npm package: @stdlib/stats-base-dists-cauchy-entropy
Description: Cauchy distribution differential entropy.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Entropy

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Cauchy][cauchy-distribution] distribution [differential entropy][entropy].

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<section class="intro">

The [differential entropy][entropy] for a [Cauchy][cauchy-distribution] random variable with location parameter `x0` and scale parameter `Ɣ > 0` is

<!-- <equation class="equation" label="eq:cauchy_entropy" align="center" raw="h\left( X \right) = \log(\gamma)\,+\,\log(4\,\pi)" alt="Differential entropy for a Cauchy distribution."> -->

<div class="equation" align="center" data-raw-text="h\left( X \right) = \log(\gamma)\,+\,\log(4\,\pi)" data-equation="eq:cauchy_entropy">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/cauchy/entropy/docs/img/equation_cauchy_entropy.svg" alt="Differential entropy for a Cauchy distribution.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-cauchy-entropy
```

</section>

<section class="usage">

## Usage

```javascript
var entropy = require( '@stdlib/stats-base-dists-cauchy-entropy' );
```

#### entropy( x0, gamma )

Returns the [differential entropy][entropy] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma` (in [nats][nats]).

```javascript
var v = entropy( 10.0, 5.0 );
// returns ~4.14

v = entropy( 7.0, 2.0 );
// returns ~3.224
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var v = entropy( NaN, 5.0 );
// returns NaN

v = entropy( 20.0, NaN );
// returns NaN
```

If provided `gamma <= 0`, the function returns `NaN`.

```javascript
var v = entropy( 1.0, -1.0 );
// returns NaN

v = entropy( 1.0, 0.0 );
// returns NaN
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var entropy = require( '@stdlib/stats-base-dists-cauchy-entropy' );

var opts = {
    'dtype': 'float64'
};
var gamma = uniform( 10, EPS, 10.0, opts );
var x0 = uniform( 10, 0.0, 100.0, opts );

logEachMap( 'x0: %0.4f, γ: %0.4f, h(X;x0,γ): %0.4f', x0, gamma, entropy );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/cauchy/entropy.h"
```

#### stdlib_base_dists_cauchy_entropy( x0, gamma )

Evaluates the [differential entropy][entropy] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma` (in [nats][nats]).

```c
double out = stdlib_base_dists_cauchy_entropy( 10.0, 5.0 );
// returns ~4.14
```

The function accepts the following arguments:

-   **x0**: `[in] double` location parameter.
-   **gamma**: `[in] double` scale parameter.

```c
double stdlib_base_dists_cauchy_entropy( const double x0, const double gamma );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/cauchy/entropy.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double gamma;
    double x0;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x0 = random_uniform( 0.0, 100.0 );
        gamma = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        y = stdlib_base_dists_cauchy_entropy( x0, gamma );
        printf( "x0: %lf, γ: %lf, h(X;x0,γ): %lf\n", x0, gamma, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-cauchy-entropy/main/LICENSE

[cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution

[entropy]: https://en.wikipedia.org/wiki/Entropy_%28information_theory%29

[nats]: https://en.wikipedia.org/wiki/Nat_%28unit%29

</section>

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