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
|
SUBROUTINE SPBCON( UPLO, N, KD, AB, LDAB, 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 SLACN2 in place of SLACON, 7 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
REAL ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL AB( LDAB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SPBCON estimates the reciprocal of the condition number (in the
* 1-norm) of a real symmetric positive definite band matrix using the
* Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF.
*
* An estimate is obtained for norm(inv(A)), and the reciprocal of the
* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangular factor stored in AB;
* = 'L': Lower triangular factor stored in AB.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input) REAL array, dimension (LDAB,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**T*U or A = L*L**T of the band matrix A, stored in the
* first KD+1 rows of the array. The j-th column of U or L is
* stored in the j-th column of the array AB as follows:
* if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
* if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* ANORM (input) REAL
* The 1-norm (or infinity-norm) of the symmetric band matrix A.
*
* RCOND (output) REAL
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
* estimate of the 1-norm of inv(A) computed in this routine.
*
* WORK (workspace) REAL 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 ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
CHARACTER NORMIN
INTEGER IX, KASE
REAL AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ISAMAX
REAL SLAMCH
EXTERNAL LSAME, ISAMAX, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SLACN2, SLATBS, SRSCL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPBCON', -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 1-norm of the inverse.
*
KASE = 0
NORMIN = 'N'
10 CONTINUE
CALL SLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( UPPER ) THEN
*
* Multiply by inv(U').
*
CALL SLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
$ INFO )
NORMIN = 'Y'
*
* Multiply by inv(U).
*
CALL SLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
$ INFO )
ELSE
*
* Multiply by inv(L).
*
CALL SLATBS( 'Lower', 'No transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
$ INFO )
NORMIN = 'Y'
*
* Multiply by inv(L').
*
CALL SLATBS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
$ KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
$ INFO )
END IF
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
SCALE = SCALEL*SCALEU
IF( SCALE.NE.ONE ) THEN
IX = ISAMAX( N, WORK, 1 )
IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL SRSCL( 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
*
20 CONTINUE
*
RETURN
*
* End of SPBCON
*
END
|
