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@stdlib/math-iter-sequences-continued-fraction

Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of provided number.

https://github.com/stdlib-js/math-iter-sequences-continued-fraction

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Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of provided number.

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README.md

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iterContinuedFractionSeq

NPM version Build Status Coverage Status <!-- dependencies -->

Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.

A **generalized continued fraction** has the form ```math x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}} ``` If `a_i = 1` for all `i`, the above expression reduces to a **simple continued fraction**. ```math x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}} ``` where the values `b_i` are called the **coefficients** or **terms** of the continued fraction and the rationals ```math b_0, b_0 + \frac{1}{b_1},b_0 + \frac{1}{b_1 + \frac{1}{b2}},b_0 + \frac{1}{b_1 + \frac{1}{b2+\frac{1}{b3}}},\ldots ``` are called **convergents**.
## Installation ```bash npm install @stdlib/math-iter-sequences-continued-fraction ``` Alternatively, - To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]). - If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions). - For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]). The [branches.md][branches-url] 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 ```javascript var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' ); ``` #### iterContinuedFractionSeq( x, \[options] ) Returns an iterator which generates a list of all continued fraction terms (`b_i`) which can be obtained given the precision of `x`. ```javascript var it = iterContinuedFractionSeq( 3.245 ); // returns var v = it.next().value; // returns 3 v = it.next().value; // returns 4 v = it.next().value; // returns 12 v = it.next().value; // returns 4 var bool = it.next().done; // returns true ``` The returned iterator protocol-compliant object has the following properties: - **next**: function which returns an iterator protocol-compliant object containing the next iterated value (if one exists) assigned to a `value` property and a `done` property having a `boolean` value indicating whether the iterator is finished. - **return**: function which closes an iterator and returns a single (optional) argument in an iterator protocol-compliant object. The function supports the following `options`: - **iter**: maximum number of iterations. Default: `1e308`. - **tol**: tolerance at which to terminate further evaluation of the continued fraction. Default: [floating-point epsilon][@stdlib/constants/float64/eps]. - **returns**: specifies the type of result to return. Must be one of the following options: - **terms**: return continued fraction terms. - **convergents**: return continued fraction convergents. - **\***: return both continued fraction terms and their associated convergents as a two-element array: `[ , ]`. Default: `'terms'`. By default, in theory, the function returns an infinite iterator; however, in practice, due to limited precision, every floating-point number is a rational number, and, thus, every returned iterator will end in a finite number of iterations. To explicitly cap the maximum number of iterations, set the `iter` option. ```javascript var opts = { 'iter': 2 }; var it = iterContinuedFractionSeq( 3.245, opts ); // returns var v = it.next().value; // returns 3 v = it.next().value; // returns 4 var bool = it.next().done; // returns true ``` The returned iterator terminates once the difference between the input value `x` and a continued fraction approximation is sufficiently small. The default tolerance is [floating-point epsilon][@stdlib/constants/float64/eps] (`~2.22e-16`). Once an update to a continued fraction approximation is less than or equal to this tolerance, the iterator terminates. To adjust the tolerance (e.g., to return a rough approximation of an input value `x`), set the `tol` option. ```javascript var opts = { 'tol': 1.0e-7 }; var it = iterContinuedFractionSeq( 3.141592653589793, opts ); // returns var v = it.next().value; // returns 3 v = it.next().value; // returns 7 v = it.next().value; // returns 16 var bool = it.next().done; // returns true // The returned terms [3; 7, 16] evaluate to 3.1415929203539825 ``` By default, the returned iterator returns continued fraction terms. To return convergents, set the `returns` option to `'convergents'`. ```javascript var it = iterContinuedFractionSeq( 3.245, { 'returns': 'convergents' }); // returns var v = it.next().value; // returns 3.0 v = it.next().value; // returns 3.25 v = it.next().value; // returns ~3.2449 v = it.next().value; // returns 3.245 var bool = it.next().done; // returns true ``` To return both continued fraction terms and their associated convergents, set the `returns` option to `*`. ```javascript var it = iterContinuedFractionSeq( 3.245, { 'returns': '*' }); // returns var v = it.next().value; // returns [ 3, 3.0 ] v = it.next().value; // returns [ 4, 3.25 ] v = it.next().value; // returns [ 12, ~3.2449 ] v = it.next().value; // returns [ 4, 3.245 ] var bool = it.next().done; // returns true ```
## Notes - The returned iterator returns the terms for a **simple continued fraction**. - For `x < 0`, the returned iterator returns negated terms for `|x|` (i.e., if the terms for `|x|` are `[b0; b1, b2, ..., bn]`, the returned iterator returns `[-b0; -b1, -b2, ..., -bn]`). While other continued fraction representations are possible, floating-point rounding error can introduce asymmetries when evaluating terms to recover the original values for `|x|` and `x < 0`. Accordingly, alternative continued fraction representations for negative input values are not supported. - If an environment supports `Symbol.iterator`, the returned iterator is iterable.
## Examples ```javascript var PI = require( '@stdlib/constants-float64-pi' ); var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' ); function evaluate( terms ) { var sum; var N; var i; N = terms.length; sum = 0.0; if ( N > 1 ) { sum = 1.0 / terms[ N-1 ]; for ( i = N-2; i > 0; i-- ) { sum = 1.0 / ( terms[ i ] + sum ); } } sum += terms[ 0 ]; return sum; } // Create an iterator: var opts = { 'iter': 20 }; var it = iterContinuedFractionSeq( PI, opts ); // Perform manual iteration... var terms = []; var v; while ( true ) { v = it.next(); if ( v.done ) { break; } terms.push( v.value ); } console.log( 'original: %d', PI ); console.log( terms ); console.log( 'computed: %d', evaluate( terms ) ); ```
* * * ## See Also - [`@stdlib/math-iter/utils/continued-fraction`][@stdlib/math/iter/utils/continued-fraction]: evaluate the terms of a continued fraction.
* * * ## Notice This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib]. #### Community [![Chat][chat-image]][chat-url] --- ## License See [LICENSE][stdlib-license]. ## Copyright Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].

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npmjs.org: @stdlib/math-iter-sequences-continued-fraction

Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.

  • Homepage: https://stdlib.io
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