138 lines
3.5 KiB
Markdown
138 lines
3.5 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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# Binet's Formula
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> Evaluate [Binet's formula][fibonacci-number] extended to real numbers.
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<section class="intro">
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[Binet's formula][fibonacci-number] refers to the closed-form solution for computing the nth [Fibonacci number][fibonacci-number] and may be expressed
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<!-- <equation class="equation" label="eq:binets_formula" align="center" raw="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" alt="Binet's formula"> -->
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<div class="equation" align="center" data-raw-text="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" data-equation="eq:binets_formula">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula.svg" alt="Binet's formula">
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<br>
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</div>
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<!-- </equation> -->
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where `φ` is the [golden ratio][golden-ratio] and `ψ` is `1 - φ`. To extend [Fibonacci numbers][fibonacci-number] to real numbers, we may express [Binet's formula][fibonacci-number] as
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<!-- <equation class="equation" label="eq:binets_formula_real_numbers" align="center" raw="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" alt="Binet's formula extended to real numbers."> -->
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<div class="equation" align="center" data-raw-text="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" data-equation="eq:binets_formula_real_numbers">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula_real_numbers.svg" alt="Binet's formula extended to real numbers.">
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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 binet = require( '@stdlib/math/base/special/binet' );
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```
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#### binet( x )
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Evaluates [Binet's formula][fibonacci-number] extended to real numbers.
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```javascript
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var v = binet( 0.0 );
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// returns 0.0
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v = binet( 1.0 );
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// returns 1.0
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v = binet( 2.0 );
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// returns 1.0
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v = binet( 3.0 );
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// returns 2.0
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v = binet( -1.0 );
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// returns 1.0
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v = binet( 3.14 );
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// returns ~2.12
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```
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If provided `NaN`, the function returns `NaN`.
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```javascript
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var v = binet( NaN );
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// returns NaN
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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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- The function returns only **approximate** [Fibonacci numbers][fibonacci-number] for nonnegative integers.
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- The function does **not** return complex numbers, guaranteeing real-valued return values.
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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 binet = require( '@stdlib/math/base/special/binet' );
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var v;
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var i;
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for ( i = 0; i < 79; i++ ) {
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v = binet( i );
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console.log( v );
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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-number]: https://en.wikipedia.org/wiki/Fibonacci_number
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[golden-ratio]: https://en.wikipedia.org/wiki/Golden_ratio
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</section>
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<!-- /.links -->
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