Ruby snippet that tries to compute and find the root for a function using the Newton-Raphson method.

newtons_method.rb

#!/usr/env ruby

def derivative_of(f:, x:)
  h = 1e-14 # 1 * 10^-14 this is the smallest number ruby can handle in calculations before defaulting result to zero.

  (f.call(x + h) - f.call(x)) / h # This is a naive approach to calculate limit of f(x) in x.
end

# Newton's Method, where we calculate f(x), f'(x) and calculate error.
def find_root(f:, seed:, iterations: nil, error_level: nil)
  if iterations.nil? && error_level.nil?
    throw 'Iterations and error level not defined. Please specify at least one'
  end

  x = nil # Root
  x0 = seed # Initial value
  i = 0 # Number of iterations made

  loop do
    break if !iterations.nil? && i >= iterations # Leave if we have reached the number of iterations.

    x = x0 - (f.call(x0) / derivative_of(f: f, x: x0))

    if x.nan?
      x = nil
      break
    end

    break if x == 0 ||  (x - x0).abs <= error_level

    i = i+1 # Iteration finished
    x0 = x # Swap the x0 value
  end

  x # Result
rescue Math::DomainError
  x = nil
end

# Example # 1
# F(x) = (e ^ x) * ln(x)
# Exact root is defined at 1
f = lambda { |x| Math.exp(x) * Math.log(x) } # f(x) = (e^x) * ln(x) defined in R

root = find_root(f: f, seed: 2, error_level: 0.00000001)

if root.nil?
  puts 'Computed root of F(x) cannot be found or do not exist.'
else
  puts "Computed root of F(x) = (e^x)ln(x) is #{root}. Exact root is 1.0."
end

# Example # 2
# F(x) = (x ^ 5) * tanh(x^3)
# Exact root is defined at 0
f = lambda { |x| (x ** 5) * Math.tanh(x ** 3) } # f(x) = (x^5) * tanh(x^3) defined in R

root = find_root(f: f, seed: 2, error_level: 0.00000001)

if root.nil?
  puts 'Computed root of F(x) cannot be found or do not exist.'
else
  puts "Computed root of F(x) = (x^5)tanh(x^3) is #{root}. Exact root is 0.0."
end

# Example # 3
# F(x) = (x ^ 2) + 2
# Exact root is not defined
f = lambda { |x| (x ** 2) + 2 } # f(x) = x^2 + 2 defined in R

root = find_root(f: f, seed: 2, error_level: 0.00000001)

if root.nil?
  puts 'Computed root of F(x) = x^2 + 2 cannot be found or do not exist.'
else
  puts "Root of F(x) is #{root}."
end

# Example # 4
# F(x) = ln^x(x) + 1
# Exact root is not defined
f = lambda { |x| (Math.log(x) ** x) + 1 } # f(x) = ln^x(x) + 1 defined in R

root = find_root(f: f, seed: 2, error_level: 0.00000001)

if root.nil?
  puts 'Computed root of F(x) = ln^x(x) + 1 cannot be found or do not exist.'
else
  puts "Computed root of F(x) is #{root}."
end

# Example # 5
# F(x) = (x - 1) / (x + 1)
# Exact root is defined at 1
f = lambda { |x| (x - 1) * (x + 1) } # f(x) = (x - 1) * (x + 1) defined in R

root = find_root(f: f, seed: 2, error_level: 0.00000001)

if root.nil?
  puts 'Computed root of F(x) cannot be found or do not exist.'
else
  puts "Computed root of F(x) = (x - 1)(x + 1) is #{root}. Exact root is 1.0"
end