151 lines
5.1 KiB
Markdown
151 lines
5.1 KiB
Markdown
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<!--
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Fibonacci Polynomial
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> Evaluate a [Fibonacci polynomial][fibonacci-polynomials].
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<section class="intro">
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A [Fibonacci polynomial][fibonacci-polynomials] is expressed according to the following recurrence relation
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<!-- <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."> -->
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<div class="equation" align="center" data-raw-text="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}" data-equation="eq:fibonacci_polynomial">
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<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.">
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<br>
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</div>
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<!-- </equation> -->
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Alternatively, if `F(n,k)` is the coefficient of `x^k` in `F_n(x)`, then
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<!-- <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."> -->
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<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">
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<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.">
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<br>
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</div>
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<!-- </equation> -->
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where
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<!-- <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."> -->
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<div class="equation" align="center" data-raw-text="F(n,k) = {{\frac{n+k-1}{2}} \choose {k}}" data-equation="eq:fibonacci_polynomial_coefficients">
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<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.">
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<br>
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</div>
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<!-- </equation> -->
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We can extend [Fibonacci polynomials][fibonacci-polynomials] to negative `n` using the identity
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<!-- <equation class="equation" label="eq:negafibonacci_polynomial" align="center" raw="F_{-n}(x) = (-1)^{n-1} F_n(x)" alt="NegaFibonacci polynomial."> -->
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<div class="equation" align="center" data-raw-text="F_{-n}(x) = (-1)^{n-1} F_n(x)" data-equation="eq:negafibonacci_polynomial">
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<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.">
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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 fibpoly = require( '@stdlib/math/base/tools/fibpoly' );
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```
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#### fibpoly( n, x )
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Evaluates a [Fibonacci polynomial][fibonacci-polynomials] at a value `x`.
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```javascript
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var v = fibpoly( 5, 2.0 ); // => 2^4 + 3*2^2 + 1
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// returns 29.0
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```
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#### fibpoly.factory( n )
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Uses code generation to generate a `function` for evaluating a [Fibonacci polynomial][fibonacci-polynomials].
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```javascript
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var polyval = fibpoly.factory( 5 );
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var v = polyval( 1.0 ); // => 1^4 + 3*1^2 + 1
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// returns 5.0
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v = polyval( 2.0 ); // => 2^4 + 3*2^2 + 1
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// returns 29.0
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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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## Notes
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- For hot code paths, a compiled function will be more performant than `fibpoly()`.
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- 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.
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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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<!-- eslint no-undef: "error" -->
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```javascript
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var fibpoly = require( '@stdlib/math/base/tools/fibpoly' );
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var i;
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// Compute the negaFibonacci and Fibonacci numbers...
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for ( i = -77; i < 78; i++ ) {
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console.log( 'F_%d = %d', i, fibpoly( 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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[mdn-csp]: https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP
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</section>
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<!-- /.links -->
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