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

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# Fibonacci 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 [Fibonacci polynomial][fibonacci-polynomials].

<section class="intro">

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

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

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

<!-- </equation> -->

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

<!-- <equation class="equation" label="eq:fibonacci_polynomial_combinatoric" align="center" raw="F_n(x) = \sum_{k = 0}^n F(n,k) x^k" alt="Combinatoric interpretation of a Fibonacci polynomial."> -->

<!-- <div class="equation" align="center" data-raw-text="F_n(x) = \sum_{k = 0}^n F(n,k) x^k" data-equation="eq:fibonacci_polynomial_combinatoric">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/fibpoly/docs/img/equation_fibonacci_polynomial_combinatoric.svg" alt="Combinatoric interpretation of a Fibonacci polynomial.">
    <br>
</div> -->

<!-- </equation> -->

where

<!-- <equation class="equation" label="eq:fibonacci_polynomial_coefficients" align="center" raw="F(n,k) = {{\frac{n+k-1}{2}} \choose {k}}" alt="Fibonacci polynomial coefficients."> -->

<!-- <div class="equation" align="center" data-raw-text="F(n,k) = {{\frac{n+k-1}{2}} \choose {k}}" data-equation="eq:fibonacci_polynomial_coefficients">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/fibpoly/docs/img/equation_fibonacci_polynomial_coefficients.svg" alt="Fibonacci polynomial coefficients.">
    <br>
</div> -->

<!-- </equation> -->

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

<!-- <equation class="equation" label="eq:negafibonacci_polynomial" align="center" raw="F_{-n}(x) = (-1)^{n-1} F_n(x)" alt="NegaFibonacci polynomial."> -->

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

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### fibpoly( n, x )

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

```javascript
var v = fibpoly( 5, 2.0 ); // => 2^4 + 3*2^2 + 1
// returns 29.0
```

#### fibpoly.factory( n )

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

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

var v = polyval( 1.0 ); // => 1^4 + 3*1^2 + 1
// returns 5.0

v = polyval( 2.0 ); // => 2^4 + 3*2^2 + 1
// returns 29.0
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   For hot code paths, a compiled function will be more performant than `fibpoly()`.
-   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 fibpoly = require( '@stdlib/math-base-tools-fibpoly' );

var i;

// Compute the negaFibonacci and Fibonacci numbers...
for ( i = -77; i < 78; i++ ) {
    console.log( 'F_%d = %d', i, fibpoly( 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/lucaspoly`][@stdlib/math/base/tools/lucaspoly]</span><span class="delimiter">: </span><span class="description">evaluate a Lucas polynomial.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## 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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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-tools-fibpoly/main/LICENSE

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

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

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