In the first part, a number of numerical approximation techniques are treated, among them interpolation and Lebesgue constants, least-squares, best approximation, near-best approximation, for polynomial, rational as well as trigonometric approximants. We will also study advanced techniques for solving structured linear systems, convergence acceleration, radial basis functions and the like.
In part two, we explore "experimental mathematics". This is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."