Algebraic Computation of Matrix-like Pad\'e Approximants

Pad\'e approximation is a fairly well known technique for constructing rational approximants to functions represented as power series. There are a number of other rational approximants, which although less well known, also have interesting applications. These include simultaneous-Pad\'e and Hermite-Pad\'e approximants along with their vector and matrix generalizations. For example, Hermite used simultaneous Pad\'e approximants in his proof of the transcendence of~$e$; Hermite-Pad\'e approximation is a valuable tool used in the~{\sc Gfun} package of Salvy and Zimmerman; vector Hermite-Pad\'e computation is a major part of the new Van Hoeij algorithm for factoring differential operators.

We give an introduction to Pad\'e and matrix-like Pad\'e approximants giving some applications and methods of computation, both primarily from an algebraic point of view. We also discuss some recent work on the computation of such approximants. In particular, we show how to obtain recursive algorithms for these approximants by building bases for the modules of solutions.