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
|
*> \brief \b SPTCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SPTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sptcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sptcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sptcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* REAL ANORM, RCOND
* ..
* .. Array Arguments ..
* REAL D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SPTCON computes the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite tridiagonal matrix
*> using the factorization A = L*D*L**T or A = U**T*D*U computed by
*> SPTTRF.
*>
*> Norm(inv(A)) is computed by a direct method, and the reciprocal of
*> the condition number is computed as
*> RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> factorization of A, as computed by SPTTRF.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is REAL array, dimension (N-1)
*> The (n-1) off-diagonal elements of the unit bidiagonal factor
*> U or L from the factorization of A, as computed by SPTTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is REAL
*> The 1-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/(ANORM * AINVNM), where AINVNM is the
*> 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK 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 December 2016
*
*> \ingroup realPTcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The method used is described in Nicholas J. Higham, "Efficient
*> Algorithms for Computing the Condition Number of a Tridiagonal
*> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, N
REAL ANORM, RCOND
* ..
* .. Array Arguments ..
REAL D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IX
REAL AINVNM
* ..
* .. External Functions ..
INTEGER ISAMAX
EXTERNAL ISAMAX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPTCON', -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
*
* Check that D(1:N) is positive.
*
DO 10 I = 1, N
IF( D( I ).LE.ZERO )
$ RETURN
10 CONTINUE
*
* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
* m(i,j) = abs(A(i,j)), i = j,
* m(i,j) = -abs(A(i,j)), i .ne. j,
*
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
*
* Solve M(L) * x = e.
*
WORK( 1 ) = ONE
DO 20 I = 2, N
WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) )
20 CONTINUE
*
* Solve D * M(L)**T * x = b.
*
WORK( N ) = WORK( N ) / D( N )
DO 30 I = N - 1, 1, -1
WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) )
30 CONTINUE
*
* Compute AINVNM = max(x(i)), 1<=i<=n.
*
IX = ISAMAX( N, WORK, 1 )
AINVNM = ABS( WORK( IX ) )
*
* Compute the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of SPTCON
*
END
|
