Rational Approximations

October 3, 2017

Finding rational approximations to real numbers may help us simplify calculations in every day life, because using

\displaystyle \pi=\frac{355}{113}

makes back-of-the-envelope estimations much easier. It also may have some application in programming, when your CPU is kind of weak and do not deal well with floating point numbers. Floating point numbers emulated in software are very slow, so if we can dispense from them an use integer arithmetic, all the better.

However, finding good rational approximations to arbitrary constant is not quite as trivial as it may seem. Indeed, we may think that using something like

\displaystyle a=\frac{1000000 c}{1000000}

will be quite sufficient as it will give you 6 digits precision, but why use 3141592/1000000 when 355/113 gives you better precision? Certainly, we must find a better way of finding approximations that are simultaneously precise and … well, let’s say cute. Well, let’s see what we could do.

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ln n!

February 23, 2016

In the course of analyzing an algorithm, I used the simplifying hypothesis that the cost function is

\displaystyle c(n)\approx\sum_{i_1}^n \lg i=\lg n!.

That expression is cumbersome but we can get a really good simplified function to use as a proxy. Let’s see how.

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