Computation of continued fractions by square-root modification: Reflection and examples
The present paper is a continuation of the author's previous work on the computation, modifications and convergence acceleration of continued fractions. For continued fractions of the form \[ (1)\quad^{\infty}_{n=1}\frac{a_ n}{1}=\frac{a_ 1}{1\quad +}\frac{a_ 2}{1\quad +}...\frac{a_ n}{1\quad +}...,\quad a_ n\in {\mathbb{C}},\quad a_ n\neq 0, \] the linear fractional transformation \(\omega =S_ n(w)\) is defined by \((2)\quad S_ n(w)=\frac{a_ 1}{1\quad +}\frac{a_ 2}{1\quad +}...\frac{a_ n}{+\quad (1+w)}.\) Although the value of the continued fraction (1) is defined as \(\lim_{n\to \infty}S_ n(0)\), one shows how it is possible to determine this value more quickly, rather than \(S_ n(0)\), by using \(S_ n(w_ n)\) for some \(\{w_ n\}\). \(S_ n(w_ n)\) is sometimes called a ``modified'' approximant and \(\{w_ n\}^ a \)``modifying'' factor. Using these notions and the concept of an ``auxiliary continued fraction'', \(K(a\) \(*_ n/1)\), being ``near'' (1) in some sense \((a\) \(*_ n-a_ n\to 0\) is mostly used), the author defines the J- and D- approximants. A general result is found by the author for any limit periodic continued fraction, \(K(a_ n/1)\), where \(a_ n\to a\not\in (- \infty,-1/4]\) and several examples concerning the computation of continued fractions by means of J- and square-root approximation are also discussed. The speed of convergence for these types of approximants is comparatively investigated.
scientific article; zbMATH DE number 4048906
scientific article