1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
|
SUBROUTINE MB03OY( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
$ TAU, DWORK, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To compute a rank-revealing QR factorization of a real general
C M-by-N matrix A, which may be rank-deficient, and estimate its
C effective rank using incremental condition estimation.
C
C The routine uses a truncated QR factorization with column pivoting
C [ R11 R12 ]
C A * P = Q * R, where R = [ ],
C [ 0 R22 ]
C with R11 defined as the largest leading upper triangular submatrix
C whose estimated condition number is less than 1/RCOND. The order
C of R11, RANK, is the effective rank of A. Condition estimation is
C performed during the QR factorization process. Matrix R22 is full
C (but of small norm), or empty.
C
C MB03OY does not perform any scaling of the matrix A.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension
C ( LDA, N )
C On entry, the leading M-by-N part of this array must
C contain the given matrix A.
C On exit, the leading RANK-by-RANK upper triangular part
C of A contains the triangular factor R11, and the elements
C below the diagonal in the first RANK columns, with the
C array TAU, represent the orthogonal matrix Q as a product
C of RANK elementary reflectors.
C The remaining N-RANK columns contain the result of the
C QR factorization process used.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= max(1,M).
C
C RCOND (input) DOUBLE PRECISION
C RCOND is used to determine the effective rank of A, which
C is defined as the order of the largest leading triangular
C submatrix R11 in the QR factorization with pivoting of A,
C whose estimated condition number is less than 1/RCOND.
C 0 <= RCOND <= 1.
C NOTE that when SVLMAX > 0, the estimated rank could be
C less than that defined above (see SVLMAX).
C
C SVLMAX (input) DOUBLE PRECISION
C If A is a submatrix of another matrix B, and the rank
C decision should be related to that matrix, then SVLMAX
C should be an estimate of the largest singular value of B
C (for instance, the Frobenius norm of B). If this is not
C the case, the input value SVLMAX = 0 should work.
C SVLMAX >= 0.
C
C RANK (output) INTEGER
C The effective (estimated) rank of A, i.e., the order of
C the submatrix R11.
C
C SVAL (output) DOUBLE PRECISION array, dimension ( 3 )
C The estimates of some of the singular values of the
C triangular factor R:
C SVAL(1): largest singular value of R(1:RANK,1:RANK);
C SVAL(2): smallest singular value of R(1:RANK,1:RANK);
C SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
C if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
C otherwise.
C If the triangular factorization is a rank-revealing one
C (which will be the case if the leading columns were well-
C conditioned), then SVAL(1) will also be an estimate for
C the largest singular value of A, and SVAL(2) and SVAL(3)
C will be estimates for the RANK-th and (RANK+1)-st singular
C values of A, respectively.
C By examining these values, one can confirm that the rank
C is well defined with respect to the chosen value of RCOND.
C The ratio SVAL(1)/SVAL(2) is an estimate of the condition
C number of R(1:RANK,1:RANK).
C
C JPVT (output) INTEGER array, dimension ( N )
C If JPVT(i) = k, then the i-th column of A*P was the k-th
C column of A.
C
C TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
C The leading RANK elements of TAU contain the scalar
C factors of the elementary reflectors.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension ( 3*N-1 )
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The routine computes a truncated QR factorization with column
C pivoting of A, A * P = Q * R, with R defined above, and,
C during this process, finds the largest leading submatrix whose
C estimated condition number is less than 1/RCOND, taking the
C possible positive value of SVLMAX into account. This is performed
C using the LAPACK incremental condition estimation scheme and a
C slightly modified rank decision test. The factorization process
C stops when RANK has been determined.
C
C The matrix Q is represented as a product of elementary reflectors
C
C Q = H(1) H(2) . . . H(k), where k = rank <= min(m,n).
C
C Each H(i) has the form
C
C H = I - tau * v * v'
C
C where tau is a real scalar, and v is a real vector with
C v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
C A(i+1:m,i), and tau in TAU(i).
C
C The matrix P is represented in jpvt as follows: If
C jpvt(j) = i
C then the jth column of P is the ith canonical unit vector.
C
C REFERENCES
C
C [1] Bischof, C.H. and P. Tang.
C Generalizing Incremental Condition Estimation.
C LAPACK Working Notes 32, Mathematics and Computer Science
C Division, Argonne National Laboratory, UT, CS-91-132,
C May 1991.
C
C [2] Bischof, C.H. and P. Tang.
C Robust Incremental Condition Estimation.
C LAPACK Working Notes 33, Mathematics and Computer Science
C Division, Argonne National Laboratory, UT, CS-91-133,
C May 1991.
C
C NUMERICAL ASPECTS
C
C The algorithm is backward stable.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1998.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Jan. 2009.
C
C KEYWORDS
C
C Eigenvalue problem, matrix operations, orthogonal transformation,
C singular values.
C
C ******************************************************************
C
C .. Parameters ..
INTEGER IMAX, IMIN
PARAMETER ( IMAX = 1, IMIN = 2 )
DOUBLE PRECISION ZERO, ONE, P05
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, P05 = 0.05D0 )
C .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), DWORK( * ), SVAL( 3 ), TAU( * )
C ..
C .. Local Scalars ..
INTEGER I, ISMAX, ISMIN, ITEMP, J, MN, PVT
DOUBLE PRECISION AII, C1, C2, S1, S2, SMAX, SMAXPR, SMIN,
$ SMINPR, TEMP, TEMP2
C ..
C .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DNRM2
EXTERNAL DNRM2, IDAMAX
C .. External Subroutines ..
EXTERNAL DLAIC1, DLARF, DLARFG, DSCAL, DSWAP, XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
C ..
C .. Executable Statements ..
C
C Test the input scalar arguments.
C
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( RCOND.LT.ZERO .OR. RCOND.GT.ONE ) THEN
INFO = -5
ELSE IF( SVLMAX.LT.ZERO ) THEN
INFO = -6
END IF
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB03OY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
MN = MIN( M, N )
IF( MN.EQ.0 ) THEN
RANK = 0
SVAL( 1 ) = ZERO
SVAL( 2 ) = ZERO
SVAL( 3 ) = ZERO
RETURN
END IF
C
ISMIN = 1
ISMAX = ISMIN + N
C
C Initialize partial column norms and pivoting vector. The first n
C elements of DWORK store the exact column norms. The already used
C leading part is then overwritten by the condition estimator.
C
DO 10 I = 1, N
DWORK( I ) = DNRM2( M, A( 1, I ), 1 )
DWORK( N+I ) = DWORK( I )
JPVT( I ) = I
10 CONTINUE
C
C Compute factorization and determine RANK using incremental
C condition estimation.
C
RANK = 0
C
20 CONTINUE
IF( RANK.LT.MN ) THEN
I = RANK + 1
C
C Determine ith pivot column and swap if necessary.
C
PVT = ( I-1 ) + IDAMAX( N-I+1, DWORK( I ), 1 )
C
IF( PVT.NE.I ) THEN
CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
DWORK( PVT ) = DWORK( I )
DWORK( N+PVT ) = DWORK( N+I )
END IF
C
C Save A(I,I) and generate elementary reflector H(i).
C
IF( I.LT.M ) THEN
AII = A( I, I )
CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) )
ELSE
TAU( M ) = ZERO
END IF
C
IF( RANK.EQ.0 ) THEN
C
C Initialize; exit if matrix is zero (RANK = 0).
C
SMAX = ABS( A( 1, 1 ) )
IF ( SMAX.EQ.ZERO ) THEN
SVAL( 1 ) = ZERO
SVAL( 2 ) = ZERO
SVAL( 3 ) = ZERO
RETURN
END IF
SMIN = SMAX
SMAXPR = SMAX
SMINPR = SMIN
C1 = ONE
C2 = ONE
ELSE
C
C One step of incremental condition estimation.
C
CALL DLAIC1( IMIN, RANK, DWORK( ISMIN ), SMIN, A( 1, I ),
$ A( I, I ), SMINPR, S1, C1 )
CALL DLAIC1( IMAX, RANK, DWORK( ISMAX ), SMAX, A( 1, I ),
$ A( I, I ), SMAXPR, S2, C2 )
END IF
C
IF( SVLMAX*RCOND.LE.SMAXPR ) THEN
IF( SVLMAX*RCOND.LE.SMINPR ) THEN
IF( SMAXPR*RCOND.LE.SMINPR ) THEN
C
C Continue factorization, as rank is at least RANK.
C
IF( I.LT.N ) THEN
C
C Apply H(i) to A(i:m,i+1:n) from the left.
C
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
$ TAU( I ), A( I, I+1 ), LDA,
$ DWORK( 2*N+1 ) )
A( I, I ) = AII
END IF
C
C Update partial column norms.
C
DO 30 J = I + 1, N
IF( DWORK( J ).NE.ZERO ) THEN
TEMP = ONE -
$ ( ABS( A( I, J ) ) / DWORK( J ) )**2
TEMP = MAX( TEMP, ZERO )
TEMP2 = ONE + P05*TEMP*
$ ( DWORK( J ) / DWORK( N+J ) )**2
IF( TEMP2.EQ.ONE ) THEN
IF( M-I.GT.0 ) THEN
DWORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
DWORK( N+J ) = DWORK( J )
ELSE
DWORK( J ) = ZERO
DWORK( N+J ) = ZERO
END IF
ELSE
DWORK( J ) = DWORK( J )*SQRT( TEMP )
END IF
END IF
30 CONTINUE
C
DO 40 I = 1, RANK
DWORK( ISMIN+I-1 ) = S1*DWORK( ISMIN+I-1 )
DWORK( ISMAX+I-1 ) = S2*DWORK( ISMAX+I-1 )
40 CONTINUE
C
DWORK( ISMIN+RANK ) = C1
DWORK( ISMAX+RANK ) = C2
SMIN = SMINPR
SMAX = SMAXPR
RANK = RANK + 1
GO TO 20
END IF
END IF
END IF
END IF
C
C Restore the changed part of the (RANK+1)-th column and set SVAL.
C
IF ( RANK.LT.N ) THEN
IF ( I.LT.M ) THEN
CALL DSCAL( M-I, -A( I, I )*TAU( I ), A( I+1, I ), 1 )
A( I, I ) = AII
END IF
END IF
IF ( RANK.EQ.0 ) THEN
SMIN = ZERO
SMINPR = ZERO
END IF
SVAL( 1 ) = SMAX
SVAL( 2 ) = SMIN
SVAL( 3 ) = SMINPR
C
RETURN
C *** Last line of MB03OY ***
END
|
