Convergence acceleration of logarithmic fixed point sequences
A real sequence \((S_ n)\) which converges to its limit S, verifying: \(\lim (S_{n+1}-S)/(S_ n-S)=1,\) is a sequence converging logarithmically. Such a sequence needs algorithms for accelerating its convergence. So, a sequence \((x_ n)\) generated by: \(x_{n+1}=f(x_ n)\), where \(f(x)=x+\sum_{i\geq 1}\alpha_{p+i}x^{p+i};\) \(p\geq 1\), \(\alpha_{p+1}<0\), for \(x_ 0>0\) small, converges to zero logarithmically i.e. lim \(x_{n+1}/x_ n=1\). The author, using asymptotic expansions of the \(\Delta^ 2\)- and \(\theta\)-algorithms, obtains modified iterated versions of the first one and combinations of it with the iterated \(\theta_ 2\)-algorithm. He explains some superconvergence phenomena and shows that the algorithms are good accelerators for large classes of slowly convergent integrals and series.
scientific article; zbMATH DE number 4009628
scientific article