<commit_before><!--

@license Apache-2.0

Copyright (c) 2018 The Stdlib Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

   http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.

-->

# 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@99e622e3907aa8bd2f252516d022fc73519b1cdf/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

-   <NAME>. 1962. "Algorithm 116: Complex Division." _Commun. ACM_ 5 (8). New York, NY, USA: ACM: 435. doi:[10.1145/368637.368661][@smith:1962a].
-   <NAME>. 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].
-   <NAME>. 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].
-   <NAME>, and <NAME>. 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 -->
<commit_msg>Add parentheses<commit_after><!--

@license Apache-2.0

Copyright (c) 2018 The Stdlib Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

   http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.

-->

# 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@99e622e3907aa8bd2f252516d022fc73519b1cdf/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

-   <NAME>. 1962. "Algorithm 116: Complex Division." _Commun. ACM_ 5 (8). New York, NY, USA: ACM: 435. doi:[10.1145/368637.368661][@smith:1962a].
-   <NAME>. 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].
-   <NAME>. 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 <NAME>. 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 -->

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