README.md

gammincinv

Inverse of incomplete gamma function.

Computes the inverse of the lower incomplete gamma function

$$P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt$$

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:

$$Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt$$

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

Usage

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.

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.

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.

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

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

Examples

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 );

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See Also

• @stdlib/math/base/special/gamma: gamma function.
• @stdlib/math/base/special/gamma1pm1: compute gamma(x+1) - 1.
• @stdlib/math/base/special/gammainc: incomplete gamma function.
• @stdlib/math/base/special/gammaln: natural logarithm of the gamma function.