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
|
SUBROUTINE MB02OD( SIDE, UPLO, TRANS, DIAG, NORM, M, N, ALPHA, A,
$ LDA, B, LDB, RCOND, TOL, IWORK, 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 solve (if well-conditioned) one of the matrix equations
C
C op( A )*X = alpha*B, or X*op( A ) = alpha*B,
C
C where alpha is a scalar, X and B are m-by-n matrices, A is a unit,
C or non-unit, upper or lower triangular matrix and op( A ) is one
C of
C
C op( A ) = A or op( A ) = A'.
C
C An estimate of the reciprocal of the condition number of the
C triangular matrix A, in either the 1-norm or the infinity-norm, is
C also computed as
C
C RCOND = 1 / ( norm(A) * norm(inv(A)) ).
C
C and the specified matrix equation is solved only if RCOND is
C larger than a given tolerance TOL. In that case, the matrix X is
C overwritten on B.
C
C ARGUMENTS
C
C Mode Parameters
C
C SIDE CHARACTER*1
C Specifies whether op( A ) appears on the left or right
C of X as follows:
C = 'L': op( A )*X = alpha*B;
C = 'R': X*op( A ) = alpha*B.
C
C UPLO CHARACTER*1
C Specifies whether the matrix A is an upper or lower
C triangular matrix as follows:
C = 'U': A is an upper triangular matrix;
C = 'L': A is a lower triangular matrix.
C
C TRANS CHARACTER*1
C Specifies the form of op( A ) to be used in the matrix
C multiplication as follows:
C = 'N': op( A ) = A;
C = 'T': op( A ) = A';
C = 'C': op( A ) = A'.
C
C DIAG CHARACTER*1
C Specifies whether or not A is unit triangular as follows:
C = 'U': A is assumed to be unit triangular;
C = 'N': A is not assumed to be unit triangular.
C
C NORM CHARACTER*1
C Specifies whether the 1-norm condition number or the
C infinity-norm condition number is required:
C = '1' or 'O': 1-norm;
C = 'I': Infinity-norm.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of B. M >= 0.
C
C N (input) INTEGER
C The number of columns of B. N >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The scalar alpha. When alpha is zero then A is not
C referenced and B need not be set before entry.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,k),
C where k is M when SIDE = 'L' and is N when SIDE = 'R'.
C On entry with UPLO = 'U', the leading k-by-k upper
C triangular part of this array must contain the upper
C triangular matrix and the strictly lower triangular part
C of A is not referenced.
C On entry with UPLO = 'L', the leading k-by-k lower
C triangular part of this array must contain the lower
C triangular matrix and the strictly upper triangular part
C of A is not referenced.
C Note that when DIAG = 'U', the diagonal elements of A are
C not referenced either, but are assumed to be unity.
C
C LDA INTEGER
C The leading dimension of array A.
C LDA >= max(1,M) when SIDE = 'L';
C LDA >= max(1,N) when SIDE = 'R'.
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
C On entry, the leading M-by-N part of this array must
C contain the right-hand side matrix B.
C On exit, if INFO = 0, the leading M-by-N part of this
C array contains the solution matrix X.
C Otherwise, this array is not modified by the routine.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= max(1,M).
C
C RCOND (output) DOUBLE PRECISION
C The reciprocal of the condition number of the matrix A,
C computed as RCOND = 1/(norm(A) * norm(inv(A))).
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used to test for near singularity of
C the matrix A. If the user sets TOL > 0, then the given
C value of TOL is used as a lower bound for the reciprocal
C condition number of that matrix; a matrix whose estimated
C condition number is less than 1/TOL is considered to be
C nonsingular. If the user sets TOL <= 0, then an implicitly
C computed, default tolerance, defined by TOLDEF = k*k*EPS,
C is used instead, where EPS is the machine precision (see
C LAPACK Library routine DLAMCH).
C
C Workspace
C
C IWORK INTEGER array, dimension (k)
C
C DWORK DOUBLE PRECISION array, dimension (3*k)
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 = 1: the matrix A is numerically singular, i.e. the
C condition number estimate of A (in the specified
C norm) exceeds 1/TOL.
C
C METHOD
C
C An estimate of the reciprocal of the condition number of the
C triangular matrix A (in the specified norm) is computed, and if
C this estimate is larger then the given (or default) tolerance,
C the specified matrix equation is solved using Level 3 BLAS
C routine DTRSM.
C
C
C REFERENCES
C
C None.
C
C NUMERICAL ASPECTS
C 2
C The algorithm requires k N/2 operations.
C
C CONTRIBUTORS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1997.
C
C REVISIONS
C
C February 20, 1998.
C
C KEYWORDS
C
C Condition number, matrix algebra, matrix operations.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER DIAG, NORM, SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDB, M, N
DOUBLE PRECISION ALPHA, RCOND, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*)
C .. Local Scalars ..
LOGICAL LSIDE, ONENRM
INTEGER NROWA
DOUBLE PRECISION TOLDEF
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, LSAME
C .. External Subroutines ..
EXTERNAL DTRCON, DTRSM, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
C .. Executable Statements ..
C
LSIDE = LSAME( SIDE, 'L' )
IF( LSIDE )THEN
NROWA = M
ELSE
NROWA = N
END IF
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
C
C Test the input scalar arguments.
C
INFO = 0
IF( ( .NOT.LSIDE ).AND.( .NOT.LSAME( SIDE, 'R' ) ) )THEN
INFO = -1
ELSE IF( ( .NOT.LSAME( UPLO, 'U' ) ).AND.
$ ( .NOT.LSAME( UPLO, 'L' ) ) )THEN
INFO = -2
ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'T' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'C' ) ) )THEN
INFO = -3
ELSE IF( ( .NOT.LSAME( DIAG, 'U' ) ).AND.
$ ( .NOT.LSAME( DIAG, 'N' ) ) )THEN
INFO = -4
ELSE IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -5
ELSE IF( M.LT.0 )THEN
INFO = -6
ELSE IF( N.LT.0 )THEN
INFO = -7
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, M ) )THEN
INFO = -12
END IF
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB02OD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( NROWA.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
C
TOLDEF = TOL
IF ( TOLDEF.LE.ZERO )
$ TOLDEF = DBLE( NROWA*NROWA )*DLAMCH( 'Epsilon' )
C
CALL DTRCON( NORM, UPLO, DIAG, NROWA, A, LDA, RCOND, DWORK,
$ IWORK, INFO )
C
IF ( RCOND.GT.TOLDEF ) THEN
CALL DTRSM( SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, LDA, B,
$ LDB )
ELSE
INFO = 1
END IF
C *** Last line of MB02OD ***
END
|
