1: *> \brief <b> DGELS solves overdetermined or underdetermined systems for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGELS + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgels.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgels.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER TRANS
26: * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DGELS solves overdetermined or underdetermined real linear systems
39: *> involving an M-by-N matrix A, or its transpose, using a QR or LQ
40: *> factorization of A. It is assumed that A has full rank.
41: *>
42: *> The following options are provided:
43: *>
44: *> 1. If TRANS = 'N' and m >= n: find the least squares solution of
45: *> an overdetermined system, i.e., solve the least squares problem
46: *> minimize || B - A*X ||.
47: *>
48: *> 2. If TRANS = 'N' and m < n: find the minimum norm solution of
49: *> an underdetermined system A * X = B.
50: *>
51: *> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
52: *> an underdetermined system A**T * X = B.
53: *>
54: *> 4. If TRANS = 'T' and m < n: find the least squares solution of
55: *> an overdetermined system, i.e., solve the least squares problem
56: *> minimize || B - A**T * X ||.
57: *>
58: *> Several right hand side vectors b and solution vectors x can be
59: *> handled in a single call; they are stored as the columns of the
60: *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
61: *> matrix X.
62: *> \endverbatim
63: *
64: * Arguments:
65: * ==========
66: *
67: *> \param[in] TRANS
68: *> \verbatim
69: *> TRANS is CHARACTER*1
70: *> = 'N': the linear system involves A;
71: *> = 'T': the linear system involves A**T.
72: *> \endverbatim
73: *>
74: *> \param[in] M
75: *> \verbatim
76: *> M is INTEGER
77: *> The number of rows of the matrix A. M >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in] N
81: *> \verbatim
82: *> N is INTEGER
83: *> The number of columns of the matrix A. N >= 0.
84: *> \endverbatim
85: *>
86: *> \param[in] NRHS
87: *> \verbatim
88: *> NRHS is INTEGER
89: *> The number of right hand sides, i.e., the number of
90: *> columns of the matrices B and X. NRHS >=0.
91: *> \endverbatim
92: *>
93: *> \param[in,out] A
94: *> \verbatim
95: *> A is DOUBLE PRECISION array, dimension (LDA,N)
96: *> On entry, the M-by-N matrix A.
97: *> On exit,
98: *> if M >= N, A is overwritten by details of its QR
99: *> factorization as returned by DGEQRF;
100: *> if M < N, A is overwritten by details of its LQ
101: *> factorization as returned by DGELQF.
102: *> \endverbatim
103: *>
104: *> \param[in] LDA
105: *> \verbatim
106: *> LDA is INTEGER
107: *> The leading dimension of the array A. LDA >= max(1,M).
108: *> \endverbatim
109: *>
110: *> \param[in,out] B
111: *> \verbatim
112: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
113: *> On entry, the matrix B of right hand side vectors, stored
114: *> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
115: *> if TRANS = 'T'.
116: *> On exit, if INFO = 0, B is overwritten by the solution
117: *> vectors, stored columnwise:
118: *> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
119: *> squares solution vectors; the residual sum of squares for the
120: *> solution in each column is given by the sum of squares of
121: *> elements N+1 to M in that column;
122: *> if TRANS = 'N' and m < n, rows 1 to N of B contain the
123: *> minimum norm solution vectors;
124: *> if TRANS = 'T' and m >= n, rows 1 to M of B contain the
125: *> minimum norm solution vectors;
126: *> if TRANS = 'T' and m < n, rows 1 to M of B contain the
127: *> least squares solution vectors; the residual sum of squares
128: *> for the solution in each column is given by the sum of
129: *> squares of elements M+1 to N in that column.
130: *> \endverbatim
131: *>
132: *> \param[in] LDB
133: *> \verbatim
134: *> LDB is INTEGER
135: *> The leading dimension of the array B. LDB >= MAX(1,M,N).
136: *> \endverbatim
137: *>
138: *> \param[out] WORK
139: *> \verbatim
140: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
141: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
142: *> \endverbatim
143: *>
144: *> \param[in] LWORK
145: *> \verbatim
146: *> LWORK is INTEGER
147: *> The dimension of the array WORK.
148: *> LWORK >= max( 1, MN + max( MN, NRHS ) ).
149: *> For optimal performance,
150: *> LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
151: *> where MN = min(M,N) and NB is the optimum block size.
152: *>
153: *> If LWORK = -1, then a workspace query is assumed; the routine
154: *> only calculates the optimal size of the WORK array, returns
155: *> this value as the first entry of the WORK array, and no error
156: *> message related to LWORK is issued by XERBLA.
157: *> \endverbatim
158: *>
159: *> \param[out] INFO
160: *> \verbatim
161: *> INFO is INTEGER
162: *> = 0: successful exit
163: *> < 0: if INFO = -i, the i-th argument had an illegal value
164: *> > 0: if INFO = i, the i-th diagonal element of the
165: *> triangular factor of A is zero, so that A does not have
166: *> full rank; the least squares solution could not be
167: *> computed.
168: *> \endverbatim
169: *
170: * Authors:
171: * ========
172: *
173: *> \author Univ. of Tennessee
174: *> \author Univ. of California Berkeley
175: *> \author Univ. of Colorado Denver
176: *> \author NAG Ltd.
177: *
178: *> \date December 2016
179: *
180: *> \ingroup doubleGEsolve
181: *
182: * =====================================================================
183: SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
184: $ INFO )
185: *
186: * -- LAPACK driver routine (version 3.7.0) --
187: * -- LAPACK is a software package provided by Univ. of Tennessee, --
188: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189: * December 2016
190: *
191: * .. Scalar Arguments ..
192: CHARACTER TRANS
193: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
194: * ..
195: * .. Array Arguments ..
196: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: DOUBLE PRECISION ZERO, ONE
203: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
204: * ..
205: * .. Local Scalars ..
206: LOGICAL LQUERY, TPSD
207: INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
208: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM
209: * ..
210: * .. Local Arrays ..
211: DOUBLE PRECISION RWORK( 1 )
212: * ..
213: * .. External Functions ..
214: LOGICAL LSAME
215: INTEGER ILAENV
216: DOUBLE PRECISION DLAMCH, DLANGE
217: EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE
218: * ..
219: * .. External Subroutines ..
220: EXTERNAL DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ, DORMQR,
221: $ DTRTRS, XERBLA
222: * ..
223: * .. Intrinsic Functions ..
224: INTRINSIC DBLE, MAX, MIN
225: * ..
226: * .. Executable Statements ..
227: *
228: * Test the input arguments.
229: *
230: INFO = 0
231: MN = MIN( M, N )
232: LQUERY = ( LWORK.EQ.-1 )
233: IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN
234: INFO = -1
235: ELSE IF( M.LT.0 ) THEN
236: INFO = -2
237: ELSE IF( N.LT.0 ) THEN
238: INFO = -3
239: ELSE IF( NRHS.LT.0 ) THEN
240: INFO = -4
241: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
242: INFO = -6
243: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
244: INFO = -8
245: ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
246: $ THEN
247: INFO = -10
248: END IF
249: *
250: * Figure out optimal block size
251: *
252: IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
253: *
254: TPSD = .TRUE.
255: IF( LSAME( TRANS, 'N' ) )
256: $ TPSD = .FALSE.
257: *
258: IF( M.GE.N ) THEN
259: NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
260: IF( TPSD ) THEN
261: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N,
262: $ -1 ) )
263: ELSE
264: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N,
265: $ -1 ) )
266: END IF
267: ELSE
268: NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
269: IF( TPSD ) THEN
270: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M,
271: $ -1 ) )
272: ELSE
273: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M,
274: $ -1 ) )
275: END IF
276: END IF
277: *
278: WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB )
279: WORK( 1 ) = DBLE( WSIZE )
280: *
281: END IF
282: *
283: IF( INFO.NE.0 ) THEN
284: CALL XERBLA( 'DGELS ', -INFO )
285: RETURN
286: ELSE IF( LQUERY ) THEN
287: RETURN
288: END IF
289: *
290: * Quick return if possible
291: *
292: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
293: CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
294: RETURN
295: END IF
296: *
297: * Get machine parameters
298: *
299: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
300: BIGNUM = ONE / SMLNUM
301: CALL DLABAD( SMLNUM, BIGNUM )
302: *
303: * Scale A, B if max element outside range [SMLNUM,BIGNUM]
304: *
305: ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
306: IASCL = 0
307: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
308: *
309: * Scale matrix norm up to SMLNUM
310: *
311: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
312: IASCL = 1
313: ELSE IF( ANRM.GT.BIGNUM ) THEN
314: *
315: * Scale matrix norm down to BIGNUM
316: *
317: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
318: IASCL = 2
319: ELSE IF( ANRM.EQ.ZERO ) THEN
320: *
321: * Matrix all zero. Return zero solution.
322: *
323: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
324: GO TO 50
325: END IF
326: *
327: BROW = M
328: IF( TPSD )
329: $ BROW = N
330: BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
331: IBSCL = 0
332: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
333: *
334: * Scale matrix norm up to SMLNUM
335: *
336: CALL DLASCL( '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 DLASCL( '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 DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
353: $ INFO )
354: *
355: * workspace at least N, optimally N*NB
356: *
357: IF( .NOT.TPSD ) THEN
358: *
359: * Least-Squares Problem min || A * X - B ||
360: *
361: * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
362: *
363: CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
364: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
365: $ INFO )
366: *
367: * workspace at least NRHS, optimally NRHS*NB
368: *
369: * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
370: *
371: CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
372: $ A, LDA, B, LDB, INFO )
373: *
374: IF( INFO.GT.0 ) THEN
375: RETURN
376: END IF
377: *
378: SCLLEN = N
379: *
380: ELSE
381: *
382: * Underdetermined system of equations A**T * X = B
383: *
384: * B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
385: *
386: CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
387: $ A, LDA, B, LDB, INFO )
388: *
389: IF( INFO.GT.0 ) THEN
390: RETURN
391: END IF
392: *
393: * B(N+1:M,1:NRHS) = ZERO
394: *
395: DO 20 J = 1, NRHS
396: DO 10 I = N + 1, M
397: B( I, J ) = ZERO
398: 10 CONTINUE
399: 20 CONTINUE
400: *
401: * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
402: *
403: CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
404: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
405: $ INFO )
406: *
407: * workspace at least NRHS, optimally NRHS*NB
408: *
409: SCLLEN = M
410: *
411: END IF
412: *
413: ELSE
414: *
415: * Compute LQ factorization of A
416: *
417: CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
418: $ INFO )
419: *
420: * workspace at least M, optimally M*NB.
421: *
422: IF( .NOT.TPSD ) THEN
423: *
424: * underdetermined system of equations A * X = B
425: *
426: * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
427: *
428: CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
429: $ A, LDA, B, LDB, INFO )
430: *
431: IF( INFO.GT.0 ) THEN
432: RETURN
433: END IF
434: *
435: * B(M+1:N,1:NRHS) = 0
436: *
437: DO 40 J = 1, NRHS
438: DO 30 I = M + 1, N
439: B( I, J ) = ZERO
440: 30 CONTINUE
441: 40 CONTINUE
442: *
443: * B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
444: *
445: CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
446: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
447: $ INFO )
448: *
449: * workspace at least NRHS, optimally NRHS*NB
450: *
451: SCLLEN = N
452: *
453: ELSE
454: *
455: * overdetermined system min || A**T * X - B ||
456: *
457: * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
458: *
459: CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
460: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
461: $ INFO )
462: *
463: * workspace at least NRHS, optimally NRHS*NB
464: *
465: * B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
466: *
467: CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
468: $ A, LDA, B, LDB, INFO )
469: *
470: IF( INFO.GT.0 ) THEN
471: RETURN
472: END IF
473: *
474: SCLLEN = M
475: *
476: END IF
477: *
478: END IF
479: *
480: * Undo scaling
481: *
482: IF( IASCL.EQ.1 ) THEN
483: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
484: $ INFO )
485: ELSE IF( IASCL.EQ.2 ) THEN
486: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
487: $ INFO )
488: END IF
489: IF( IBSCL.EQ.1 ) THEN
490: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
491: $ INFO )
492: ELSE IF( IBSCL.EQ.2 ) THEN
493: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
494: $ INFO )
495: END IF
496: *
497: 50 CONTINUE
498: WORK( 1 ) = DBLE( WSIZE )
499: *
500: RETURN
501: *
502: * End of DGELS
503: *
504: END
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