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Binet's Formula

Evaluate Binet's formula extended to real numbers.

Binet's formula refers to the closed-form solution for computing the nth Fibonacci number and may be expressed

$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$

where φ is the golden ratio and ψ is 1 - φ. To extend Fibonacci numbers to real numbers, we may express Binet's formula as

$$F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}$$

Installation

npm install @stdlib/math-base-special-binet

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var binet = require( '@stdlib/math-base-special-binet' );

binet( x )

Evaluates Binet's formula extended to real numbers.

var v = binet( 0.0 );
// returns 0.0

v = binet( 1.0 );
// returns 1.0

v = binet( 2.0 );
// returns 1.0

v = binet( 3.0 );
// returns 2.0

v = binet( -1.0 );
// returns 1.0

v = binet( 3.14 );
// returns ~2.12

If provided NaN, the function returns NaN.

var v = binet( NaN );
// returns NaN

Notes

• The function returns only approximate Fibonacci numbers for nonnegative integers.
• The function does not return complex numbers, guaranteeing real-valued return values.

Examples

var binet = require( '@stdlib/math-base-special-binet' );

var v;
var i;

for ( i = 0; i < 79; i++ ) {
    v = binet( i );
    console.log( v );
}

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C APIs

Usage

#include "stdlib/math/base/special/binet.h"

stdlib_base_binet( x )

Evaluates Binet's formula extended to real numbers.

double out = stdlib_base_binet( 0.0 );
// returns 0.0

out = stdlib_base_binet( 1.0 );
// returns 1.0

The function accepts the following arguments:

• x: [in] double input value.

double stdlib_base_binet( const double x );

Examples

#include "stdlib/math/base/special/binet.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 1.0, 2.0, 3.0, 4.0 };

    double y;
    int i;
    for ( i = 0; i < 5; i++ ) {
        y = stdlib_base_binet( x[ i ] );
        printf( "binet(%lf) = %lf\n", x[ i ], y );
    }
}

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See Also

• @stdlib/math-base/special/fibonacci: compute the nth Fibonacci number.
• @stdlib/math-base/special/negafibonacci: compute the nth negaFibonacci number.

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.