Series Prediction Based on Algebraic Approximants

Using sequence transformation and extrapolation algorithms for the prediction of further sequence elements from a finite number of known sequence elements is a topic of growing importance in applied mathematics. For a short introduction, see the book of Brezinski and Redivo Zaglia 1, Section 6.8 . We mention theoretical work on prediction properties of Padé approximants and related algorithms like the epsilon algorithm, and the iterated Aitken and Theta algorithms 2–5 , Levin-type sequence transformations 6, 7 , the E algorithm 4, 8 , and applications on perturbation series of physical problems 7, 9 . Here, we will concentrate on a different class of approximants, namely, the algebraic approximants. For a general introduction to these approximants and the related HermitePadé polynomials see 10 . Programs for these approximants are available 11 . We summarize those properties that are important for the following. Consider a function f of complex variable zwith a known formal power series