npm package: @stdlib/math-base-special-cinv
Description: Compute the inverse of a double-precision complex floating-point number.
Author: The Stdlib Authors
Latest version: 0.2.3

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# cinv

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> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-special-cinv
```

</section>

<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 -->

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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].

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## License

See [LICENSE][stdlib-license].


## Copyright

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

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[@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

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