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
|
// Copyright Bruno Pinçon, ESIAL-IECN, Inria CORIDA project
// <bruno.pincon@iecn.u-nancy.fr>
//
// This set of scilab 's macros provide a few sparse utilities.
//
// This software is governed by the CeCILL license under French law and
// abiding by the rules of distribution of free software. You can use,
// modify and/ or redistribute the software under the terms of the CeCILL
// license as circulated by CEA, CNRS and INRIA at the following URL
// "http://www.cecill.info".
//
// As a counterpart to the access to the source code and rights to copy,
// modify and redistribute granted by the license, users are provided only
// with a limited warranty and the software's author, the holder of the
// economic rights, and the successive licensors have only limited
// liability.
//
// In this respect, the user's attention is drawn to the risks associated
// with loading, using, modifying and/or developing or reproducing the
// software by the user in light of its specific status of free software,
// that may mean that it is complicated to manipulate, and that also
// therefore means that it is reserved for developers and experienced
// professionals having in-depth computer knowledge. Users are therefore
// encouraged to load and test the software's suitability as regards their
// requirements in conditions enabling the security of their systems and/or
// data to be ensured and, more generally, to use and operate it in the
// same conditions as regards security.
//
// The fact that you are presently reading this means that you have had
// knowledge of the CeCILL license and that you accept its terms.
function [K2, lm, vm, lM, vM] = cond2sp(A, C, rtol, itermax, verb)
//
// PURPOSE
// for a s.p.d. matrix computes the maximum and minimum
// eigen element (value and vector) with the power and
// inverse power method then the 2-norm condition number
// K2 = lM / lm
//
// PARAMETERS
// inputs
// ------
// A : a sparse s.p.d. matrix
// C : pointer onto a Cholesky factorization (gotten with
// taucs_chfact)
// rtol : (optional) relative precision for the output test
// (l_new - l_old)/l_new < rtol
// itermax : (optional) maximum number of iteration in each step
// verb : (optional) a boolean must be %t for display result
// for each iteration
//
// outputs
// -------
// K2 : 2-norm condition number
// lm : min eigenvalue
// vm : associated eigenvector
// lM : max eigenvalue
// vM : associated eigenvector
//
[lhs, rhs] = argn()
// no verif
if ~exists("verb", "local") then , verb = %f , end
if ~exists("rtol", "local") then , rtol = 1.e-3, end
if ~exists("itermax","local") then , itermax = 30 , end
itermax = max(4,itermax) // 4 iterations are forced
// 1) computes (with "direct Rayleigh power method") lM, vM
n = size(A,1)
x = rand(n,1) ; x = x / norm(x)
y = A*x
lM_old = x'*y
iter = 0
if verb then
mprintf(gettext("\n approximate (lM,vM) with the iterative power method \n"));
mprintf(gettext(" ----------------------------------------------------- \n"));
end
while %t
iter = iter + 1
x = y / norm(y)
y = A*x
lM = x'*y
if verb then
mprintf(gettext(" iteration %3d : lM = %e \n"), iter, lM);
end
crit = abs((lM - lM_old)/lM)
if crit < rtol & iter > 3 then
break
else
lM_old = lM
end
if iter >= itermax then
mprintf(gettext(" Warning : for lM ""convergence"" at rtol = %e \n"), rtol);
mprintf(gettext(" don''t reached after %d iterations (got only %e) \n"), itermax, crit);
break
end
end
vM = x
// 2) computes (with "inverse Rayleigh power method") lm, vm
x = rand(n,1) ; x = x / norm(x)
y = taucs_chsolve(C,x)
lm_old = x'*y
iter = 0
if verb then
mprintf(gettext("\n approximate (lm,vm) with the inverse iterative power method \n"));
mprintf(gettext(" ------------------------------------------------------------\n"));
end
while %t
iter = iter + 1
x = y / norm(y)
y = taucs_chsolve(C,x)
lm = x'*y
if verb then
mprintf(gettext(" iteration %3d : lm = %e \n"), iter, 1/lm)
end
crit = abs((lm - lm_old)/lm)
if crit < rtol & iter > 3 then
break
else
lm_old = lm
end
if iter >= itermax then
mprintf(gettext(" Warning : for lm ""convergence"" at rtol = %e \n"), rtol);
mprintf(gettext(" don''t reached after %d iterations (got only %e) \n"),itermax, crit);
break
end
end
vm = x
lm = 1/lm;
K2 = lM/lm;
endfunction
|
