Generalizations of Shanks transformation and corresponding convergence acceleration algorithms via pfaffians

Firstly, a new sequence transformation that can be expressed in terms of a ratio of two pfaffians is derived based on a special kernel. It can be regarded as a direct generalization of Aitken’s Δ 2 process from the point of view of pfaffians and then the corresponding convergence acceleration algorithm is constructed. Numerical examples with applications of this algorithm are also presented. Secondly, we find a way to generalize the Shanks transformation via pfaffians so that a larger class of new sequence transformations are derived. The corresponding recursive algorithms are also proposed.