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!

ellipj

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

> Compute the [Jacobi elliptic functions][jacobi-elliptic] sn, cn, and dn.

The [Jacobi elliptic functions][jacobi-elliptic] may be defined as the inverse of the [incomplete elliptic integral of the first kind][incomplete-elliptic]. Accordingly, they compute the value φ which satisfies the equation

where the parameter m is related to the modulus k by m = k^2.

Installation

``bash npm install @stdlib/math-base-special-ellipj`

`Usage`

`javascript var ellipj = require( '@stdlib/math-base-special-ellipj' );`

#### ellipj( u, m )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions sn, cn, and dn, and the Jacobi amplitude am.

`javascript var v = ellipj( 0.3, 0.5 ); // returns [ ~0.293, ~0.956, ~0.978, ~0.298 ]

v = ellipj( 0.0, 0.0 ); // returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

v = ellipj( Infinity, 1.0 ); // returns [ ~1.0, ~0.0, ~0.0, ~1.571 ]

v = ellipj( 0.0, -2.0 ); // returns [ ~0.0, ~1.0, ~1.0, NaN ]

v = ellipj( NaN, NaN ); // returns [ NaN, NaN, NaN, NaN ]`

#### ellipj.assign( u, m, out, stride, offset )

Computes the Jacobi elliptic functions sn, cn, dn, and Jacobi amplitude am and assigns results to a provided output array.

`javascript var Float64Array = require( '@stdlib/array-float64' );

var out = new Float64Array( 4 );

var v = ellipj.assign( 0.0, 0.0, out, 1, 0 ); // returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

var bool = ( v === out ); // returns true`

#### ellipj.sn( u, m )

Computes the Jacobi elliptic function sn of value u with modulus m.

`javascript var v = ellipj.sn( 0.3, 0.5 ); // returns ~0.293`

#### ellipj.cn( u, m )

Computes the Jacobi elliptic function cn of value u with modulus m.

`javascript var v = ellipj.cn( 0.3, 0.5 ); // returns ~0.956`

#### ellipj.dn( u, m )

Computes the Jacobi elliptic function dn of value u with modulus m.

`javascript var v = ellipj.dn( 0.3, 0.5 ); // returns ~0.978`

#### ellipj.am( u, m )

Computes the Jacobi amplitude am of value u with modulus m.

`javascript var v = ellipj.am( 0.3, 0.5 ); // returns ~0.298

v = ellipj.am( 0.3, 2.0 ); // returns NaN`

Although sn, cn, and dn may be computed for -∞ < m < ∞, the domain of am is 0 ≤ m ≤ 1. For m < 0 or m > 1, the function returns NaN.

`Notes`

- Functions sn, cn, and dn are valid for -∞ < m < ∞. Values for m < 0 or m > 1 are computed in terms of Jacobi elliptic functions with 0 < m < 1via the transformations outlined in Equations 16.13 and 16.15 from _The Handbook of Mathematical Functions_ (Abramowitz and Stegun). - If more than one ofsn, cn, dn, or am is to be computed, preferring using ellipj to compute all four values simultaneously.

`Examples`

`javascript var linspace = require( '@stdlib/array-base-linspace' ); var ellipk = require( '@stdlib/math-base-special-ellipk' ); var ellipj = require( '@stdlib/math-base-special-ellipj' );

var m = 0.7; var u = linspace( 0.0, ellipk( m ), 100 );

var out; var i; for ( i = 0; i < 100; i++ ) { out = ellipj( u[ i ], m ); console.log( 'sn(%d, %d) = %d', u[ i ], m, out[ 0 ] ); console.log( 'cn(%d, %d) = %d', u[ i ], m, out[ 1 ] ); console.log( 'dn(%d, %d) = %d', u[ i ], m, out[ 2 ] ); console.log( 'am(%d, %d) = %d', u[ i ], m, out[ 3 ] ); }`

*

`C APIs`

`$3`

`c #include "stdlib/math/base/special/ellipj.h"`

#### stdlib_base_ellipj( x, m, &sn, &cn, &dn, &am )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions sn, cn, and dn, and the Jacobi amplitude am.

`c double sn; double cn; double dn; double am;

stdlib_base_ellipj( 0.3, 0.5, &sn, &cn, &dn, &am );`

The function accepts the following arguments:

- x: [in] doubleinput value. - m:[in] double modulus m, equivalent to k². - sn:[out] double*destination for the sine amplitude. - cn:[out] double*destination for the cosine amplitude. - dn:[out] double*destination for the delta amplitude. - am:[out] double* destination for the Jacobi amplitude.

`c void stdlib_base_ellipj( const double u, const double m, double sn, double cn, double dn, double am );`

`$3`

`c #include "stdlib/math/base/special/ellipj.h" #include #include

int main( void ) { double sn; double cn; double dn; double am; double x; int i;

for ( i = 0; i < 100; i++ ) { x = 2.0 * ( (double)rand() / (double)RAND_MAX ); stdlib_base_ellipj( x, 0.7, &sn, &cn, &dn, &am ); printf( "x: %lf, m: %lf => sn: %lf, cn: %lf, dn: %lf, am: %lf\n", x, 0.7, sn, cn, dn, am ); } }`

*

`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].

*

`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]

---

`License`

See [LICENSE][stdlib-license].

`Copyright`

[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-ellipj.svg [npm-url]: https://npmjs.org/package/@stdlib/math-base-special-ellipj

[test-image]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml/badge.svg?branch=v0.4.1 [test-url]: https://github.com/stdlib-js/math-base-special-ellipj/actions/workflows/test.yml?query=branch:v0.4.1

[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-ellipj/main.svg [coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-ellipj?branch=main

[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg [chat-url]: https://stdlib.zulipchat.com

[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-ellipj/tree/deno [deno-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/deno/README.md [umd-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/umd [umd-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/umd/README.md [esm-url]: https://github.com/stdlib-js/math-base-special-ellipj/tree/esm [esm-readme]: https://github.com/stdlib-js/math-base-special-ellipj/blob/esm/README.md [branches-url]: https://github.com/stdlib-js/math-base-special-ellipj/blob/main/branches.md

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-ellipj/main/LICENSE

[jacobi-elliptic]: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions

[incomplete-elliptic]: 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/ellipe]: https://www.npmjs.com/package/@stdlib/math-base-special-ellipe

[@stdlib/math/base/special/ellipk]: https://www.npmjs.com/package/@stdlib/math-base-special-ellipk


  
    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!

`ellipj`

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

> Compute the [Jacobi elliptic functions][jacobi-elliptic] sn, cn, and dn.

The [Jacobi elliptic functions][jacobi-elliptic] may be defined as the inverse of the [incomplete elliptic integral of the first kind][incomplete-elliptic]. Accordingly, they compute the value φ which satisfies the equation

where the parameter m is related to the modulus k by m = k^2.

`Installation`

``bash npm install @stdlib/math-base-special-ellipj `

`Usage`

`javascript var ellipj = require( '@stdlib/math-base-special-ellipj' ); `

#### ellipj( u, m )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions sn, cn, and dn, and the Jacobi amplitude am.

`javascript var v = ellipj( 0.3, 0.5 ); // returns [ ~0.293, ~0.956, ~0.978, ~0.298 ]

v = ellipj( 0.0, 0.0 ); // returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

v = ellipj( Infinity, 1.0 ); // returns [ ~1.0, ~0.0, ~0.0, ~1.571 ]

v = ellipj( 0.0, -2.0 ); // returns [ ~0.0, ~1.0, ~1.0, NaN ]

v = ellipj( NaN, NaN ); // returns [ NaN, NaN, NaN, NaN ] `

#### ellipj.assign( u, m, out, stride, offset )

Computes the Jacobi elliptic functions sn, cn, dn, and Jacobi amplitude am and assigns results to a provided output array.

`javascript var Float64Array = require( '@stdlib/array-float64' );

var out = new Float64Array( 4 );

var v = ellipj.assign( 0.0, 0.0, out, 1, 0 ); // returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

var bool = ( v === out ); // returns true `

#### ellipj.sn( u, m )

Computes the Jacobi elliptic function sn of value u with modulus m.

`javascript var v = ellipj.sn( 0.3, 0.5 ); // returns ~0.293 `

#### ellipj.cn( u, m )

Computes the Jacobi elliptic function cn of value u with modulus m.

`javascript var v = ellipj.cn( 0.3, 0.5 ); // returns ~0.956 `

#### ellipj.dn( u, m )

Computes the Jacobi elliptic function dn of value u with modulus m.

`javascript var v = ellipj.dn( 0.3, 0.5 ); // returns ~0.978 `

#### ellipj.am( u, m )

Computes the Jacobi amplitude am of value u with modulus m.

`javascript var v = ellipj.am( 0.3, 0.5 ); // returns ~0.298

v = ellipj.am( 0.3, 2.0 ); // returns NaN `

Although sn, cn, and dn may be computed for -∞ < m < ∞, the domain of am is 0 ≤ m ≤ 1. For m < 0 or m > 1, the function returns NaN.

`Notes`

- Functions sn, cn, and dn are valid for -∞ < m < ∞. Values for m < 0 or m > 1 are computed in terms of Jacobi elliptic functions with 0 < m < 1 via the transformations outlined in Equations 16.13 and 16.15 from _The Handbook of Mathematical Functions_ (Abramowitz and Stegun). - If more than one of sn, cn, dn, or am is to be computed, preferring using ellipj to compute all four values simultaneously.

`Examples`

`javascript var linspace = require( '@stdlib/array-base-linspace' ); var ellipk = require( '@stdlib/math-base-special-ellipk' ); var ellipj = require( '@stdlib/math-base-special-ellipj' );

var m = 0.7; var u = linspace( 0.0, ellipk( m ), 100 );

var out; var i; for ( i = 0; i < 100; i++ ) { out = ellipj( u[ i ], m ); console.log( 'sn(%d, %d) = %d', u[ i ], m, out[ 0 ] ); console.log( 'cn(%d, %d) = %d', u[ i ], m, out[ 1 ] ); console.log( 'dn(%d, %d) = %d', u[ i ], m, out[ 2 ] ); console.log( 'am(%d, %d) = %d', u[ i ], m, out[ 3 ] ); } `

*

`C APIs`

`$3`

`c #include "stdlib/math/base/special/ellipj.h" `

#### stdlib_base_ellipj( x, m, &sn, &cn, &dn, &am )

Computes the [Jacobi elliptic functions][jacobi-elliptic] functions sn, cn, and dn, and the Jacobi amplitude am.

`c double sn; double cn; double dn; double am;

stdlib_base_ellipj( 0.3, 0.5, &sn, &cn, &dn, &am ); `

The function accepts the following arguments:

- x: [in] double input value. - m: [in] double modulus m, equivalent to k². - sn: [out] double* destination for the sine amplitude. - cn: [out] double* destination for the cosine amplitude. - dn: [out] double* destination for the delta amplitude. - am: [out] double* destination for the Jacobi amplitude.

`c void stdlib_base_ellipj( const double u, const double m, double sn, double cn, double dn, double am ); `

`$3`

`c #include "stdlib/math/base/special/ellipj.h" #include #include

int main( void ) { double sn; double cn; double dn; double am; double x; int i;

for ( i = 0; i < 100; i++ ) { x = 2.0 * ( (double)rand() / (double)RAND_MAX ); stdlib_base_ellipj( x, 0.7, &sn, &cn, &dn, &am ); printf( "x: %lf, m: %lf => sn: %lf, cn: %lf, dn: %lf, am: %lf\n", x, 0.7, sn, cn, dn, am ); } } `