The epsilon algorithm in a non-commutative algebra

摘要

In the case of a non-commutative algebra, the epsilon algorithm is deduced from the Padé approximants at t = 1, and from the use of the cross rule; their algebraic properties are a consequence of those verified by the Padé approximants. The computation of the coefficients is particularly studied. It is shown, that it does not exist any non-invertible needed elements if and only if the Hankel matrices Mk(ΔlSn) = (ΔlSn+i+j)k−1i=j=0, for l = 1, 2 and 3, have an inverse. Some results of convergence and convergence acceleration are also given.