Padé approximants and related matters

Abstract

The aim of this chapter is to show how the theory of general orthogonal polynomials can be used to derive old and new results for Padé approximants and related matters as continued fractions and the ε-algorithm. All the known results are obtained much more easily by using general orthogonal polynomials than they were previously obtained; no special prerequisite is needed such as Schwein’s development or extensional identities. Moreover everything follows in a very natural way and a unified approach of Padé approximants and related subjects is obtained. New results are also extracted, such as recursive methods to compute Padé approximants or the connection between the topological ε -algorithm and the biconjugate gradient method. The first attempt to use the theory of general orthogonal polynomials in Padé approximants seems to be due to Wall [133]. This problem was also tackled in [3, 4, 25, 101, 125, 145].