Special Functions

Special functions is a Nim library for calculating 'special' mathematical functions that are not found in Nim math.

Supported Functions

Below are the supported gamma functions:

import special_functions/gamma

var x1: float

# Incomplete gamma functions
x1 = lower_incomplete_gamma(2.0, 0.25)
x1 = upper_incomplete_gamma(2.0, 0.25)
x1 = regularized_lower_incomplete_gamma(2.0, 0.25)
x1 = regularized_upper_incomplete_gamma(2.0, 0.25)

# Inverse Incomplete gamma functions
x1 = inverse_lower_incomplete_gamma(2.0, 0.25)
x1 = inverse_upper_incomplete_gamma(2.0, 0.25)
x1 = inverse_regularized_lower_incomplete_gamma(2.0, 0.25)
x1 = inverse_regularized_upper_incomplete_gamma(2.0, 0.25)

Below are the supported beta functions:

import special_functions/beta

var x1: float

# Incomplete beta functions
x1 = lower_incomplete_beta(2.0, 2.0, 0.25)
x1 = upper_incomplete_beta(2.0, 2.0, 0.25)
x1 = regularized_lower_incomplete_beta(2.0, 2.0, 0.25)
x1 = regularized_upper_incomplete_beta(2.0, 2.0, 0.25)

# Inverse Incomplete beta functions
x1 = inverse_lower_incomplete_beta(2.0, 2.0, 0.25)
x1 = inverse_upper_incomplete_beta(2.0, 2.0, 0.25)
x1 = inverse_regularized_lower_incomplete_beta(2.0, 2.0, 0.25)
x1 = inverse_regularized_upper_incomplete_beta(2.0, 2.0, 0.25)

Below are the supported erf functions:

import special_functions/erf

var x1: float

# Erf functions
x1 = inverse_erf(0.25)

Future Directions:

There are really two directions this library can grow.

1. Write more special functions in Nim.
   • Pros: Library is in Nim.
   • Cons: Longer dev time, calculations are more error prone.
2. Implement all special functions using a C-wrapper around the Rmath library.
   • Pros: Reduced dev time, calculations are less error prone.
   • Cons: Library is written in C and hard to read (its filled with macros).

I don't have a preferred direction and I prefer to leave the answer for the Nim scientific community to answer.

Accuracy

All functions in this library are accurate up-to 14 decimal places (float64).

Performance

This library was written with accuracy as a top priority as opposed to performance, however almost all implementations here are faster than SciPy's implementations and equal to, slower, or faster than Julia's Special Functions implementations.

TODO

List is organized from most important to least important:

• Add more special functions on an as-need-bases.
• Some functions do not have a unittest because I couldn't find a source of truth to compare them against. Finding a source of truth and adding a unittest is the next step.

Performance, feature, and documentation PR's are always welcome.

Acknowledgments

Many of the algorithms used here are based off of different peoples' code. Thank you to:

• https://github.com/jkovacic/math/
• http://libit.sourceforge.net/math_8c-source.html

Contact

I can be reached at aymanalbaz98@gmail.com