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# cinv

> Compute the inverse of a double-precision complex floating-point 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( z )

Computes the inverse of a double-precision complex floating-point number.

```javascript
var Complex128 = require( '@stdlib/complex/float64/ctor' );

var v = cinv( new Complex128( 2.0, 4.0 ) );
// returns <Complex128>[ 0.1, -0.2 ]
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var Complex128Array = require( '@stdlib/array/complex128' );
var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var cinv = require( '@stdlib/math/base/special/cinv' );

// Create an array of random numbers:
var arr = new Complex128Array( uniform( 200, -100.0, 100.0 ) );

// Compute the inverse of each number in the array:
logEachMap( '1.0 / (%s) = %s', arr, cinv );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- 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">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/cinv.h"
```

#### stdlib_base_cinv( z )

Computes the inverse of a double-precision complex floating-point number.

```c
#include "stdlib/complex/float64/ctor.h"
#include "stdlib/complex/float64/real.h"
#include "stdlib/complex/float64/imag.h"

stdlib_complex128_t z = stdlib_complex128( 2.0, 4.0 );

stdlib_complex128_t out = stdlib_base_cinv( z );

double re = stdlib_complex128_real( out );
// returns 0.1

double im = stdlib_complex128_imag( out );
// returns -0.2
```

The function accepts the following arguments:

-   **z**: `[in] stdlib_complex128_t` input value.

```c
stdlib_complex128_t stdlib_base_cinv( const stdlib_complex128_t z );
```

</section>

<!-- /.usage -->

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

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/cinv.h"
#include "stdlib/complex/float64/ctor.h"
#include "stdlib/complex/float64/reim.h"
#include <stdio.h>

int main() {
    const stdlib_complex128_t x[] = {
        stdlib_complex128( 3.14, 1.5 ),
        stdlib_complex128( -3.14, -1.5 ),
        stdlib_complex128( 0.0, 0.0 ),
        stdlib_complex128( 0.0/0.0, 0.0/0.0 )
    };

    stdlib_complex128_t v;
    stdlib_complex128_t y;
    double re1;
    double im1;
    double re2;
    double im2;
    int i;
    for ( i = 0; i < 4; i++ ) {
        v = x[ i ];
        y = stdlib_base_cinv( v );
        stdlib_complex128_reim( v, &re1, &im1 );
        stdlib_complex128_reim( y, &re2, &im2 );
        printf( "cinv(%lf + %lfi) = %lf + %lfi\n", re1, im1, re2, im2 );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

* * *

<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 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/complex/float64/base/div`][@stdlib/complex/float64/base/div]</span><span class="delimiter">: </span><span class="description">divide two complex numbers.</span>
-   <span class="package-name">[`@stdlib/math/base/special/cinvf`][@stdlib/math/base/special/cinvf]</span><span class="delimiter">: </span><span class="description">compute the inverse of a single-precision complex floating-point number.</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">

[@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

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

[@stdlib/complex/float64/base/div]: https://www.npmjs.com/package/@stdlib/complex-float64-base-div

[@stdlib/math/base/special/cinvf]: https://github.com/stdlib-js/math/tree/main/base/special/cinvf

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

</section>

<!-- /.links -->
