Module: Newton

Relationships & Source Files
Super Chains via Extension / Inclusion / Inheritance
Instance Chain: self, `::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

.nlsolve(f, x) mod_func

See also Newton.
.norm(fv, zero = 0.0) mod_func Internal use only

Instance Method Summary

`::Jacobian` - Included

#dfdxi	Computes the derivative of `i` at `i`.
#isEqual	Determines the equality of two numbers by comparing to zero, or using the epsilon value.
#jacobian	Computes the `::Jacobian` of `f` at `x`.

`::LUSolve` - Included

#ludecomp	Performs LU decomposition of the n by n matrix a.
#lusolve	Solves a*x = b for x, using LU decomposition.

Class Method Details

.nlsolve(f, x) (mod_func)

See also Newton

[ GitHub ]

# 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) (mod_func)

This method is for internal use only.