Annotation of rpl/lapack/lapack/dgelsy.f, revision 1.2
1.1 bertrand 1: SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
2: $ WORK, LWORK, 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: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
11: DOUBLE PRECISION RCOND
12: * ..
13: * .. Array Arguments ..
14: INTEGER JPVT( * )
15: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * DGELSY computes the minimum-norm solution to a real linear least
22: * squares problem:
23: * minimize || A * X - B ||
24: * using a complete orthogonal factorization of A. A is an M-by-N
25: * matrix which may be rank-deficient.
26: *
27: * Several right hand side vectors b and solution vectors x can be
28: * handled in a single call; they are stored as the columns of the
29: * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
30: * matrix X.
31: *
32: * The routine first computes a QR factorization with column pivoting:
33: * A * P = Q * [ R11 R12 ]
34: * [ 0 R22 ]
35: * with R11 defined as the largest leading submatrix whose estimated
36: * condition number is less than 1/RCOND. The order of R11, RANK,
37: * is the effective rank of A.
38: *
39: * Then, R22 is considered to be negligible, and R12 is annihilated
40: * by orthogonal transformations from the right, arriving at the
41: * complete orthogonal factorization:
42: * A * P = Q * [ T11 0 ] * Z
43: * [ 0 0 ]
44: * The minimum-norm solution is then
45: * X = P * Z' [ inv(T11)*Q1'*B ]
46: * [ 0 ]
47: * where Q1 consists of the first RANK columns of Q.
48: *
49: * This routine is basically identical to the original xGELSX except
50: * three differences:
51: * o The call to the subroutine xGEQPF has been substituted by the
52: * the call to the subroutine xGEQP3. This subroutine is a Blas-3
53: * version of the QR factorization with column pivoting.
54: * o Matrix B (the right hand side) is updated with Blas-3.
55: * o The permutation of matrix B (the right hand side) is faster and
56: * more simple.
57: *
58: * Arguments
59: * =========
60: *
61: * M (input) INTEGER
62: * The number of rows of the matrix A. M >= 0.
63: *
64: * N (input) INTEGER
65: * The number of columns of the matrix A. N >= 0.
66: *
67: * NRHS (input) INTEGER
68: * The number of right hand sides, i.e., the number of
69: * columns of matrices B and X. NRHS >= 0.
70: *
71: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
72: * On entry, the M-by-N matrix A.
73: * On exit, A has been overwritten by details of its
74: * complete orthogonal factorization.
75: *
76: * LDA (input) INTEGER
77: * The leading dimension of the array A. LDA >= max(1,M).
78: *
79: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
80: * On entry, the M-by-NRHS right hand side matrix B.
81: * On exit, the N-by-NRHS solution matrix X.
82: *
83: * LDB (input) INTEGER
84: * The leading dimension of the array B. LDB >= max(1,M,N).
85: *
86: * JPVT (input/output) INTEGER array, dimension (N)
87: * On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
88: * to the front of AP, otherwise column i is a free column.
89: * On exit, if JPVT(i) = k, then the i-th column of AP
90: * was the k-th column of A.
91: *
92: * RCOND (input) DOUBLE PRECISION
93: * RCOND is used to determine the effective rank of A, which
94: * is defined as the order of the largest leading triangular
95: * submatrix R11 in the QR factorization with pivoting of A,
96: * whose estimated condition number < 1/RCOND.
97: *
98: * RANK (output) INTEGER
99: * The effective rank of A, i.e., the order of the submatrix
100: * R11. This is the same as the order of the submatrix T11
101: * in the complete orthogonal factorization of A.
102: *
103: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
104: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
105: *
106: * LWORK (input) INTEGER
107: * The dimension of the array WORK.
108: * The unblocked strategy requires that:
109: * LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
110: * where MN = min( M, N ).
111: * The block algorithm requires that:
112: * LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
113: * where NB is an upper bound on the blocksize returned
114: * by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
115: * and DORMRZ.
116: *
117: * If LWORK = -1, then a workspace query is assumed; the routine
118: * only calculates the optimal size of the WORK array, returns
119: * this value as the first entry of the WORK array, and no error
120: * message related to LWORK is issued by XERBLA.
121: *
122: * INFO (output) INTEGER
123: * = 0: successful exit
124: * < 0: If INFO = -i, the i-th argument had an illegal value.
125: *
126: * Further Details
127: * ===============
128: *
129: * Based on contributions by
130: * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
131: * E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
132: * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
133: *
134: * =====================================================================
135: *
136: * .. Parameters ..
137: INTEGER IMAX, IMIN
138: PARAMETER ( IMAX = 1, IMIN = 2 )
139: DOUBLE PRECISION ZERO, ONE
140: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
141: * ..
142: * .. Local Scalars ..
143: LOGICAL LQUERY
144: INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
145: $ LWKOPT, MN, NB, NB1, NB2, NB3, NB4
146: DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
147: $ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
148: * ..
149: * .. External Functions ..
150: INTEGER ILAENV
151: DOUBLE PRECISION DLAMCH, DLANGE
152: EXTERNAL ILAENV, DLAMCH, DLANGE
153: * ..
154: * .. External Subroutines ..
155: EXTERNAL DCOPY, DGEQP3, DLABAD, DLAIC1, DLASCL, DLASET,
156: $ DORMQR, DORMRZ, DTRSM, DTZRZF, XERBLA
157: * ..
158: * .. Intrinsic Functions ..
159: INTRINSIC ABS, MAX, MIN
160: * ..
161: * .. Executable Statements ..
162: *
163: MN = MIN( M, N )
164: ISMIN = MN + 1
165: ISMAX = 2*MN + 1
166: *
167: * Test the input arguments.
168: *
169: INFO = 0
170: LQUERY = ( LWORK.EQ.-1 )
171: IF( M.LT.0 ) THEN
172: INFO = -1
173: ELSE IF( N.LT.0 ) THEN
174: INFO = -2
175: ELSE IF( NRHS.LT.0 ) THEN
176: INFO = -3
177: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
178: INFO = -5
179: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
180: INFO = -7
181: END IF
182: *
183: * Figure out optimal block size
184: *
185: IF( INFO.EQ.0 ) THEN
186: IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
187: LWKMIN = 1
188: LWKOPT = 1
189: ELSE
190: NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
191: NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
192: NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 )
193: NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, NRHS, -1 )
194: NB = MAX( NB1, NB2, NB3, NB4 )
195: LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS )
196: LWKOPT = MAX( LWKMIN,
197: $ MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS )
198: END IF
199: WORK( 1 ) = LWKOPT
200: *
201: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
202: INFO = -12
203: END IF
204: END IF
205: *
206: IF( INFO.NE.0 ) THEN
207: CALL XERBLA( 'DGELSY', -INFO )
208: RETURN
209: ELSE IF( LQUERY ) THEN
210: RETURN
211: END IF
212: *
213: * Quick return if possible
214: *
215: IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
216: RANK = 0
217: RETURN
218: END IF
219: *
220: * Get machine parameters
221: *
222: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
223: BIGNUM = ONE / SMLNUM
224: CALL DLABAD( SMLNUM, BIGNUM )
225: *
226: * Scale A, B if max entries outside range [SMLNUM,BIGNUM]
227: *
228: ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
229: IASCL = 0
230: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
231: *
232: * Scale matrix norm up to SMLNUM
233: *
234: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
235: IASCL = 1
236: ELSE IF( ANRM.GT.BIGNUM ) THEN
237: *
238: * Scale matrix norm down to BIGNUM
239: *
240: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
241: IASCL = 2
242: ELSE IF( ANRM.EQ.ZERO ) THEN
243: *
244: * Matrix all zero. Return zero solution.
245: *
246: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
247: RANK = 0
248: GO TO 70
249: END IF
250: *
251: BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
252: IBSCL = 0
253: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
254: *
255: * Scale matrix norm up to SMLNUM
256: *
257: CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
258: IBSCL = 1
259: ELSE IF( BNRM.GT.BIGNUM ) THEN
260: *
261: * Scale matrix norm down to BIGNUM
262: *
263: CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
264: IBSCL = 2
265: END IF
266: *
267: * Compute QR factorization with column pivoting of A:
268: * A * P = Q * R
269: *
270: CALL DGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
271: $ LWORK-MN, INFO )
272: WSIZE = MN + WORK( MN+1 )
273: *
274: * workspace: MN+2*N+NB*(N+1).
275: * Details of Householder rotations stored in WORK(1:MN).
276: *
277: * Determine RANK using incremental condition estimation
278: *
279: WORK( ISMIN ) = ONE
280: WORK( ISMAX ) = ONE
281: SMAX = ABS( A( 1, 1 ) )
282: SMIN = SMAX
283: IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
284: RANK = 0
285: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
286: GO TO 70
287: ELSE
288: RANK = 1
289: END IF
290: *
291: 10 CONTINUE
292: IF( RANK.LT.MN ) THEN
293: I = RANK + 1
294: CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
295: $ A( I, I ), SMINPR, S1, C1 )
296: CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
297: $ A( I, I ), SMAXPR, S2, C2 )
298: *
299: IF( SMAXPR*RCOND.LE.SMINPR ) THEN
300: DO 20 I = 1, RANK
301: WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
302: WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
303: 20 CONTINUE
304: WORK( ISMIN+RANK ) = C1
305: WORK( ISMAX+RANK ) = C2
306: SMIN = SMINPR
307: SMAX = SMAXPR
308: RANK = RANK + 1
309: GO TO 10
310: END IF
311: END IF
312: *
313: * workspace: 3*MN.
314: *
315: * Logically partition R = [ R11 R12 ]
316: * [ 0 R22 ]
317: * where R11 = R(1:RANK,1:RANK)
318: *
319: * [R11,R12] = [ T11, 0 ] * Y
320: *
321: IF( RANK.LT.N )
322: $ CALL DTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
323: $ LWORK-2*MN, INFO )
324: *
325: * workspace: 2*MN.
326: * Details of Householder rotations stored in WORK(MN+1:2*MN)
327: *
328: * B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
329: *
330: CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
331: $ B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
332: WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) )
333: *
334: * workspace: 2*MN+NB*NRHS.
335: *
336: * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
337: *
338: CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
339: $ NRHS, ONE, A, LDA, B, LDB )
340: *
341: DO 40 J = 1, NRHS
342: DO 30 I = RANK + 1, N
343: B( I, J ) = ZERO
344: 30 CONTINUE
345: 40 CONTINUE
346: *
347: * B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS)
348: *
349: IF( RANK.LT.N ) THEN
350: CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,
351: $ LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ),
352: $ LWORK-2*MN, INFO )
353: END IF
354: *
355: * workspace: 2*MN+NRHS.
356: *
357: * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
358: *
359: DO 60 J = 1, NRHS
360: DO 50 I = 1, N
361: WORK( JPVT( I ) ) = B( I, J )
362: 50 CONTINUE
363: CALL DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
364: 60 CONTINUE
365: *
366: * workspace: N.
367: *
368: * Undo scaling
369: *
370: IF( IASCL.EQ.1 ) THEN
371: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
372: CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
373: $ INFO )
374: ELSE IF( IASCL.EQ.2 ) THEN
375: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
376: CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
377: $ INFO )
378: END IF
379: IF( IBSCL.EQ.1 ) THEN
380: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
381: ELSE IF( IBSCL.EQ.2 ) THEN
382: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
383: END IF
384: *
385: 70 CONTINUE
386: WORK( 1 ) = LWKOPT
387: *
388: RETURN
389: *
390: * End of DGELSY
391: *
392: END
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