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# gammaincinv

> Inverse of [incomplete gamma function][incomplete-gamma-function].

<section class="intro">

Computes the inverse of the lower [incomplete gamma function][incomplete-gamma-function]

<!-- <equation class="equation" label="eq:lower_incomplete_gamma" align="center" raw="P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt" alt="Regularized lower incomplete gamma function."> -->

<div class="equation" align="center" data-raw-text="P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt" data-equation="eq:lower_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammaincinv/docs/img/equation_lower_incomplete_gamma.svg" alt="Regularized lower incomplete gamma function.">
    <br>
</div> -->

<!-- </equation> -->

Specifically, for given `p` and `a` it finds the `x` such that `p =  P(x, a)`.

The function can also be used to invert the upper incomplete gamma function, which is defined as follows:

<!-- <equation class="equation" label="eq:upper_incomplete_gamma" align="center" raw="Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt" alt="Regularized upper incomplete gamma function."> -->

<!-- <div class="equation" align="center" data-raw-text="Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt" data-equation="eq:upper_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammaincinv/docs/img/equation_upper_incomplete_gamma.svg" alt="Regularized upper incomplete gamma function.">
    <br>
</div>

<!-- </equation> -->

Again, for given `p` and `a` the function returns the `x` which satisfies `p = Q(x, a)`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### gammaincinv( p, s\[, upper ] )

Inverts the regularized incomplete gamma function. Contrary to the more commonly used definition, in this implementation the first argument is the probability `p` and the second argument is the scale factor `a`. By default, the function inverts the _lower_ regularized incomplete gamma function, `P(x,a)`. To invert the _upper_ function instead, i.e. `Q(x,a)`, set the `upper` argument to `true`.

```javascript
var y = gammaincinv( 0.5, 2.0 );
// returns ~1.678

y = gammaincinv( 0.1, 10.0 );
// returns ~6.221

y = gammaincinv( 0.75, 3.0 );
// returns ~3.92

y = gammaincinv( 0.75, 3.0, true );
// returns ~1.727
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = gammaincinv( NaN, 1.0 );
// returns NaN

y = gammaincinv( 0.5, NaN );
// returns NaN
```

If provided a value outside `[0,1]` for `p`, the function returns `NaN`.

```javascript
var y = gammaincinv( 1.2, 1.0 );
// returns NaN

y = gammaincinv( -0.5, 1.0 );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

var opts = {
    'dtype': 'float64'
};
var p = uniform( 100, 0.0, 1.0, opts );
var a = uniform( 100, 0.0, 50.0, opts );

logEachMap( 'p: %0.4f, \t a: %0.4f, \t P^(-1)(p, a): %0.4f', p, a, gammaincinv );
```

</section>

<!-- /.examples -->

<!-- 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/math/base/special/gamma`][@stdlib/math/base/special/gamma]</span><span class="delimiter">: </span><span class="description">gamma function.</span>
-   <span class="package-name">[`@stdlib/math/base/special/gamma1pm1`][@stdlib/math/base/special/gamma1pm1]</span><span class="delimiter">: </span><span class="description">compute gamma(x+1) - 1.</span>
-   <span class="package-name">[`@stdlib/math/base/special/gammainc`][@stdlib/math/base/special/gammainc]</span><span class="delimiter">: </span><span class="description">incomplete gamma function.</span>
-   <span class="package-name">[`@stdlib/math/base/special/gammaln`][@stdlib/math/base/special/gammaln]</span><span class="delimiter">: </span><span class="description">natural logarithm of the gamma function.</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">

[incomplete-gamma-function]: https://en.wikipedia.org/wiki/Incomplete_gamma_function

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

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

[@stdlib/math/base/special/gamma1pm1]: https://github.com/stdlib-js/math/tree/main/base/special/gamma1pm1

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

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

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

</section>

<!-- /.links -->
