You could say that some of the equations that you encounter in describing physical systems can�t be solved in terms of familiar functions and that they require numerical calculations to solve. It would be misleading to say this however, because the reality is quite the opposite. Most of the equations that describe the real world are su_ciently complex that your only hope of solving them is to use numericalmethods. The simple equations that you �nd in introductory texts are there because they can be solvedin terms of elementary calculations. When you start to add reality, you quickly reach a point at which no amount of clever analytical ability will get you a solution. That becomes the subject of this chapter. In all of the examples that I present I�m just showing you a taste of the subject, but I hope that you will see the essential ideas of how to extend the concepts.