Remarks to a definition of convergence acceleration illustrated by means of continued fractions \(K(a_ n/1)\) where \(a_ n\to 0\) (Q1095560)

If there are two sequences \(\{c_ n\}\) and \(\{c^*_ n\}\) converging to the same finite limit c, we usually say that \(\{c^*_ n\}\) converges faster to c than \(\{c_ n\}\) when \((1)\quad \alpha_ n=(c^*_ n-c)/(c_ n-c)\to 0.\) The author neither discards the definition (1) nor introduces a new concept on the acceleration, but she gives several examples to deserve attention. Her main remarks seem to be the following. First, since (1) is an asymptotic property, it may happen that \(| \alpha_ n|\) eventually tends to 0, but is very large for certain small n's. Second, when the computation of \(c^*_ n\) needs much more operations than that of \(c_ n\), it is reasonable to assume \((c^*_ n-c)/(c_{n+k(n)}-c)\to 0\) for some k(n) instead of (1), but the latter does not always hold intrinsically. She mainly discusses a continued fraction \(K(a_ n/1)\), \(a_ n\neq 0\), \(a_ n\to 0\) (but very slowly). A standard algorithm is the so-called square-root modification: \(S_ n(w_ n)=a_ 1/1+a_ 2/1+...+a_ n/(1+w_ n),\) \(w_ n=\sqrt{a_ n+(1/4)}-1/2,\) instead of \(f_ n=S_ n(0)\to f\). She gives a necessary and sufficient condition to be \(| S_ n(w_ n)-f| /| f_{n+k}-f| \to 0\) for each fixed k. She also gives a similar theorem for Aitken's \(\Delta^ 2\)-procedure. She illustrates her assertions through numerical examples in the final section.