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Add tool to evaluate a Lucas polynomial
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‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/README.md

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# Lucas Polynomial
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> Evaluate a [Lucas polynomial][fibonacci-polynomials].
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<section class="intro">
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A [Lucas polynomial][fibonacci-polynomials] is expressed according to the following recurrence relation
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<!-- <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."> -->
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<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">
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<img src="" alt="Lucas polynomial.">
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<br>
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</div>
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<!-- </equation> -->
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Alternatively, if `L(n,k)` is the coefficient of `x^k` in `L_n(x)`, then
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<!-- <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."> -->
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<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">
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<img src="" alt="Lucas polynomial expressed as a sum.">
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<br>
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</div>
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<!-- </equation> -->
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We can extend [Lucas polynomials][fibonacci-polynomials] to negative `n` using the identity
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<!-- <equation class="equation" label="eq:negalucas_polynomial" align="center" raw="L_{-n}(x) = (-1)^{n} L_n(x)" alt="NegaLucas polynomial."> -->
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<div class="equation" align="center" data-raw-text="L_{-n}(x) = (-1)^{n} L_n(x)" data-equation="eq:negalucas_polynomial">
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<img src="" alt="NegaLucas polynomial.">
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<br>
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</div>
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<!-- </equation> -->
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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``` javascript
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var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' );
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```
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#### lucaspoly( n, x )
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Evaluates a [Lucas polynomial][fibonacci-polynomials] at a value `x`.
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``` javascript
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var v = lucaspoly( 5, 2.0 ); // => 2^5 + 5*2^3 + 5*2
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// returns 82.0
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```
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#### lucaspoly.factory( n )
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Returns a `function` for evaluating a [Lucas polynomial][fibonacci-polynomials].
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``` javascript
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var polyval = lucaspoly.factory( 5 );
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var v = polyval( 1.0 ); // => 1^5 + 5*1^3 + 5
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// returns 11.0
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v = polyval( 2.0 ); // => 2^5 + 5*2^3 + 5*2
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// returns 82.0
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```
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</section>
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<!-- /.usage -->
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<section class="notes">
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</section>
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<!-- /.notes -->
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<section class="examples">
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## Examples
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``` javascript
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var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' );
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var i;
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// Compute the negaLucas and Lucas numbers...
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for ( i = -76; i < 77; i++ ) {
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console.log( 'L_%d = %d', i, lucaspoly( i, 1.0 ) );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[fibonacci-polynomials]: https://en.wikipedia.org/wiki/Fibonacci_polynomials
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/benchmark/benchmark.factory.js

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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var randu = require( '@stdlib/math/base/random/randu' );
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var round = require( '@stdlib/math/base/special/round' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var factory = require( './../lib/factory.js' );
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// MAIN //
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bench( pkg+'::create:factory', function benchmark( b ) {
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var n;
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var f;
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var i;
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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n = round( randu()*172.0 ) - 76;
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f = factory( n );
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if ( typeof f !== 'function' ) {
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b.fail( 'should return a function' );
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}
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}
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b.toc();
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if ( typeof f !== 'function' ) {
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b.fail( 'should return a function' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});
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bench( pkg+'::evaluate:factory', function benchmark( b ) {
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var x;
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var v;
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var f;
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var i;
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f = factory( 76.0 );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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x = randu() * 100.0;
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v = f( x );
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/benchmark/benchmark.js

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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var randu = require( '@stdlib/math/base/random/randu' );
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var round = require( '@stdlib/math/base/special/round' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var lucaspoly = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var n;
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var x;
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var v;
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var i;
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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n = round( randu()*172.0 ) - 76;
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x = randu() * 100.0;
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v = lucaspoly( n, x );
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( v ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/docs/repl.txt

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{{alias}}( n, x )
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Evaluates a Lucas polynomial.
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Parameters
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----------
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n: integer
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Lucas polynomial to evaluate.
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x: number
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Value at which to evaluate the Lucas polynomial.
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Returns
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-------
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out: number
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Evaluated Lucas polynomial.
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Examples
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--------
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// 2^5 + 5*2^3 + 5*2
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> var v = {{alias}}( 5, 2.0 )
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82.0
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{{alias}}.factory( n )
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Returns a function for evaluating a Lucas polynomial.
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Parameters
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----------
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n: integer
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Lucas polynomial to evaluate.
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Returns
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-------
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fcn: Function
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Function for evaluating a Lucas polynomial.
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Examples
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--------
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> var polyval = {{alias}}.factory( 5 );
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// 1^5 + 5*1^2 + 5
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> var v = polyval( 1.0 )
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11.0
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// 2^5 + 5*2^3 + 5*2
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> v = polyval( 2.0 )
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82.0
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See Also
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--------
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‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/examples/index.js

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

‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/lib/cache.js

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'use strict';
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// EXPORTS //
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module.exports = {};

‎lib/node_modules/@stdlib/math/base/tools/lucaspoly/lib/coefficients.js

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'use strict';
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// MODULES //
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var binomcoef = require( '@stdlib/math/base/special/binomcoef' );
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var floor = require( '@stdlib/math/base/special/floor' );
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var ceil = require( '@stdlib/math/base/special/ceil' );
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var cache = require( './cache.js' );
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// MAIN //
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/**
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* Computes polynomial coefficients.
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*
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* ## Notes
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*
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* * Coefficients are computed via a (1,2)-Pascal triangle (i.e., Lucas triangle). For more details, see [Lucas polynomials]{@link https://oeis.org/wiki/Lucas_polynomials} and [Lucas triangle]{@link https://oeis.org/wiki/Lucas_triangle}.
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*
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*
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* @private
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* @param {NonNegativeInteger} n - Lucas polynomial for which to compute coefficients
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* @returns {NonNegativeIntegerArray} polynomial coefficients
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*/
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function coefficients( n ) {
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var coefs;
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var half;
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var high;
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var low;
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var p;
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var a;
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var b;
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var m;
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var i;
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coefs = cache[ n ];
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if ( coefs === void 0 ) {
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m = n + 1;
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coefs = new Array( m );
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if ( n === 0 ) {
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coefs[ 0 ] = 2.0;
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} else {
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for ( i = 0; i < m; i++ ) {
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coefs[ i ] = 0.0;
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}
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half = n / 2;
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high = ceil( half );
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low = floor( half );
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for ( i = 0; i <= low; i++ ) {
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p = (2*i) + (n%2);
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a = 2.0 * binomcoef( high+i-1, low-i-1 );
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b = binomcoef( high+i-1, low-i );
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coefs[ p ] += a + b;
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}
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}
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// Memoize the coefficients:
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cache[ n ] = coefs;
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}
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return coefs;
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} // end FUNCTION coefficients()
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// EXPORTS //
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module.exports = coefficients;

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