# gammincinv
> Inverse of [incomplete gamma function][incomplete-gamma-function].
Computes the inverse of the lower [incomplete gamma function][incomplete-gamma-function]
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:
Again, for given `p` and `a` the function returns the `x` which satisfies `p = Q(x, a)`.
## 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
```
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var gammaincinv = require( '@stdlib/math/base/special/gammaincinv' );
var a;
var i;
var p;
for ( i = 0; i < 100; i++ ) {
p = randu();
a = randu() * 50.0;
console.log( 'p: %d, \t a: %d, \t P^(-1)(p, a): %d', p.toFixed( 4 ), a.toFixed( 4 ), gammaincinv( p, a ).toFixed( 4 ) );
}
```
[incomplete-gamma-function]: https://en.wikipedia.org/wiki/Incomplete_gamma_function
