Padé approximants for operators: theory and applications
The subject of this lecture note is of great importance nowadays, but space does not allow one to go into the details needed to give full credit to the achievement of the author. The book is divided into three chapters. The first treats the generalization of the concept of Padé approximants to the situation of nonlinear operators from a Banach space to a commutative Banach algebra (if the space and the algebra both are \({\mathbb{R}}\), the classical univariate approximant is recovered). The second chapter focuses its attention on maps from \({\mathbb{R}}^ p\) to \({\mathbb{R}}\), leading to multivariate approximants. It further gives an extensive comparison of the approximants proposed and those defined by Levin, Karlsson-Wallin, Lutterodt and Hillion. Finally, the third chapter is devoted to the (approximate) solution of nonlinear operator equations and gives a proof of the numerical stability of the Halley-iteration for the solution of a system of nonlinear equations. This well-written lecture note, which is lavishly endowed with numerical examples, is a must for workers on and users of abstract and multivariate Padé approximation!
scientific article; zbMATH DE number 3855666