newton-raphson-method

> Find zeros of a function using Newton's Method

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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:

``javascript 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`

`bash $ 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.

`See Also`

- modified-newton-raphson: A simple modification of Newton-Raphson that may exhibit improved convergence. -newton-raphson`: A similar and lovely implementation that differs (only?) in requiring a first derivative.

License

Authors

Ricky Reusser

[npm-image]: https://badge.fury.io/js/newton-raphson-method.svg
[npm-url]: https://npmjs.org/package/newton-raphson-method
[travis-image]: https://travis-ci.org/scijs/newton-raphson-method.svg?branch=master
[travis-url]: https://travis-ci.org/scijs/newton-raphson-method
[daviddm-image]: https://david-dm.org/scijs/newton-raphson-method.svg?theme=shields.io
[daviddm-url]: https://david-dm.org/scijs/newton-raphson-method
[semistandard-image]: https://img.shields.io/badge/code%20style-semistandard-brightgreen.svg?style=flat-square
[semistandard-url]: https://github.com/Flet/semistandard

Example

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

javascript
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)

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:
-

: 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

.
-

: 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.

Authors

Ricky Reusser