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
|
SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
$ WORK, IWORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER INFO, KL, KU, LDAB, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGBCON estimates the reciprocal of the condition number of a real
* general band matrix A, in either the 1-norm or the infinity-norm,
* using the LU factorization computed by DGBTRF.
*
* 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)) ).
*
* Arguments
* =========
*
* NORM (input) 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.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KL (input) INTEGER
* The number of subdiagonals within the band of A. KL >= 0.
*
* KU (input) INTEGER
* The number of superdiagonals within the band of A. KU >= 0.
*
* AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
* Details of the LU factorization of the band matrix A, as
* computed by DGBTRF. 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.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= N, row i of the matrix was
* interchanged with row IPIV(i).
*
* ANORM (input) DOUBLE PRECISION
* If NORM = '1' or 'O', the 1-norm of the original matrix A.
* If NORM = 'I', the infinity-norm of the original matrix A.
*
* RCOND (output) DOUBLE PRECISION
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LNOTI, ONENRM
CHARACTER NORMIN
INTEGER IX, J, JP, KASE, KASE1, KD, LM
DOUBLE PRECISION AINVNM, SCALE, SMLNUM, T
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLACN2, DLATBS, DRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MIN
* ..
* .. 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( 'DGBCON', -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 = DLAMCH( '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 DLACN2( N, WORK( N+1 ), WORK, IWORK, 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 DAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ), 1 )
20 CONTINUE
END IF
*
* Multiply by inv(U).
*
CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N,
$ KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
$ INFO )
ELSE
*
* Multiply by inv(U').
*
CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
$ INFO )
*
* Multiply by inv(L').
*
IF( LNOTI ) THEN
DO 30 J = N - 1, 1, -1
LM = MIN( KL, N-J )
WORK( J ) = WORK( J ) - DDOT( 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 = IDAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 40
CALL DRSCL( 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 DGBCON
*
END
|
