npm package: @stdlib/math-base-tools-lucaspoly
Description: Evaluate a Lucas polynomial.
Author: The Stdlib Authors
Latest version: 0.2.3

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</details>

# Lucas Polynomial

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

> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-tools-lucaspoly
```

</section>

<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 -->

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* * *

## 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

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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

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