Mixed precision arithmetic: hardware, algorithms, and analysis

This lecture is concerned with floating-point arithmetic and its effect on numerical algorithms. After briefly reviewing its basic properties, we will discuss recent evolutions of floating-point arithmetic, notably the emergence of very low precisions on modern computer hardware. Low precision arithmetics have shown great potential in terms of speed but, when used on their own, they can only yield results of correspondingly low accuracy. We will present a variety of mixed precision algorithms that combine low and high precisions in order to achieve both high speed and high accuracy.