time-to-botec/squiggle/node_modules/@stdlib/math/base/special/cinv/README.md

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# inv

> Compute the inverse of a complex number.

<section class="intro">

The inverse (or reciprocal) of a non-zero complex number `z = a + bi` is defined as

<!-- <equation class="equation" label="eq:complex_inverse" align="center" raw="{\frac {1}{z}}=\frac{\bar{z}}{z{\bar{z}}} = \frac{a}{a^{2}+b^{2}} - \frac{b}{a^2+b^2}i." alt="Complex Inverse" > -->

<div class="equation" align="center" data-raw-text="{\frac {1}{z}}=\frac{\bar{z}}{z{\bar{z}}} = \frac{a}{a^{2}+b^{2}} - \frac{b}{a^2+b^2}i." data-equation="eq:complex_inverse">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@026bc0ee34051ddb44f3222f620bc7a300b9799e/lib/node_modules/@stdlib/math/base/special/cinv/docs/img/equation_complex_inverse.svg" alt="Complex Inverse">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var cinv = require( '@stdlib/math/base/special/cinv' );
```

#### cinv( \[out,] re1, im1 )

Computes the inverse of a `complex` number comprised of a **real** component `re` and an **imaginary** component `im`.

```javascript
var v = cinv( 2.0, 4.0 );
// returns [ 0.1, -0.2 ]
```

By default, the function returns real and imaginary components as a two-element `array`. To avoid unnecessary memory allocation, the function supports providing an output (destination) object.

```javascript
var Float64Array = require( '@stdlib/array/float64' );

var out = new Float64Array( 2 );

var v = cinv( out, 2.0, 4.0 );
// returns <Float64Array>[ 0.1, -0.2 ]

var bool = ( v === out );
// returns true
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var Complex128 = require( '@stdlib/complex/float64' );
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var real = require( '@stdlib/complex/real' );
var imag = require( '@stdlib/complex/imag' );
var cinv = require( '@stdlib/math/base/special/cinv' );

var re;
var im;
var z1;
var z2;
var o;
var i;

for ( i = 0; i < 100; i++ ) {
    re = round( randu()*100.0 ) - 50.0;
    im = round( randu()*100.0 ) - 50.0;
    z1 = new Complex128( re, im );

    o = cinv( real(z1), imag(z1) );
    z2 = new Complex128( o[ 0 ], o[ 1 ] );

    console.log( '1.0 / (%s) = %s', z1.toString(), z2.toString() );
}
```

</section>

<!-- /.examples -->

* * *

<section class="references">

## References

-   Smith, Robert L. 1962. "Algorithm 116: Complex Division." _Commun. ACM_ 5 (8). New York, NY, USA: ACM: 435. doi:[10.1145/368637.368661][@smith:1962a].
-   Stewart, G. W. 1985. "A Note on Complex Division." _ACM Trans. Math. Softw._ 11 (3). New York, NY, USA: ACM: 238–41. doi:[10.1145/214408.214414][@stewart:1985a].
-   Priest, Douglas M. 2004. "Efficient Scaling for Complex Division." _ACM Trans. Math. Softw._ 30 (4). New York, NY, USA: ACM: 389–401. doi:[10.1145/1039813.1039814][@priest:2004a].
-   Baudin, Michael, and Robert L. Smith. 2012. "A Robust Complex Division in Scilab." _arXiv_ abs/1210.4539 \[cs.MS] (October): 1–25. [&lt;https://arxiv.org/abs/1210.4539>][@baudin:2012a].

</section>

<!-- /.references -->

<section class="links">

[@smith:1962a]: https://doi.org/10.1145/368637.368661

[@stewart:1985a]: https://doi.org/10.1145/214408.214414

[@priest:2004a]: https://doi.org/10.1145/1039813.1039814

[@baudin:2012a]: https://arxiv.org/abs/1210.4539

</section>

<!-- /.links -->