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
|
require "bigdecimal/ludcmp"
require "bigdecimal/jacobian"
#
# newton.rb
#
# Solves the nonlinear algebraic equation system f = 0 by Newton's method.
# This program is not dependent on BigDecimal.
#
# To call:
# n = nlsolve(f,x)
# where n is the number of iterations required,
# x is the initial value vector
# f is an Object which is used to compute the values of the equations to be solved.
# It must provide the following methods:
#
# f.values(x):: returns the values of all functions at x
#
# f.zero:: returns 0.0
# f.one:: returns 1.0
# f.two:: returns 1.0
# f.ten:: returns 10.0
#
# f.eps:: returns the convergence criterion (epsilon value) used to determine whether two values are considered equal. If |a-b| < epsilon, the two values are considered equal.
#
# On exit, x is the solution vector.
#
module Newton
include LUSolve
include Jacobian
module_function
def norm(fv,zero=0.0)
s = zero
n = fv.size
for i in 0...n do
s += fv[i]*fv[i]
end
s
end
def nlsolve(f,x)
nRetry = 0
n = x.size
f0 = f.values(x)
zero = f.zero
one = f.one
two = f.two
p5 = one/two
d = norm(f0,zero)
minfact = f.ten*f.ten*f.ten
minfact = one/minfact
e = f.eps
while d >= e do
nRetry += 1
# Not yet converged. => Compute Jacobian matrix
dfdx = jacobian(f,f0,x)
# Solve dfdx*dx = -f0 to estimate dx
dx = lusolve(dfdx,f0,ludecomp(dfdx,n,zero,one),zero)
fact = two
xs = x.dup
begin
fact *= p5
if fact < minfact then
raise "Failed to reduce function values."
end
for i in 0...n do
x[i] = xs[i] - dx[i]*fact
end
f0 = f.values(x)
dn = norm(f0,zero)
end while(dn>=d)
d = dn
end
nRetry
end
end
|
