Some basic but difficult to implement mathmatical functions

Note: for distribution functions please see`distributions`

`npm install mathfn`

`var mathfn = ; console; // 0.0 `

`mathfn`

is a slowly growing collection of some difficult mathmatical functions
there should be included in `Math.`

but isn't. This is a list of the currently
implemented functions and a few details.

`p = erf(x)`

- The error functionThis function is implemented using the "Abramowitz & Stegun" approximation
its theortical accuracy is `1.5 * 10^-7`

. However the limitations of JavaScript
might result in a lower accuracy.

`p = erfc(x)`

The complementary error functionUnlike most implementation of `erfc(x)`

, this is not calculated using `1 - erf(x)`

,
but is an acutall approximation of `erfc(x)`

.

`p = invErf(p)`

The inverse error functionThis is calculated using `inverf(p) = -inverfc(p + 1)`

, if you known of specific
approximation please file an issue or pull request.

`p = invErfc(p)`

The inverse complementary error functionThis uses a very common approximation of `inverfc(p)`

, see source code for more
details.

`p = gamma(x)`

The gamma functionThis acutally contains 3 diffrent approximations of `gamma(x)`

which one is
automatically determined by the `x`

value.

`p = logGamma(x)`

The logarithmic gamma functionFor values less than `12`

the result is calculated using `log(gamma(x))`

, in
any other case a specific approximation is used.

These are taken from the `jstat`

library and modified to fit intro the API
pattern used in this module. Futhermore they also take advanges of the special
`log1p`

function implemented in this module.

`p = beta(x)`

- The beta function`p = logBeta(x)`

- The logarithmic beta function`p = incBeta(x)`

- The incomplete beta function`p = invIncBeta(x)`

- The inverse incomplete beta function`y = log1p(x)`

- Calculates `y = ln(1 + x)`

When `x`

is a very small number computers calculates `ln(1 + x)`

as `ln(1)`

which
is `zero`

and then every thing is lost. This is a specific approximation of
`ln(1 + x)`

and should be used only in case of small values.

`y = logFactorial(x)`

- Calculates `y = ln(x!)`

`x!`

can quickly get very big, and exceed the limitation of the float value,
approimating `ln(x!)`

instead can in some cases solve this problem.

All functions are tested by comparing with a mathematical reference
either *MatLab*, *Maple* or *R*.

A special thank to John D. Cook, who writes a very good blog about some of these functions, and maintains a stand alone implementation catalog. See also this article about regarding floating point errors in some mathematical function: http://www.johndcook.com/blog/2010/06/07/math-library-functions-that-seem-unnecessary/

**The software is license under "MIT"**

Copyright (c) 2013 Andreas Madsen

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.