README.mdtex

# newton-raphson-method

> Find zeros of a function using [Newton's Method](https://en.wikipedia.org/wiki/Newton%27s_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`](https://github.com/scijs/modified-newton-raphson): A simple modification of Newton-Raphson that may exhibit improved convergence.
- [`newton-raphson`](https://www.npmjs.com/package/newton-raphson): A similar and lovely implementation that differs (only?) in requiring a first derivative.

## License

&copy; 2016 Scijs Authors. MIT License.

## Authors

Ricky Reusser

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