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Module: Newton

Relationships & Source Files
Super Chains via Extension / Inclusion / Inheritance
Instance Chain:
Defined in: ext/bigdecimal/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

Instance Method Summary

::Jacobian - Included

#dfdxi

Computes the derivative of f at x.

#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 'ext/bigdecimal/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.
[ GitHub ]

  
# File 'ext/bigdecimal/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