The epsilon algorithm and related topics

The epsilon algorithm is recommended as the best all-purpose acceleration method for slowly converging sequences. It exploits the numerical precision of the data to extrapolate the sequence to its limit. We explain its connections with Padé approximation and continued fractions which underpin its theoretical base. Then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector-valued sequences, and some related topics. In this paper, we consider the class of methods based on using generalised inverses of vectors, and the formulation specifically includes the complex case wherever possible.