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Lucas Polynomial

Evaluate a Lucas polynomial.

A Lucas polynomial is expressed according to the following recurrence relation

$$L_n(x) = \begin{cases}2 & \textrm{if}\ n = 0\\x & \textrm{if}\ n = 1\\x \cdot L_{n-1}(x) + L_{n-2}(x) & \textrm{if}\ n \geq 2\end{cases}$$

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

$$L_n(x) = \sum_{k = 0}^n L(n,k) x^k$$

We can extend Lucas polynomials to negative n using the identity

$$L_{-n}(x) = (-1)^{n} L_n(x)$$

Installation

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

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

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

lucaspoly( n, x )

Evaluates a Lucas polynomial at a value x.

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.

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

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 (CSP). If running in or targeting an environment with a CSP, avoid using code generation.

Examples

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 ) );
}

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See Also

• @stdlib/math-base/tools/evalpoly: evaluate a polynomial using double-precision floating-point arithmetic.
• @stdlib/math-base/tools/fibpoly: evaluate a Fibonacci polynomial.

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Notice

This package is part of 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, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.