Module: Newton

Includes:
Jacobian, LUSolve
Defined in:
lib/bigdecimal/newton.rb

Overview

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 2.0

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.

Class Method Summary collapse

Methods included from Jacobian

dfdxi, isEqual, jacobian

Methods included from LUSolve

ludecomp, lusolve

Class Method Details

.nlsolve(f, x) ⇒ Object

See also Newton



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
 
# File 'lib/bigdecimal/newton.rb', line 44

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

.norm(fv, zero = 0.0) ⇒ Object

:nodoc:



34
35
36
37
38
39
40
41
 
# File 'lib/bigdecimal/newton.rb', line 34

def norm(fv,zero=0.0) # :nodoc:
  s = zero
  n = fv.size
  for i in 0...n do
    s += fv[i]*fv[i]
  end
  s
end