https://github.com/stdlib-js/math-base-special-gammainc
Incomplete gamma function.
https://github.com/stdlib-js/math-base-special-gammainc
approximation function gamma incomplete-gamma javascript math mathematics node node-js nodejs number scalar special special-function stdlib
Last synced: about 1 month ago
JSON representation
Incomplete gamma function.
- Host: GitHub
- URL: https://github.com/stdlib-js/math-base-special-gammainc
- Owner: stdlib-js
- License: apache-2.0
- Created: 2021-06-15T17:34:37.000Z (over 4 years ago)
- Default Branch: main
- Last Pushed: 2025-07-20T04:46:02.000Z (3 months ago)
- Last Synced: 2025-07-20T06:39:05.395Z (3 months ago)
- Topics: approximation, function, gamma, incomplete-gamma, javascript, math, mathematics, node, node-js, nodejs, number, scalar, special, special-function, stdlib
- Language: JavaScript
- Homepage: https://github.com/stdlib-js/stdlib
- Size: 2.31 MB
- Stars: 1
- Watchers: 3
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Citation: CITATION.cff
- Security: SECURITY.md
Awesome Lists containing this project
README
About stdlib...
We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.
The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.
When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.
To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!
# gammainc
[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url]
> [Incomplete gamma function][incomplete-gamma-function].
Evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function]:
```math
P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt
```
The function can also be used to evaluate the regularized upper incomplete gamma function, which is defined as follows:
```math
Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt
```
The two functions have the relationship `Q(x,a) = 1 - P(x,a)`.
In addition, this routine can be used to evaluate the _unregularized_ gamma functions. The range of above functions is `[0,1]`, which is not the case for the _unregularized_ versions. The unregularized lower incomplete gamma function is defined as
```math
\gamma(a,x) = \int_0^x t^{a-1} e^{-t} \; dt
```
and the upper unregularized incomplete gamma function is
```math
\Gamma(a,x)= \int_x^\infty t^{a-1} e^{-t} \; dt
```
The relationship between the two functions is `Γ(a,x) = γ(a,x) + Γ(a)`.
## Installation
```bash
npm install @stdlib/math-base-special-gammainc
```
Alternatively,
- To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).
- If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).
- For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).
The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.
To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.
## Usage
```javascript
var gammainc = require( '@stdlib/math-base-special-gammainc' );
```
#### gammainc( x, s\[, regularized\[, upper ]] )
By default, evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `s`. The third and fourth parameters of the function can be used to specify whether instead to evaluate the non-regularized and/or upper incomplete gamma functions, respectively.
```javascript
var y = gammainc( 6.0, 2.0 );
// returns ~0.9826
y = gammainc( 1.0, 2.0, true, true );
// returns ~0.7358
y = gammainc( 7.0, 5.0 );
// returns ~0.8270
y = gammainc( 7.0, 5.0, false );
// returns ~19.8482
```
If provided `NaN` as any argument, the function returns `NaN`.
```javascript
var y = gammainc( NaN, 1.0 );
// returns NaN
y = gammainc( 1.0, NaN );
// returns NaN
```
If provided a negative `x`, the function returns `NaN`.
```javascript
var y = gammainc( -2.0, 2.0 );
// returns NaN
```
If provided a nonpositive `s`, the function returns `NaN`.
```javascript
var y = gammainc( 2.0, -1.0 );
// returns NaN
y = gammainc( 2.0, 0.0 );
// returns NaN
```
## Examples
```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var gammainc = require( '@stdlib/math-base-special-gammainc' );
var opts = {
'dtype': 'float64'
};
var x = uniform( 100, 0.0, 10.0, opts );
var s = uniform( 100, 0.0, 10.0, opts );
logEachMap( 'x: %0.4f, \t s: %0.4f, \t f(x,s): %0.4f', x, s, gammainc );
```
* * *
## C APIs
### Usage
```c
#include "stdlib/math/base/special/gammainc.h"
```
#### stdlib_base_gammainc( x, a, regularized, upper )
Evaluates the [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `a`.
```c
double out = stdlib_base_gammainc( 0.0, 1.0, true, false );
// returns 0.0
out = stdlib_base_gammainc( 6.0, 2.0, true, false );
// returns ~0.9826
```
The function accepts the following arguments:
- **x**: `[in] double` input value.
- **a**: `[in] double` input value.
- **regularized**: `[in] bool` boolean indicating whether to evaluate a regularized incomplete gamma function.
- **upper**: `[in] bool` boolean indicating whether to evaluate the upper incomplete gamma function.
```c
double stdlib_base_gammainc( const double x, const double a, const bool regularized, const bool upper );
```
### Examples
```c
#include "stdlib/math/base/special/gammainc.h"
#include
#include
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 x;
double a;
double y;
int i;
for ( i = 0; i < 10; i++ ) {
x = random_uniform( 0.0, 1000.0 );
a = random_uniform( 1.0, 1000.0 );
y = stdlib_base_gammainc( x, a, true, false );
printf( "x: %lf, s: %lf, f(x,s): %lf\n", x, a, y );
}
}
```
* * *
## See Also
- [`@stdlib/math-base/special/gamma`][@stdlib/math/base/special/gamma]: gamma function.
- [`@stdlib/math-base/special/gamma1pm1`][@stdlib/math/base/special/gamma1pm1]: compute gamma(x+1) - 1.
- [`@stdlib/math-base/special/gammaincinv`][@stdlib/math/base/special/gammaincinv]: inverse incomplete gamma function.
- [`@stdlib/math-base/special/gammaln`][@stdlib/math/base/special/gammaln]: natural logarithm of the gamma function.
* * *
## 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].
#### Community
[![Chat][chat-image]][chat-url]
---
## Copyright
Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].
[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-gammainc.svg
[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-gammainc
[test-image]: https://github.com/stdlib-js/math-base-special-gammainc/actions/workflows/test.yml/badge.svg?branch=main
[test-url]: https://github.com/stdlib-js/math-base-special-gammainc/actions/workflows/test.yml?query=branch:main
[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-gammainc/main.svg
[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-gammainc?branch=main
[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg
[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im
[stdlib]: https://github.com/stdlib-js/stdlib
[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors
[umd]: https://github.com/umdjs/umd
[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules
[deno-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/deno
[deno-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/deno/README.md
[umd-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/umd
[umd-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/umd/README.md
[esm-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/esm
[esm-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/esm/README.md
[branches-url]: https://github.com/stdlib-js/math-base-special-gammainc/blob/main/branches.md
[incomplete-gamma-function]: https://en.wikipedia.org/wiki/Incomplete_gamma_function
[@stdlib/math/base/special/gamma]: https://github.com/stdlib-js/math-base-special-gamma
[@stdlib/math/base/special/gamma1pm1]: https://github.com/stdlib-js/math-base-special-gamma1pm1
[@stdlib/math/base/special/gammaincinv]: https://github.com/stdlib-js/math-base-special-gammaincinv
[@stdlib/math/base/special/gammaln]: https://github.com/stdlib-js/math-base-special-gammaln