README.md (4426B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2018 The Stdlib Authors. 6 7 Licensed under the Apache License, Version 2.0 (the "License"); 8 you may not use this file except in compliance with the License. 9 You may obtain a copy of the License at 10 11 http://www.apache.org/licenses/LICENSE-2.0 12 13 Unless required by applicable law or agreed to in writing, software 14 distributed under the License is distributed on an "AS IS" BASIS, 15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 16 See the License for the specific language governing permissions and 17 limitations under the License. 18 19 --> 20 21 # Lucas Polynomial 22 23 > Evaluate a [Lucas polynomial][fibonacci-polynomials]. 24 25 <section class="intro"> 26 27 A [Lucas polynomial][fibonacci-polynomials] is expressed according to the following recurrence relation 28 29 <!-- <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."> --> 30 31 <div class="equation" align="center" data-raw-text="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}" data-equation="eq:lucas_polynomial"> 32 <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."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 Alternatively, if `L(n,k)` is the coefficient of `x^k` in `L_n(x)`, then 39 40 <!-- <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."> --> 41 42 <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"> 43 <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."> 44 <br> 45 </div> 46 47 <!-- </equation> --> 48 49 We can extend [Lucas polynomials][fibonacci-polynomials] to negative `n` using the identity 50 51 <!-- <equation class="equation" label="eq:negalucas_polynomial" align="center" raw="L_{-n}(x) = (-1)^{n} L_n(x)" alt="NegaLucas polynomial."> --> 52 53 <div class="equation" align="center" data-raw-text="L_{-n}(x) = (-1)^{n} L_n(x)" data-equation="eq:negalucas_polynomial"> 54 <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."> 55 <br> 56 </div> 57 58 <!-- </equation> --> 59 60 </section> 61 62 <!-- /.intro --> 63 64 <section class="usage"> 65 66 ## Usage 67 68 ```javascript 69 var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' ); 70 ``` 71 72 #### lucaspoly( n, x ) 73 74 Evaluates a [Lucas polynomial][fibonacci-polynomials] at a value `x`. 75 76 ```javascript 77 var v = lucaspoly( 5, 2.0 ); // => 2^5 + 5*2^3 + 5*2 78 // returns 82.0 79 ``` 80 81 #### lucaspoly.factory( n ) 82 83 Uses code generation to generate a `function` for evaluating a [Lucas polynomial][fibonacci-polynomials]. 84 85 ```javascript 86 var polyval = lucaspoly.factory( 5 ); 87 88 var v = polyval( 1.0 ); // => 1^5 + 5*1^3 + 5 89 // returns 11.0 90 91 v = polyval( 2.0 ); // => 2^5 + 5*2^3 + 5*2 92 // returns 82.0 93 ``` 94 95 </section> 96 97 <!-- /.usage --> 98 99 <section class="notes"> 100 101 ## Notes 102 103 - For hot code paths, a compiled function will be more performant than `lucaspoly()`. 104 - 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. 105 106 </section> 107 108 <!-- /.notes --> 109 110 <section class="examples"> 111 112 ## Examples 113 114 <!-- eslint no-undef: "error" --> 115 116 ```javascript 117 var lucaspoly = require( '@stdlib/math/base/tools/lucaspoly' ); 118 119 var i; 120 121 // Compute the negaLucas and Lucas numbers... 122 for ( i = -76; i < 77; i++ ) { 123 console.log( 'L_%d = %d', i, lucaspoly( i, 1.0 ) ); 124 } 125 ``` 126 127 </section> 128 129 <!-- /.examples --> 130 131 <section class="links"> 132 133 [fibonacci-polynomials]: https://en.wikipedia.org/wiki/Fibonacci_polynomials 134 135 [mdn-csp]: https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP 136 137 </section> 138 139 <!-- /.links -->