The coolest piece of mathematical software I’ve come across in the last few years is Chebfun, <a href="http://www.chebfun.org" rel="nofollow">http://www.chebfun.org</a><p>Basic idea is to represent functions by high-degree polynomials which approximate them to near machine precision. After each operation, any terms which contribute at less than about 10^-15 are dropped, in the same way as bits get dropped after a standard floating point operation. This keeps the degree of the polynomial from growing exponentially.<p>The Chebfun team has found or invented a number of efficient algorithms for interacting with such polynomials, which make it practical to find roots, take integrals and derivatives, solve differential equations, combine multiple functions in various ways, and so on.<p>I think it must be the largest Matlab library anyone has yet built, and while I’m not the biggest fan of Matlab, they’ve done a great job integrating with the environment. Turns out Matlab has a bunch of object oriented features with operator and function overloading, so Chebfun can adopt most of the operators and functions for operating on regular Matlab arrays and overload them to work on their continuous function approximations. This makes for very clean and readable code for their users, though the internals get pretty complex.<p>Also recommended is Trefethen’s book Approximation Theory and Approximation Practice, which explains the mathematical ideas underlying the library. First 6 chapters are freely available online: <a href="http://www.chebfun.org/ATAP/" rel="nofollow">http://www.chebfun.org/ATAP/</a>