npm package: @stdlib/math-iter-sequences-continued-fraction
Description: Create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.
Author: The Stdlib Authors
Latest version: 0.2.3

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# iterContinuedFractionSeq

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

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<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@b05b08e74e5706e091431f99b26166caea11fed9/lib/node_modules/@stdlib/math/iter/sequences/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@b05b08e74e5706e091431f99b26166caea11fed9/lib/node_modules/@stdlib/math/iter/sequences/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 and the rationals

<!-- <equation class="equation" label="eq:simple_continued_fraction_convergents" align="center" raw="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" alt="Simple continued fraction convergents"> -->

<!-- <div class="equation" align="center" data-raw-text="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" data-equation="eq:simple_continued_fraction_convergents">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6be2d95d94757e01f8e07117664fdf14b5f82777/lib/node_modules/@stdlib/math/iter/sequences/continued-fraction/docs/img/equation_simple_continued_fraction_convergents.svg" alt="Simple continued fraction convergents">
    <br>
</div>

<!-- </equation> -->

are called **convergents**.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## 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 <Object>

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: `[ <term>, <convergent> ]`.

    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 <Object>

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 <Object>

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 <Object>

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 <Object>

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
```

</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

-   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.

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```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 ) );
```

</section>

<!-- /.examples -->

<!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="references">

</section>

<!-- /.references -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-iter/utils/continued-fraction`][@stdlib/math/iter/utils/continued-fraction]</span><span class="delimiter">: </span><span class="description">evaluate the terms of a continued fraction.</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

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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