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
|
SUBROUTINE MB03OD( JOBQR, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
$ RANK, SVAL, DWORK, LDWORK, 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 (optionally) a rank-revealing QR factorization of a
C real general M-by-N matrix A, which may be rank-deficient,
C and estimate its effective rank using incremental condition
C estimation.
C
C The routine uses a QR factorization with column pivoting:
C A * P = Q * R, where R = [ R11 R12 ],
C [ 0 R22 ]
C with R11 defined as the largest leading submatrix whose estimated
C condition number is less than 1/RCOND. The order of R11, RANK,
C is the effective rank of A.
C
C MB03OD does not perform any scaling of the matrix A.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOBQR CHARACTER*1
C = 'Q': Perform a QR factorization with column pivoting;
C = 'N': Do not perform the QR factorization (but assume
C that it has been done outside).
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 with JOBQR = 'Q', the leading M by N part of this
C array must contain the given matrix A.
C On exit with JOBQR = 'Q', the leading min(M,N) by N upper
C triangular part of A contains the triangular factor R,
C and the elements below the diagonal, with the array TAU,
C represent the orthogonal matrix Q as a product of
C min(M,N) elementary reflectors.
C On entry and on exit with JOBQR = 'N', the leading
C min(M,N) by N upper triangular part of A contains the
C triangular factor R, as determined by the QR factorization
C with pivoting. The elements below the diagonal of A are
C not referenced.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= max(1,M).
C
C JPVT (input/output) INTEGER array, dimension ( N )
C On entry with JOBQR = 'Q', if JPVT(i) <> 0, the i-th
C column of A is an initial column, otherwise it is a free
C column. Before the QR factorization of A, all initial
C columns are permuted to the leading positions; only the
C remaining free columns are moved as a result of column
C pivoting during the factorization. For rank determination
C it is preferable that all columns be free.
C On exit with JOBQR = 'Q', if JPVT(i) = k, then the i-th
C column of A*P was the k-th column of A.
C Array JPVT is not referenced when JOBQR = 'N'.
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 RCOND >= 0.
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 TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
C On exit with JOBQR = 'Q', the leading min(M,N) elements of
C TAU contain the scalar factors of the elementary
C reflectors.
C Array TAU is not referenced when JOBQR = 'N'.
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 Workspace
C
C DWORK DOUBLE PRECISION array, dimension ( LDWORK )
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= 3*N + 1, if JOBQR = 'Q';
C LDWORK >= max( 1, 2*min( M, N ) ), if JOBQR = 'N'.
C For good performance when JOBQR = 'Q', LDWORK should be
C larger. Specifically, LDWORK >= 2*N + ( N + 1 )*NB, where
C NB is the optimal block size for the LAPACK Library
C routine DGEQP3.
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 or uses a QR factorization with column
C pivoting of A, A * P = Q * R, with R defined above, and then
C finds the largest leading submatrix whose estimated condition
C number is less than 1/RCOND, taking the possible positive value of
C SVLMAX into account. This is performed using the LAPACK
C incremental condition estimation scheme and a slightly modified
C rank decision test.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Nov. 1996.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2005.
C
C ******************************************************************
C
C .. Parameters ..
INTEGER IMAX, IMIN
PARAMETER ( IMAX = 1, IMIN = 2 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER JOBQR
INTEGER INFO, LDA, LDWORK, M, N, RANK
DOUBLE PRECISION RCOND, SVLMAX
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )
C .. Local Scalars ..
LOGICAL LJOBQR
INTEGER I, ISMAX, ISMIN, MAXWRK, MINWRK, MN
DOUBLE PRECISION C1, C2, S1, S2, SMAX, SMAXPR, SMIN, SMINPR
C ..
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DGEQP3, DLAIC1, XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX, MIN
C ..
C .. Executable Statements ..
C
LJOBQR = LSAME( JOBQR, 'Q' )
MN = MIN( M, N )
ISMIN = 1
ISMAX = MN + 1
IF( LJOBQR ) THEN
MINWRK = 3*N + 1
ELSE
MINWRK = MAX( 1, 2*MN )
END IF
MAXWRK = MINWRK
C
C Test the input scalar arguments.
C
INFO = 0
IF( .NOT.LJOBQR .AND. .NOT.LSAME( JOBQR, 'N' ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( RCOND.LT.ZERO ) THEN
INFO = -7
ELSE IF( SVLMAX.LT.ZERO ) THEN
INFO = -8
ELSE IF( LDWORK.LT.MINWRK ) THEN
INFO = -13
END IF
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB03OD', -INFO )
RETURN
END IF
C
C Quick return if possible
C
IF( MN.EQ.0 ) THEN
RANK = 0
SVAL( 1 ) = ZERO
SVAL( 2 ) = ZERO
SVAL( 3 ) = ZERO
DWORK( 1 ) = ONE
RETURN
END IF
C
IF ( LJOBQR ) THEN
C
C Compute QR factorization with column pivoting of A:
C A * P = Q * R
C Workspace need 3*N + 1;
C prefer 2*N + (N+1)*NB.
C Details of Householder rotations stored in TAU.
C
CALL DGEQP3( M, N, A, LDA, JPVT, TAU, DWORK, LDWORK, INFO )
MAXWRK = MAX( MAXWRK, INT( DWORK(1) ) )
END IF
C
C Determine RANK using incremental condition estimation
C
DWORK( ISMIN ) = ONE
DWORK( ISMAX ) = ONE
SMAX = ABS( A( 1, 1 ) )
SMIN = SMAX
IF( SMAX.EQ.ZERO .OR. SVLMAX*RCOND.GT.SMAX ) THEN
RANK = 0
SVAL( 1 ) = SMAX
SVAL( 2 ) = ZERO
SVAL( 3 ) = ZERO
ELSE
RANK = 1
SMINPR = SMIN
C
10 CONTINUE
IF( RANK.LT.MN ) THEN
I = RANK + 1
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 )
C
IF( SVLMAX*RCOND.LE.SMAXPR ) THEN
IF( SVLMAX*RCOND.LE.SMINPR ) THEN
IF( SMAXPR*RCOND.LE.SMINPR ) THEN
DO 20 I = 1, RANK
DWORK( ISMIN+I-1 ) = S1*DWORK( ISMIN+I-1 )
DWORK( ISMAX+I-1 ) = S2*DWORK( ISMAX+I-1 )
20 CONTINUE
DWORK( ISMIN+RANK ) = C1
DWORK( ISMAX+RANK ) = C2
SMIN = SMINPR
SMAX = SMAXPR
RANK = RANK + 1
GO TO 10
END IF
END IF
END IF
END IF
SVAL( 1 ) = SMAX
SVAL( 2 ) = SMIN
SVAL( 3 ) = SMINPR
END IF
C
DWORK( 1 ) = MAXWRK
RETURN
C *** Last line of MB03OD ***
END
|
