rical

RICAL

Root and Inverse CALculator

Using Newton-Raphson or Secant methods, and within an arbitrary error:

• Computes the values of a root of an arbitrary function f(x, a, b, ...)
• Computes the value of inverse function f^-1(y, a, b, ...) = x at point y
• Handles functions f(x, a, b, ...), when fixed f_args = a, b, ... are set
• Handles additional function arguments either as Array or Hash (matching positional or named parameters)

The Newton-Raphson requires:

• one "close enough" estimate - x₀ - of the root or inverse value
• the derivative function f'(x, a, b, ...)
• f_args containing a, b, ... when existing
• f'_args containing a, b, ... when existing (Note f_args are independent of f'_args)

The Secant method requires:

• two "close enough" estimates - x₀, x₁ - of the root or inverse value
• f_args containing a, b, ... when existing

If the estimates are not "close enough" (and what that means depends on f(x, a, b, ...)), the methods may not converge to a solution. This may also happen for a "low" number of iterations and/or a "low" error limit. When no convergence is possible on the set conditions, a NoConvergenceError is raised.

Installation

Add this line to your application's Gemfile:

gem 'rical'

And then execute:

$ bundle

Or install it yourself as:

$ gem install rical

Usage examples

• Set function(s) as lambdas

fx1  = -> (x) { x**2 - 2 }
dfx1 = -> (x) { 2*x }

or set function(s) as Proc

fx2 = Proc.new do |x, a, b|
  a*x**2 + b
end

dfx2 = Proc.new do |x, a|
  2*a*x
end

or set function(s) as existing method(s)

def f3(x, a:, b:)
  a*x**2 + b
end

fx3 = method(:f3)

• Call root_for of inverse_for passing function(s) and estimate(s):

Newton-Raphson's Method

Rical.root_for f: fx1, df: dfx1, x0: 1.0, method: :newton_raphson    # :newton_raphson aliased to :n, :newton
Rical.inverse_for f: fx1, df: dfx1, x0: 1.0, y: 2.0, method: :newton

# with fargs and dfargs
Rical.root_for f: fx2, fargs: [1, -2], df: dfx2, dfargs: 2, x0: 1.0, method: :newton
Rical.root_for f: fx3, fargs: { a: 1, b: -2 }, df: dfx3, dfargs: { a: 2 }, x0: 1.0, method: :newton

Secant's Method

Rical.root_for f: fx, x0: 0.0, x1: 1.0, method: :secant              # :secant aliased to :sec, :s
Rical.inverse_for f: fx, x0: 0.0, x1: 1.0, y: 2.0, method: :secant

# with fargs and dfargs
Rical.inverse_for f: fx, fargs: [1, -2], x0: 0.0, x1: 1.0, y: 2.0, method: :secant

• Change number of iterations - num - or maximum allowable error - err - if needed:

Rical.root_for f: fx, ......, num: 1e3, err: 1e-12      # defaults num: 100, err: 1e-6
Rical.inverse_for f: fx, ......, num: 1e3, err: 1e-12   # defaults num: 100, err: 1e-6

Contributing

• Fork it ( https://github.com/[my-github-username]/rical/fork )
• Create your feature branch (git checkout -b my-new-feature)
• Commit your changes (git commit -am 'Add some feature')
• Push to the branch (git push origin my-new-feature)
• Create a new Pull Request