Mathematics and Algorithms for Computer Algebra

I now want to apply the basic notions of computational representations and abstract algebra that I have developed so far to concrete algorithms, and briefly to consider their complexity. As I explained earlier, underlying all of computer algebra are the integers, so I will begin by considering algorithms for performing arithmetic on integers of unlimited magnitude, which I will call long integers. It is then a very small step to extend this to arbitrary rational numbers. I will not consider any approximate representations, such as rounded representations of the reals, because they are not fundamental to CA and they lie more within the domain of numerical analysis. Given algorithms to manipulate long integers it is a small extension to manipulate rounded real representations of unlimited magnitude and precision. I will not explicitly consider complex numbers because they are a trivial extension of the real numbers, and similarly arithmetic on the integers mod m, which as we have seen is very important, is essentially a trivial extension of integer arithmetic. The definitive presentation of the material discussed in this set of notes is in Chapter 4 “Arithmetic” in Vol. 2 “Seminumerical Algorithms” of the classic book The Art of Computer Programming by Donald E. Knuth (AddisonWesley, 1981), which henceforth I will refer to simply as “Knuth”, and most of it is in section 4.3.1 “Multiple-precision arithmetic: The classical algorithms”.