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# Lucas Polynomial

> Evaluate a [Lucas polynomial][fibonacci-polynomials].

<section class="intro">

A [Lucas polynomial][fibonacci-polynomials] is expressed according to the following recurrence relation

<!-- <equation class="equation" label="eq:lucas_polynomial" align="center" raw="L_n(x) = \begin{cases}2 & \textrm{if}\ n = 0\\x & \textrm{if}\ n = 1\\x \cdot L_{n-1}(x) + L_{n-2}(x) & \textrm{if}\ n \geq 2\end{cases}" alt="Lucas polynomial."> -->

<div class="equation" align="center" data-raw-text="L_n(x) = \begin{cases}2 &amp; \textrm{if}\ n = 0\\x &amp; \textrm{if}\ n = 1\\x \cdot L_{n-1}(x) + L_{n-2}(x) &amp; \textrm{if}\ n \geq 2\end{cases}" data-equation="eq:lucas_polynomial">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/lucaspoly/docs/img/equation_lucas_polynomial.svg" alt="Lucas polynomial.">
    <br>
</div> -->

<!-- </equation> -->

Alternatively, if `L(n,k)` is the coefficient of `x^k` in `L_n(x)`, then

<!-- <equation class="equation" label="eq:lucas_polynomial_sum" align="center" raw="L_n(x) = \sum_{k = 0}^n L(n,k) x^k" alt="Lucas polynomial expressed as a sum."> -->

<!-- <div class="equation" align="center" data-raw-text="L_n(x) = \sum_{k = 0}^n L(n,k) x^k" data-equation="eq:lucas_polynomial_sum">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/lucaspoly/docs/img/equation_lucas_polynomial_sum.svg" alt="Lucas polynomial expressed as a sum.">
    <br>
</div> -->

<!-- </equation> -->

We can extend [Lucas polynomials][fibonacci-polynomials] to negative `n` using the identity

<!-- <equation class="equation" label="eq:negalucas_polynomial" align="center" raw="L_{-n}(x) = (-1)^{n} L_n(x)" alt="NegaLucas polynomial."> -->

<!-- <div class="equation" align="center" data-raw-text="L_{-n}(x) = (-1)^{n} L_n(x)" data-equation="eq:negalucas_polynomial">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/lucaspoly/docs/img/equation_negalucas_polynomial.svg" alt="NegaLucas polynomial.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' );
```

#### lucaspoly( n, x )

Evaluates a [Lucas polynomial][fibonacci-polynomials] at a value `x`.

```javascript
var v = lucaspoly( 5, 2.0 ); // => 2^5 + 5*2^3 + 5*2
// returns 82.0
```

#### lucaspoly.factory( n )

Uses code generation to generate a `function` for evaluating a [Lucas polynomial][fibonacci-polynomials].

```javascript
var polyval = lucaspoly.factory( 5 );

var v = polyval( 1.0 ); // => 1^5 + 5*1^3 + 5
// returns 11.0

v = polyval( 2.0 ); // => 2^5 + 5*2^3 + 5*2
// returns 82.0
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   For hot code paths, a compiled function will be more performant than `lucaspoly()`.
-   While code generation can boost performance, its use may be problematic in browser contexts enforcing a strict [content security policy][mdn-csp] (CSP). If running in or targeting an environment with a CSP, avoid using code generation.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' );

var i;

// Compute the negaLucas and Lucas numbers...
for ( i = -76; i < 77; i++ ) {
    console.log( 'L_%d = %d', i, lucaspoly( i, 1.0 ) );
}
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/tools/evalpoly`][@stdlib/math/base/tools/evalpoly]</span><span class="delimiter">: </span><span class="description">evaluate a polynomial using double-precision floating-point arithmetic.</span>
-   <span class="package-name">[`@stdlib/math/base/tools/fibpoly`][@stdlib/math/base/tools/fibpoly]</span><span class="delimiter">: </span><span class="description">evaluate a Fibonacci polynomial.</span>

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[fibonacci-polynomials]: https://en.wikipedia.org/wiki/Fibonacci_polynomials

[mdn-csp]: https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP

<!-- <related-links> -->

[@stdlib/math/base/tools/evalpoly]: https://github.com/stdlib-js/math/tree/main/base/tools/evalpoly

[@stdlib/math/base/tools/fibpoly]: https://github.com/stdlib-js/math/tree/main/base/tools/fibpoly

<!-- </related-links> -->

</section>

<!-- /.links -->
