Basic Computational Algorithms

Numbers are the lifeblood of statistics, and computational statistics relies heavily on how numbers are represented and manipulated on a computer. Computer hardware and statistical software handle numbers well, and the methodology of computer arithmetic is rarely a concern. However, whenever we push hardware and software to their limits with difficult problems, we can see signs of the mechanics of floating point arithmetic around the frayed edges. To work on difficult problems with confidence and explore the frontiers of statistical methods and software, we need to have a sound understanding of the foundations of computer arithmetic. We need to know how arithmetic works and why things are designed the way they are. As scientific computation began to rely heavily on computers, a monumental decision was made during the 1960s to change from base ten arithmetic to base two. Humans had been doing base ten arithmetic for only a few hundred years, during which time great advances were possible in science in a short period of time. Consequently, the resistance to this change was strong and understandable. The motivation behind the change to base two arithmetic is merely that it is so very easy to do addition (and subtraction) and multiplication in base two arithmetic. The steps are easy enough that a machine can be designed – wire a board of relays – or design a silicon chip – to do base two arithmetic. Base ten arithmetic is comparatively quite difficult, as its recent mathematical creation would suggest. However two big problems arise in changing from base ten to base two: (1) we need to constantly convert numbers written in base ten by humans to base two number system and then back again to base ten for humans to read the results, and (2) we need to understand the limits of arithmetic in a different number system.