gammainc

Incomplete gamma function.

Evaluates the unregularized or unregularized gamma functions.

This package is a rewrite of @compute-io/gammainc in Typescript. This package supports both CommonJs and ES Modules.

Computes the regularized 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

The function can also be used to evaluate the regularized 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

The two functions have the relationship Q(x,a) = 1 - P(x,a).

In addition, this package can be used to evaluate the unregularized gamma functions. The range of above functions is [0, 1], which is not the case fo the unregularized versions. The unregularized lower incomplete gamma function is defined as

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

and the upper unregularized incomplete gamma function is

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

The relationship between the two functions is γ(a,x) + Γ(a,x) = Γ(a).

Installation

$ npm install @toshiara/special-gammainc

Usage

// for CommonJs
const { gammainc } = require('@toshiara/special-gammainc');

// for ES Modules
import { gammainc } from '@toshiara/special-gammainc';

gammainc(x, a[, options])

The domain of the function are the non-negative real numbers for x and the positve real numbers for a. If supplied a value outside the domain, the function returns NaN. For both the regularized and unregularized versions of the incomplete gamma function, in this implementation the first argument is x and the second argument is the scale factor a.

The function accepts the following options:

• upper:boolean indicating whether to evaluate the lower (false) or upper (true) incomplete gamma function. Default: false.
• regularized: boolean indicating if the function should compute the regularized (true) or unregularized (false) incomplete gamma functions. Default: true.

By default, the function evaluates the lower regularized incomplete gamma function, P(x,a). To evaluate the upper function instead, i.e. Q(x,a), set the lower option to false.

//// Regularized
gammainc(9, 3);
// returns 0.9937678048936227

gammainc(9, 3, { upper: true });
// returns 0.006232195106377313

//// Unregularized
gammainc(9, 3, { regularized: false });
// returns 1.9875356097872454

gammainc(9, 3, { upper: true, regularized: false });
// returns 0.012464390212754625

Notes

If an element is not a numeric value, the returned value is NaN.

gammainc(null, 1);
// returns NaN

Implementation

All of the four functions (regularized and non-regularized, upper and lower) share a common implementation as they are all related to each other (see the Boost C++ library documentation for a good discussion of the functions and implementation strategies).

To evaluate the regularized lower incomplete gamma function, this package uses the following representation of the integral as a power series in its implementation:

P(x, a) = \frac{1}{\Gamma(a)}\sum_{k=0}^\infty \frac{x^a e^{-x} x^k}{a(a+1)...(a+k)}

This series is evaluated for all inputs x and s unless x > 1.1 and x > s, in which case the function is evaluated using the upper incomplete gamma function as P(x,s) = 1 - Q(x,s). To evaluate the upper incomplete gamma function, Gauss' continued fraction expansion is used:

Q(x, a) = \dfrac{1}{\Gamma(a)}\dfrac{x^a e^{-x}}{1+x-a+ \dfrac{a-1}{3+x-a+ \dfrac{2(a-2)}{5+x-a+ \dfrac{3(a-3)} {7+x-a+ \dfrac{4(a-4)}{9+x-a+ \ddots}}}}}

To compute the continued fractions, the modified Lentz's method is implemented. For a discussion of this method, see section 5.2 of "Numerical Recipes in C (2nd Ed.): The Art of Scientific Computing".

References

• Lentz, W. J. (1976). Generating bessel functions in mie scattering calculations using continued fractions. Applied Optics, 15(3), 668–671. doi:10.1364/AO.15.000668
• William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. 1992. Numerical Recipes in C (2nd Ed.): The Art of Scientific Computing. Cambridge University Press, New York, NY, USA.

License

MIT license.

Copyright

Copyright © 2015. The Compute.io Authors.