January 2015 Archives

Gamma Gamma Hey!

One of the frustrating things about mathematics is that there are many important and popular functions that are easy to formulate but are extremely difficult to evaluate. We've already come across just such a function in the normal CDF, Φ, for which we were forced to resort to an arcane numerical approximation, filled to the brim with magic numbers. In this post, we shall take a look at a whole family of such functions; the gamma function and its relatives.

So without further ado; hey ho, let's go!

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The Tip Of The Romberg

Last time we took a first look at the numerical approximation of integrals with the trapezium rule, a rather simplistic algorithm with the unfortunate property that its accuracy was implicitly, rather than explicitly, specified.
As a user of numerical libraries, including my own, I would much rather have an algorithm that works to a given accuracy, or at the very least tries to, than have to figure it out for myself and I suggested that we might apply the lessons learned from our attempts to numerically approximate derivatives to do just that.

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Trapezium Artistry

We have covered in some detail the numerical approximation of differentiation but have yet to consider the inverse operation of integration and I think that it's high time that we got around to it...

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