Partial Padé prediction

When the first terms of a sequence (called sequence to predict) are known, a prediction method is a method which gives us an approximation of the following terms (the so-constructed sequence is called the predicted sequence). In this paper, we state two prediction methods, respectively called εp-prediction and partial Padé prediction, which are generalizations of Aitken's Δ2-prediction of Brezinski and Redivo-Zaglia [6] and Padé prediction of Gilewicz [8] which are very simple to use. In order to choose among the different partial Padé predictions, we study some of their properties. The most important points of this paper are:• the use of an extrapolation algorithm (the ε-algorithm), to obtain a prediction algorithm for each partial Padé prediction (which avoids solving a system); • the results about consistency obtained for the partial Padé prediction (i.e., under certain conditions, each term of the predicted sequence converges to the analogous term of the sequence to predict).