Convergence acceleration of logarithmic fixed point sequences

摘要

Let (xn) be some sequence generated by xn+1 = ƒ(xn) where ƒ(x)=(x) + ∑i ⩾ 1α p+1xp+i, p ⩾ 1, αp+1<0. For x0 > 0 small, it converges to zero logarithmically, i.e. limnxn+1xn = 1, thus we need algorithms for accelerating its convergence. Using asymptotic expansions in the analysis of the Δ2 and θ2-algorithms leads to modified iterated versions of the first one and to combinations with the iterated θ2-algorithm. In particular some superconvergence phenomena can be explained in this framework. A similar study can be made for other nonlinear algorithms known at present. Moreover, the above algorithms are also good accelerators for large classes of slowly convergent integrals and series.