1: SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
2: $ INFO )
3: *
4: * -- LAPACK driver routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: CHARACTER TRANS
11: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DGELS solves overdetermined or underdetermined real linear systems
21: * involving an M-by-N matrix A, or its transpose, using a QR or LQ
22: * factorization of A. It is assumed that A has full rank.
23: *
24: * The following options are provided:
25: *
26: * 1. If TRANS = 'N' and m >= n: find the least squares solution of
27: * an overdetermined system, i.e., solve the least squares problem
28: * minimize || B - A*X ||.
29: *
30: * 2. If TRANS = 'N' and m < n: find the minimum norm solution of
31: * an underdetermined system A * X = B.
32: *
33: * 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
34: * an undetermined system A**T * X = B.
35: *
36: * 4. If TRANS = 'T' and m < n: find the least squares solution of
37: * an overdetermined system, i.e., solve the least squares problem
38: * minimize || B - A**T * X ||.
39: *
40: * Several right hand side vectors b and solution vectors x can be
41: * handled in a single call; they are stored as the columns of the
42: * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
43: * matrix X.
44: *
45: * Arguments
46: * =========
47: *
48: * TRANS (input) CHARACTER*1
49: * = 'N': the linear system involves A;
50: * = 'T': the linear system involves A**T.
51: *
52: * M (input) INTEGER
53: * The number of rows of the matrix A. M >= 0.
54: *
55: * N (input) INTEGER
56: * The number of columns of the matrix A. N >= 0.
57: *
58: * NRHS (input) INTEGER
59: * The number of right hand sides, i.e., the number of
60: * columns of the matrices B and X. NRHS >=0.
61: *
62: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
63: * On entry, the M-by-N matrix A.
64: * On exit,
65: * if M >= N, A is overwritten by details of its QR
66: * factorization as returned by DGEQRF;
67: * if M < N, A is overwritten by details of its LQ
68: * factorization as returned by DGELQF.
69: *
70: * LDA (input) INTEGER
71: * The leading dimension of the array A. LDA >= max(1,M).
72: *
73: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
74: * On entry, the matrix B of right hand side vectors, stored
75: * columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
76: * if TRANS = 'T'.
77: * On exit, if INFO = 0, B is overwritten by the solution
78: * vectors, stored columnwise:
79: * if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
80: * squares solution vectors; the residual sum of squares for the
81: * solution in each column is given by the sum of squares of
82: * elements N+1 to M in that column;
83: * if TRANS = 'N' and m < n, rows 1 to N of B contain the
84: * minimum norm solution vectors;
85: * if TRANS = 'T' and m >= n, rows 1 to M of B contain the
86: * minimum norm solution vectors;
87: * if TRANS = 'T' and m < n, rows 1 to M of B contain the
88: * least squares solution vectors; the residual sum of squares
89: * for the solution in each column is given by the sum of
90: * squares of elements M+1 to N in that column.
91: *
92: * LDB (input) INTEGER
93: * The leading dimension of the array B. LDB >= MAX(1,M,N).
94: *
95: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
96: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
97: *
98: * LWORK (input) INTEGER
99: * The dimension of the array WORK.
100: * LWORK >= max( 1, MN + max( MN, NRHS ) ).
101: * For optimal performance,
102: * LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
103: * where MN = min(M,N) and NB is the optimum block size.
104: *
105: * If LWORK = -1, then a workspace query is assumed; the routine
106: * only calculates the optimal size of the WORK array, returns
107: * this value as the first entry of the WORK array, and no error
108: * message related to LWORK is issued by XERBLA.
109: *
110: * INFO (output) INTEGER
111: * = 0: successful exit
112: * < 0: if INFO = -i, the i-th argument had an illegal value
113: * > 0: if INFO = i, the i-th diagonal element of the
114: * triangular factor of A is zero, so that A does not have
115: * full rank; the least squares solution could not be
116: * computed.
117: *
118: * =====================================================================
119: *
120: * .. Parameters ..
121: DOUBLE PRECISION ZERO, ONE
122: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
123: * ..
124: * .. Local Scalars ..
125: LOGICAL LQUERY, TPSD
126: INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
127: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM
128: * ..
129: * .. Local Arrays ..
130: DOUBLE PRECISION RWORK( 1 )
131: * ..
132: * .. External Functions ..
133: LOGICAL LSAME
134: INTEGER ILAENV
135: DOUBLE PRECISION DLAMCH, DLANGE
136: EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE
137: * ..
138: * .. External Subroutines ..
139: EXTERNAL DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ, DORMQR,
140: $ DTRTRS, XERBLA
141: * ..
142: * .. Intrinsic Functions ..
143: INTRINSIC DBLE, MAX, MIN
144: * ..
145: * .. Executable Statements ..
146: *
147: * Test the input arguments.
148: *
149: INFO = 0
150: MN = MIN( M, N )
151: LQUERY = ( LWORK.EQ.-1 )
152: IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN
153: INFO = -1
154: ELSE IF( M.LT.0 ) THEN
155: INFO = -2
156: ELSE IF( N.LT.0 ) THEN
157: INFO = -3
158: ELSE IF( NRHS.LT.0 ) THEN
159: INFO = -4
160: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
161: INFO = -6
162: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
163: INFO = -8
164: ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
165: $ THEN
166: INFO = -10
167: END IF
168: *
169: * Figure out optimal block size
170: *
171: IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
172: *
173: TPSD = .TRUE.
174: IF( LSAME( TRANS, 'N' ) )
175: $ TPSD = .FALSE.
176: *
177: IF( M.GE.N ) THEN
178: NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
179: IF( TPSD ) THEN
180: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N,
181: $ -1 ) )
182: ELSE
183: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N,
184: $ -1 ) )
185: END IF
186: ELSE
187: NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
188: IF( TPSD ) THEN
189: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M,
190: $ -1 ) )
191: ELSE
192: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M,
193: $ -1 ) )
194: END IF
195: END IF
196: *
197: WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB )
198: WORK( 1 ) = DBLE( WSIZE )
199: *
200: END IF
201: *
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'DGELS ', -INFO )
204: RETURN
205: ELSE IF( LQUERY ) THEN
206: RETURN
207: END IF
208: *
209: * Quick return if possible
210: *
211: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
212: CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
213: RETURN
214: END IF
215: *
216: * Get machine parameters
217: *
218: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
219: BIGNUM = ONE / SMLNUM
220: CALL DLABAD( SMLNUM, BIGNUM )
221: *
222: * Scale A, B if max element outside range [SMLNUM,BIGNUM]
223: *
224: ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
225: IASCL = 0
226: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
227: *
228: * Scale matrix norm up to SMLNUM
229: *
230: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
231: IASCL = 1
232: ELSE IF( ANRM.GT.BIGNUM ) THEN
233: *
234: * Scale matrix norm down to BIGNUM
235: *
236: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
237: IASCL = 2
238: ELSE IF( ANRM.EQ.ZERO ) THEN
239: *
240: * Matrix all zero. Return zero solution.
241: *
242: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
243: GO TO 50
244: END IF
245: *
246: BROW = M
247: IF( TPSD )
248: $ BROW = N
249: BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
250: IBSCL = 0
251: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
252: *
253: * Scale matrix norm up to SMLNUM
254: *
255: CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
256: $ INFO )
257: IBSCL = 1
258: ELSE IF( BNRM.GT.BIGNUM ) THEN
259: *
260: * Scale matrix norm down to BIGNUM
261: *
262: CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
263: $ INFO )
264: IBSCL = 2
265: END IF
266: *
267: IF( M.GE.N ) THEN
268: *
269: * compute QR factorization of A
270: *
271: CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
272: $ INFO )
273: *
274: * workspace at least N, optimally N*NB
275: *
276: IF( .NOT.TPSD ) THEN
277: *
278: * Least-Squares Problem min || A * X - B ||
279: *
280: * B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
281: *
282: CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
283: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
284: $ INFO )
285: *
286: * workspace at least NRHS, optimally NRHS*NB
287: *
288: * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
289: *
290: CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
291: $ A, LDA, B, LDB, INFO )
292: *
293: IF( INFO.GT.0 ) THEN
294: RETURN
295: END IF
296: *
297: SCLLEN = N
298: *
299: ELSE
300: *
301: * Overdetermined system of equations A' * X = B
302: *
303: * B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS)
304: *
305: CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
306: $ A, LDA, B, LDB, INFO )
307: *
308: IF( INFO.GT.0 ) THEN
309: RETURN
310: END IF
311: *
312: * B(N+1:M,1:NRHS) = ZERO
313: *
314: DO 20 J = 1, NRHS
315: DO 10 I = N + 1, M
316: B( I, J ) = ZERO
317: 10 CONTINUE
318: 20 CONTINUE
319: *
320: * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
321: *
322: CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
323: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
324: $ INFO )
325: *
326: * workspace at least NRHS, optimally NRHS*NB
327: *
328: SCLLEN = M
329: *
330: END IF
331: *
332: ELSE
333: *
334: * Compute LQ factorization of A
335: *
336: CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
337: $ INFO )
338: *
339: * workspace at least M, optimally M*NB.
340: *
341: IF( .NOT.TPSD ) THEN
342: *
343: * underdetermined system of equations A * X = B
344: *
345: * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
346: *
347: CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
348: $ A, LDA, B, LDB, INFO )
349: *
350: IF( INFO.GT.0 ) THEN
351: RETURN
352: END IF
353: *
354: * B(M+1:N,1:NRHS) = 0
355: *
356: DO 40 J = 1, NRHS
357: DO 30 I = M + 1, N
358: B( I, J ) = ZERO
359: 30 CONTINUE
360: 40 CONTINUE
361: *
362: * B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS)
363: *
364: CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
365: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
366: $ INFO )
367: *
368: * workspace at least NRHS, optimally NRHS*NB
369: *
370: SCLLEN = N
371: *
372: ELSE
373: *
374: * overdetermined system min || A' * X - B ||
375: *
376: * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
377: *
378: CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
379: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
380: $ INFO )
381: *
382: * workspace at least NRHS, optimally NRHS*NB
383: *
384: * B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS)
385: *
386: CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
387: $ A, LDA, B, LDB, INFO )
388: *
389: IF( INFO.GT.0 ) THEN
390: RETURN
391: END IF
392: *
393: SCLLEN = M
394: *
395: END IF
396: *
397: END IF
398: *
399: * Undo scaling
400: *
401: IF( IASCL.EQ.1 ) THEN
402: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
403: $ INFO )
404: ELSE IF( IASCL.EQ.2 ) THEN
405: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
406: $ INFO )
407: END IF
408: IF( IBSCL.EQ.1 ) THEN
409: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
410: $ INFO )
411: ELSE IF( IBSCL.EQ.2 ) THEN
412: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
413: $ INFO )
414: END IF
415: *
416: 50 CONTINUE
417: WORK( 1 ) = DBLE( WSIZE )
418: *
419: RETURN
420: *
421: * End of DGELS
422: *
423: END
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