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‎lib/node_modules/@stdlib/math/base/special/legendre-polynomial/LICENSE‎

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‎lib/node_modules/@stdlib/math/base/special/legendre-polynomial/README.md‎

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<section class="intro">
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The [Legendre polynomials][legendre-polynomial] $P_n(x)$ are classical orthogonal polynomials which are solutions to the Legendre differential equation
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The [Legendre polynomials][legendre-polynomial] `P_n(x)` are classical orthogonal polynomials which are solutions to the [Legendre differential equation][legendre-differential-equation]
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<!-- <equation class="equation" label="eq:legendre_ode" align="center" raw="(1 - x^2) y'' - 2x y' + n(n+1) y = 0" alt="Legendre differential equation"> -->
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<!-- </equation> -->
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For **integer** non-negative $n$, they satisfy the three-term recurrence relation
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For **integer** non-negative `n`, they satisfy the [three-term recurrence relation][three-term-recurrence]
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<!-- <equation class="equation" label="eq:legendre_recurrence" align="center" raw="(n+1) P_{n+1}(x) = (2n+1)\, x\, P_n(x) - n\, P_{n-1}(x)" alt="Legendre polynomial recurrence relation"> -->
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<!-- </equation> -->
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with $P_0(x) = 1$ and $P_1(x) = x$. For **non-integer** $n$, the polynomial is generalized via the relation to the Gauss hypergeometric function
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with `P_0(x) = 1` and `P_1(x) = x`. For **non-integer** `n`, the polynomial is generalized via the relation to the [Gauss hypergeometric function][@stdlib/math/base/special/hyp2f1]
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<!-- <equation class="equation" label="eq:legendre_hyp2f1" align="center" raw="P_n(x) = {}_2F_1 \left( -n,\, n+1;\, 1;\, \tfrac{1-x}{2} \right)" alt="Legendre polynomial via hypergeometric function"> -->
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<!-- eslint no-undef: "error" -->
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```javascript
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var uniform = require( '@stdlib/random/base/uniform' );
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var logEachMap = require( '@stdlib/console/log-each-map' );
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var uniform = require( '@stdlib/random/array/uniform' );
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var legendrepoly = require( '@stdlib/math/base/special/legendre-polynomial' );
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var x;
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var n;
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var v;
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var i;
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for ( i = 0; i < 100; i++ ) {
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n = uniform( -50.0, 50.0 );
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x = uniform( -1.0, 1.0 );
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v = legendrepoly( n, x );
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console.log( 'P_%d(%d) = %d', n, x, v );
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}
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var n = uniform( 100, -50.0, 50.0 );
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var x = uniform( 100, -1.0, 1.0 );
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logEachMap( 'P_%0.4f(%0.4f) = %0.4f', n, x, legendrepoly );
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```
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</section>
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[legendre-polynomial]: https://en.wikipedia.org/wiki/Legendre_polynomials
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[legendre-differential-equation]: https://en.wikipedia.org/wiki/Legendre_polynomials
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[three-term-recurrence]: https://en.wikipedia.org/wiki/Legendre_polynomials#Recurrence_relations
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[@stdlib/math/base/special/hyp2f1]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/math/base/special/hyp2f1
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<!-- <related-links> -->
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<!-- </related-links> -->
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/math/base/special/legendre-polynomial/docs/repl.txt‎

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{{alias}}( n, x )
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Evaluates the Legendre polynomial of degree n.
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Parameters
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----------
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n: number
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Polynomial degree.
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x: number
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Point at which to evaluate the polynomial.
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Returns
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-------
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out: number
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Evaluated Legendre polynomial.
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Examples
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--------
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> var v = {{alias}}( 2, 0.5 )
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-0.125
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> v = {{alias}}( 1, 0.5 )
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0.5
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> v = {{alias}}( -1.5, 0.5 )
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~0.795
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> v = {{alias}}( NaN, 0.5 )
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NaN
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See Also
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--------

‎lib/node_modules/@stdlib/math/base/special/legendre-polynomial/examples/index.js‎

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'use strict';
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var uniform = require( '@stdlib/random/base/uniform' );
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var uniform = require( '@stdlib/random/array/uniform' );
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var logEachMap = require( '@stdlib/console/log-each-map' );
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var legendrepoly = require( './../lib' );
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var x;
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var n;
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var v;
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var i;
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var n = uniform( 100, -50.0, 50.0 );
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var x = uniform( 100, -1.0, 1.0 );
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for ( i = 0; i < 100; i++ ) {
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n = uniform( -50.0, 50.0 );
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x = uniform( -1.0, 1.0 );
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v = legendrepoly( n, x );
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console.log( 'P_%d(%d) = %d', n, x, v );
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}
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logEachMap( 'P_%0.4f(%0.4f) = %0.4f', n, x, legendrepoly );

‎lib/node_modules/@stdlib/math/base/special/legendre-polynomial/lib/main.js‎

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var isnan = require( '@stdlib/assert/is-nan' );
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var hyp2f1 = require( '@stdlib/math/base/special/hyp2f1' );
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var floor = require( '@stdlib/math/base/special/floor' );
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var gamma = require( '@stdlib/math/base/special/gamma' );
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var abs = require( '@stdlib/math/base/special/abs' );
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var floor = require( '@stdlib/math/base/special/floor' );
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// MAIN //

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