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https://github.com/stdlib-js/math-base-special-ellipe
Compute the complete elliptic integral of the second kind.
https://github.com/stdlib-js/math-base-special-ellipe
elliptic javascript math mathematics node node-js nodejs number special stdlib
Last synced: about 1 month ago
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Compute the complete elliptic integral of the second kind.
- Host: GitHub
- URL: https://github.com/stdlib-js/math-base-special-ellipe
- Owner: stdlib-js
- License: apache-2.0
- Created: 2021-06-15T17:01:20.000Z (over 3 years ago)
- Default Branch: main
- Last Pushed: 2024-11-01T09:02:35.000Z (about 2 months ago)
- Last Synced: 2024-11-08T11:20:16.858Z (about 1 month ago)
- Topics: elliptic, javascript, math, mathematics, node, node-js, nodejs, number, special, stdlib
- Language: JavaScript
- Homepage: https://github.com/stdlib-js/stdlib
- Size: 947 KB
- Stars: 2
- 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!
# ellipe
[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url]
> Compute the [complete elliptic integral of the second kind][elliptic-integral].
The [complete elliptic integral of the second kind][elliptic-integral] is defined as
```math
E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta
```
where the parameter `m` is related to the modulus `k` by `m = k^2`.
## Installation
```bash
npm install @stdlib/math-base-special-ellipe
```
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 ellipe = require( '@stdlib/math-base-special-ellipe' );
```
#### ellipe( m )
Computes the [complete elliptic integral of the second kind][elliptic-integral].
```javascript
var v = ellipe( 0.5 );
// returns ~1.351
v = ellipe( -1.0 );
// returns ~1.910
v = ellipe( 2.0 );
// returns NaN
v = ellipe( Infinity );
// returns NaN
v = ellipe( -Infinity );
// returns NaN
v = ellipe( NaN );
// returns NaN
```
## Notes
- This function is valid for `-∞ < m <= 1`.
## Examples
```javascript
var randu = require( '@stdlib/random-base-randu' );
var ellipe = require( '@stdlib/math-base-special-ellipe' );
var m;
var i;
for ( i = 0; i < 100; i++ ) {
m = -1.0 + ( randu() * 2.0 );
console.log( 'ellipe(%d) = %d', m, ellipe( m ) );
}
```
* * *
## C APIs
### Usage
```c
#include "stdlib/math/base/special/ellipe.h"
```
#### stdlib_base_ellipe( m )
Computes the [complete elliptic integral of the second kind][elliptic-integral].
```c
double out = stdlib_base_ellipe( 0.5 );
// returns ~1.351
out = stdlib_base_ellipe( -1.0 );
// returns ~1.910
```
The function accepts the following arguments:
- **x**: `[in] double` input value.
```c
double stdlib_base_ellipe( const double m );
```
### Examples
```c
#include "stdlib/math/base/special/ellipe.h"
#include
#include
int main( void ) {
double m;
double v;
int i;
for ( i = 0; i < 100; i++ ) {
m = -1.0 + ( ( (double)rand() / (double)RAND_MAX ) * 2.0 );
v = stdlib_base_ellipe( m );
printf( "ellipe(%lf) = %lf\n", m, v );
}
}
```
* * *
## References
- Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." _Celestial Mechanics and Dynamical Astronomy_ 105 (4): 305. doi:[10.1007/s10569-009-9228-z][@fukushima:2009a].
- Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." _Journal of Computational and Applied Mathematics_ 282 (July): 71–76. doi:[10.1016/j.cam.2014.12.038][@fukushima:2015a].
* * *
## See Also
- [`@stdlib/math-base/special/ellipj`][@stdlib/math/base/special/ellipj]: compute the Jacobi elliptic functions sn, cn, and dn.
- [`@stdlib/math-base/special/ellipk`][@stdlib/math/base/special/ellipk]: compute the complete elliptic integral of the first kind.
* * *
## 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 © 2016-2024. The Stdlib [Authors][stdlib-authors].
[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-ellipe.svg
[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-ellipe
[test-image]: https://github.com/stdlib-js/math-base-special-ellipe/actions/workflows/test.yml/badge.svg?branch=main
[test-url]: https://github.com/stdlib-js/math-base-special-ellipe/actions/workflows/test.yml?query=branch:main
[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-ellipe/main.svg
[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-ellipe?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-ellipe/tree/deno
[deno-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/deno/README.md
[umd-url]: https://github.com/stdlib-js/math-base-special-ellipe/tree/umd
[umd-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/umd/README.md
[esm-url]: https://github.com/stdlib-js/math-base-special-ellipe/tree/esm
[esm-readme]: https://github.com/stdlib-js/math-base-special-ellipe/blob/esm/README.md
[branches-url]: https://github.com/stdlib-js/math-base-special-ellipe/blob/main/branches.md
[elliptic-integral]: https://en.wikipedia.org/wiki/Elliptic_integral
[@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z
[@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038
[@stdlib/math/base/special/ellipj]: https://github.com/stdlib-js/math-base-special-ellipj
[@stdlib/math/base/special/ellipk]: https://github.com/stdlib-js/math-base-special-ellipk