20 i = 1, rank
385 work( ismin+i-1 ) = s1*work( ismin+i-1 )
386 work( ismax+i-1 ) = s2*work( ismax+i-1 )
387 20 CONTINUE
388 work( ismin+rank ) = c1
389 work( ismax+rank ) = c2
390 smin = sminpr
391 smax = smaxpr
392 rank = rank + 1
393 GO TO 10
394 END IF
395 END IF
396*
397* workspace: 3*MN.
398*
399* Logically partition R = [ R11 R12 ]
400* [ 0 R22 ]
401* where R11 = R(1:RANK,1:RANK)
402*
403* [R11,R12] = [ T11, 0 ] * Y
404*
405 IF( rank.LT.n )
406 \$ CALL dtzrzf( rank, n, a, lda, work( mn+1 ), work( 2*mn+1 ),
407 \$ lwork-2*mn, info )
408*
409* workspace: 2*MN.
410* Details of Householder rotations stored in WORK(MN+1:2*MN)
411*
412* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
413*
414 CALL dormqr( 'Left', 'Transpose', m, nrhs, mn, a, lda, work( 1 ),
415 \$ b, ldb, work( 2*mn+1 ), lwork-2*mn, info )
416 wsize = max( wsize, 2*mn+work( 2*mn+1 ) )
417*
418* workspace: 2*MN+NB*NRHS.
419*
420* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
421*
422 CALL dtrsm( 'Left', 'Upper', 'No transpose', 'Non-unit', rank,
423 \$ nrhs, one, a, lda, b, ldb )
424*
425 DO 40 j = 1, nrhs
426 DO 30 i = rank + 1, n
427 b( i, j ) = zero
428 30 CONTINUE
429 40 CONTINUE
430*
431* B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
432*
433 IF( rank.LT.n ) THEN
434 CALL dormrz( 'Left', 'Transpose', n, nrhs, rank, n-rank, a,
435 \$ lda, work( mn+1 ), b, ldb, work( 2*mn+1 ),
436 \$ lwork-2*mn, info )
437 END IF
438*
439* workspace: 2*MN+NRHS.
440*
441* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
442*
443 DO 60 j = 1, nrhs
444 DO 50 i = 1, n
445 work( jpvt( i ) ) = b( i, j )
446 50 CONTINUE
447 CALL dcopy( n, work( 1 ), 1, b( 1, j ), 1 )
448 60 CONTINUE
449*
450* workspace: N.
451*
452* Undo scaling
453*
454 IF( iascl.EQ.1 ) THEN
455 CALL dlascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
456 CALL dlascl( 'U', 0, 0, smlnum, anrm, rank, rank, a, lda,
457 \$ info )
458 ELSE IF( iascl.EQ.2 ) THEN
459 CALL dlascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
460 CALL dlascl( 'U', 0, 0, bignum, anrm, rank, rank, a, lda,
461 \$ info )
462 END IF
463 IF( ibscl.EQ.1 ) THEN
464 CALL dlascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
465 ELSE IF( ibscl.EQ.2 ) THEN
466 CALL dlascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
467 END IF
468*
469 70 CONTINUE
470 work( 1 ) = lwkopt
471*
472 RETURN
473*
474* End of DGELSY
475*
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
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dtrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRSM
Definition: dtrsm.f:181
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 dgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
DGEQP3
Definition: dgeqp3.f:151
subroutine dlaic1(JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
DLAIC1 applies one step of incremental condition estimation.
Definition: dlaic1.f:134
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:167
subroutine dtzrzf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DTZRZF
Definition: dtzrzf.f:151
subroutine dormrz(SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMRZ
Definition: dormrz.f:187
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