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newton-raphson-method

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newton-raphson-method

Find zeros of a function using Newton's Method

Introduction

The Newton-Raphson method uses the tangent of a curve to iteratively approximate a zero of a function, f(x). This yields the update:

$x_{i + 1} = x_i -\frac{f(x_i)}{f'(x_i)}.$

Example

Consider the zero of (x + 2) * (x - 1) at x = 1:

var nr = require('newton-raphson-method');

function f (x) { return (x - 1) * (x + 2); }
function fp (x) { return (x - 1) + (x + 2); }

// Using the derivative:
nr(f, fp, 2)
// => 1.0000000000000000 (6 iterations)

// Using a numerical derivative:
nr(f, 2)
// => 1.0000000000000000 (6 iterations)

Installation

$ npm install newton-raphson-method

API

`require('newton-raphson-method')(f[, fp], x0[, options])`

Given a real-valued function of one variable, iteratively improves and returns a guess of a zero.

Parameters:

f: The numerical function of one variable of which to compute the zero.
fp (optional): The first derivative of f. If not provided, is computed numerically using a fourth order central difference with step size h.
x0: A number representing the intial guess of the zero.
options (optional): An object permitting the following options:
- tolerance (default: 1e-7): The tolerance by which convergence is measured. Convergence is met if |x[n+1] - x[n]| <= tolerance * |x[n+1]|.
- epsilon (default: 2.220446049250313e-16 (double-precision epsilon)): A threshold against which the first derivative is tested. Algorithm fails if |y'| < epsilon * |y|.
- maxIterations (default: 20): Maximum permitted iterations.
- h (default: 1e-4): Step size for numerical differentiation.
- verbose (default: false): Output additional information about guesses, convergence, and failure.

Returns: If convergence is achieved, returns an approximation of the zero. If the algorithm fails, returns false.

License

Authors

Ricky Reusser