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

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@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
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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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: 23841. 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: 389401. 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): 125. [&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 -->