npm package: @stdlib/math-iter-utils-continued-fraction
Description: Evaluate the terms of a continued fraction.
Author: The Stdlib Authors
Latest version: 0.2.3

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# iterContinuedFraction

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> Evaluate the terms of a continued fraction.

<section class="intro">

A **generalized continued fraction** has the form

<!-- <equation class="equation" label="eq:continued_fraction" align="center" raw="x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}}" alt="Continued fraction"> -->

<div class="equation" align="center" data-raw-text="x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}}" data-equation="eq:continued_fraction">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@d3cef42cbeb8499623814933ed4d906a345cec32/lib/node_modules/@stdlib/math/iter/utils/continued-fraction/docs/img/equation_continued_fraction.svg" alt="Continued fraction">
    <br>
</div> -->

<!-- </equation> -->

If `a_i = 1` for all `i`, the above expression reduces to a **simple continued fraction**.

<!-- <equation class="equation" label="eq:simple_continued_fraction" align="center" raw="x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}}" alt="Simple continued fraction"> -->

<!-- <div class="equation" align="center" data-raw-text="x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}}" data-equation="eq:simple_continued_fraction">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@d3cef42cbeb8499623814933ed4d906a345cec32/lib/node_modules/@stdlib/math/iter/utils/continued-fraction/docs/img/equation_simple_continued_fraction.svg" alt="Simple continued fraction">
    <br>
</div>

<!-- </equation> -->

where the values `b_i` are called the **coefficients** or **terms** of the continued fraction.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-iter-utils-continued-fraction
```

</section>

<section class="usage">

## Usage

```javascript
var iterContinuedFraction = require( '@stdlib/math-iter-utils-continued-fraction' );
```

#### iterContinuedFraction( iterator\[, options] )

Evaluates the terms of a continued fraction.

```javascript
var array2iterator = require( '@stdlib/array-to-iterator' );

var arr = array2iterator( [ 3, 4, 12, 4 ] );

var v = iterContinuedFraction( arr );
// returns ~3.245
```

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].

To explicitly cap the maximum number of iterations, set the `iter` option.

```javascript
var array2iterator = require( '@stdlib/array-to-iterator' );

var arr = array2iterator( [ 3, 4, 12, 4 ] );

var opts = {
    'iter': 2
};
var v = iterContinuedFraction( arr, opts );
// returns 3.25
```

The function terminates continued fraction evaluation once the difference between continued fraction approximations 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 function terminates evaluation. To adjust the tolerance (e.g., to return a rough approximation), set the `tol` option.

```javascript
var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' );

var opts = {
    'tol': 1.0e-7
};
var it = iterContinuedFractionSeq( 3.141592653589793, opts );
// returns <Object>

var v = iterContinuedFraction( it );
// returns ~3.14159292
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

## Notes

-   If an iterated value is non-numeric (including `NaN`), the function returns `NaN`. If non-numeric iterated values are possible, you are advised to provide an [`iterator`][mdn-iterator-protocol] which type checks and handles non-numeric values accordingly.

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var iterContinuedFractionSeq = require( '@stdlib/math-iter-sequences-continued-fraction' );
var PI = require( '@stdlib/constants-float64-pi' );
var iterContinuedFraction = require( '@stdlib/math-iter-utils-continued-fraction' );

// Create an iterator for returning the list of continued fraction terms:
var it = iterContinuedFractionSeq( PI );

// Reconstitute the original value:
var v = iterContinuedFraction( it );
// returns <number>

console.log( 'Value: %d.', v );
```

</section>

<!-- /.examples -->

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<section class="references">

</section>

<!-- /.references -->

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<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-iter/sequences/continued-fraction`][@stdlib/math/iter/sequences/continued-fraction]</span><span class="delimiter">: </span><span class="description">create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## 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 &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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