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The water in the ocean, the air in the room, and a rubber ball have a common characteristic, they appear to completely occupy their respective domains. What this means is that the material occupies every point in the domain.

In this chapter we derive numerical methods to solve the first-order differential equation $$\displaystyle{ \frac{dy} {dt} = f(t,y),\;\;\text{ for }\;0

This chapter provides a brief introduction to the floating-point number system used in most scientific and engineering applications. A few examples are given in the next section illustrating some of the challenges using finite precision arithmetic, but it is worth quoting Donald Knuth to get things started. If you are unfamiliar with him, he was instrumental in the development of the analysis of algorithms, and is the creator of TeX. Anyway, here are the relevant quotes (Knuth [1997]

In this chapter one of the more common mathematical problems is studied, which is to find the solution, or solutions, of an equation of the form f(x) = 0. To illustrate the situation, we begin with a few examples.