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
|
SUBROUTINE CTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK,
$ RWORK, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORM, UPLO
INTEGER INFO, KD, LDAB, N
REAL RCOND
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX AB( LDAB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CTBCON estimates the reciprocal of the condition number of a
* triangular band matrix A, in either the 1-norm or the infinity-norm.
*
* The norm of A is computed and an estimate is obtained for
* norm(inv(A)), then 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.
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals or subdiagonals of the
* triangular band matrix A. KD >= 0.
*
* AB (input) COMPLEX array, dimension (LDAB,N)
* The upper or lower triangular band matrix A, stored in the
* first kd+1 rows of the array. The j-th column of A is stored
* in the j-th column of the array AB as follows:
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
* If DIAG = 'U', the diagonal elements of A are not referenced
* and are assumed to be 1.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* RCOND (output) REAL
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
* WORK (workspace) COMPLEX array, dimension (2*N)
*
* RWORK (workspace) REAL array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT, ONENRM, UPPER
CHARACTER NORMIN
INTEGER IX, KASE, KASE1
REAL AINVNM, ANORM, SCALE, SMLNUM, XNORM
COMPLEX ZDUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICAMAX
REAL CLANTB, SLAMCH
EXTERNAL LSAME, ICAMAX, CLANTB, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CLACON, CLATBS, CSRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTBCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
*
RCOND = ZERO
SMLNUM = SLAMCH( 'Safe minimum' )*REAL( MAX( N, 1 ) )
*
* Compute the 1-norm of the triangular matrix A or A'.
*
ANORM = CLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, RWORK )
*
* Continue only if ANORM > 0.
*
IF( ANORM.GT.ZERO ) THEN
*
* Estimate the 1-norm of the inverse of A.
*
AINVNM = ZERO
NORMIN = 'N'
IF( ONENRM ) THEN
KASE1 = 1
ELSE
KASE1 = 2
END IF
KASE = 0
10 CONTINUE
CALL CLACON( N, WORK( N+1 ), WORK, AINVNM, KASE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.KASE1 ) THEN
*
* Multiply by inv(A).
*
CALL CLATBS( UPLO, 'No transpose', DIAG, NORMIN, N, KD,
$ AB, LDAB, WORK, SCALE, RWORK, INFO )
ELSE
*
* Multiply by inv(A').
*
CALL CLATBS( UPLO, 'Conjugate transpose', DIAG, NORMIN,
$ N, KD, AB, LDAB, WORK, SCALE, RWORK, INFO )
END IF
NORMIN = 'Y'
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
IF( SCALE.NE.ONE ) THEN
IX = ICAMAX( N, WORK, 1 )
XNORM = CABS1( WORK( IX ) )
IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
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 / ANORM ) / AINVNM
END IF
*
20 CONTINUE
RETURN
*
* End of CTBCON
*
END
|
