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
|
*> \brief \b CGBCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download CGBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgbcon.f">
*> [TXT]</a>
*
* Definition
* ==========
*
* SUBROUTINE CGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
* WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER INFO, KL, KU, LDAB, N
* REAL ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* REAL RWORK( * )
* COMPLEX AB( LDAB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> CGBCON estimates the reciprocal of the condition number of a complex
*> general band matrix A, in either the 1-norm or the infinity-norm,
*> using the LU factorization computed by CGBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*> RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies whether the 1-norm condition number or the
*> infinity-norm condition number is required:
*> = '1' or 'O': 1-norm;
*> = 'I': Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is COMPLEX array, dimension (LDAB,N)
*> Details of the LU factorization of the band matrix A, as
*> computed by CGBTRF. U is stored as an upper triangular band
*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*> the multipliers used during the factorization are stored in
*> rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices; for 1 <= i <= N, row i of the matrix was
*> interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is REAL
*> If NORM = '1' or 'O', the 1-norm of the original matrix A.
*> If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is REAL
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complexGBcomputational
*
* =====================================================================
SUBROUTINE CGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
$ WORK, RWORK, INFO )
*
* -- LAPACK computational routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, KL, KU, LDAB, N
REAL ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
REAL RWORK( * )
COMPLEX AB( LDAB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LNOTI, ONENRM
CHARACTER NORMIN
INTEGER IX, J, JP, KASE, KASE1, KD, LM
REAL AINVNM, SCALE, SMLNUM
COMPLEX T, ZDUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICAMAX
REAL SLAMCH
COMPLEX CDOTC
EXTERNAL LSAME, ICAMAX, SLAMCH, CDOTC
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CLACN2, CLATBS, CSRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MIN, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
INFO = -6
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGBCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
SMLNUM = SLAMCH( 'Safe minimum' )
*
* Estimate the norm of inv(A).
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KD = KL + KU + 1
LNOTI = KL.GT.0
KASE = 0
10 CONTINUE
CALL CLACN2( N, WORK( N+1 ), WORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(L).
*
IF( LNOTI ) THEN
DO 20 J = 1, N - 1
LM = MIN( KL, N-J )
JP = IPIV( J )
T = WORK( JP )
IF( JP.NE.J ) THEN
WORK( JP ) = WORK( J )
WORK( J ) = T
END IF
CALL CAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ), 1 )
20 CONTINUE
END IF
*
* Multiply by inv(U).
*
CALL CLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N,
$ KL+KU, AB, LDAB, WORK, SCALE, RWORK, INFO )
ELSE
*
* Multiply by inv(U**H).
*
CALL CLATBS( 'Upper', 'Conjugate transpose', 'Non-unit',
$ NORMIN, N, KL+KU, AB, LDAB, WORK, SCALE, RWORK,
$ INFO )
*
* Multiply by inv(L**H).
*
IF( LNOTI ) THEN
DO 30 J = N - 1, 1, -1
LM = MIN( KL, N-J )
WORK( J ) = WORK( J ) - CDOTC( LM, AB( KD+1, J ), 1,
$ WORK( J+1 ), 1 )
JP = IPIV( J )
IF( JP.NE.J ) THEN
T = WORK( JP )
WORK( JP ) = WORK( J )
WORK( J ) = T
END IF
30 CONTINUE
END IF
END IF
*
* Divide X by 1/SCALE if doing so will not cause overflow.
*
NORMIN = 'Y'
IF( SCALE.NE.ONE ) THEN
IX = ICAMAX( N, WORK, 1 )
IF( SCALE.LT.CABS1( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 40
CALL CSRSCL( N, SCALE, WORK, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
40 CONTINUE
RETURN
*
* End of CGBCON
*
END
|
