Exact real arithmetic
Exact real arithmetic is an interesting area: it is a deep connection between
- numeric methods
- and deep theoretic fondations of algorithms (and mathematics).
Its topic: computable real numbers raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also computability theory (see numbers-as-programs approaches).
Computable reals can be achieved by many approaches -- it is not one single theory.
Jean Vuillemin's Exact real computer arithmetic with continued fractions is very good article on the topic itself. It can serve also as a good introductory article, too, because it presents the connections to both mathematical analysis and computability theory. It discusses several methods, and it describes some of them in more details.
There are functional programming materials too, even with downloadable Haskell source.