Lucas Polynomial

Evaluate a Lucas polynomial.

A Lucas polynomial is expressed according to the following recurrence relation

upper L Subscript n Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 2 2nd Column if n equals 0 2nd Row 1st Column x 2nd Column if n equals 1 3rd Row 1st Column x dot upper L Subscript n minus 1 Baseline left-parenthesis x right-parenthesis plus upper L Subscript n minus 2 Baseline left-parenthesis x right-parenthesis 2nd Column if n greater-than-or-equal-to 2 EndLayout

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

upper L Subscript n Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript n Endscripts upper L left-parenthesis n comma k right-parenthesis x Superscript k

We can extend Lucas polynomials to negative n using the identity

upper L Subscript negative n Baseline left-parenthesis x right-parenthesis equals left-parenthesis negative 1 right-parenthesis Superscript n Baseline upper L Subscript n Baseline left-parenthesis x right-parenthesis

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 ) );
}