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# iterContinuedFractionSeq

> 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="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 -->

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<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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[@stdlib/constants/float64/eps]: https://www.npmjs.com/package/@stdlib/constants-float64-eps

<!-- <related-links> -->

[@stdlib/math/iter/utils/continued-fraction]: https://github.com/stdlib-js/math/tree/main/iter/utils/continued-fraction

<!-- </related-links> -->

</section>

<!-- /.links -->
