Computation of continued fractions by square-root modification: Reflection and examples
type: Item
instance of: scholarly article
title: Computation of continued fractions by square-root modification: Reflection and examples
zbMATH Open document ID: 0643.65009
DOI: 10.1016/0168-9274(83)90013-2
author: Haakon Waadeland
published in: Applied Numerical Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 4048906
zbMATH Keywords: incomplete gamma function
zbMATH Keywords: right tail
zbMATH Keywords: wrong tail
zbMATH Keywords: regular C-fraction expansion
zbMATH Keywords: square-root modification
zbMATH Keywords: J-modification
zbMATH Keywords: D-modification
zbMATH Keywords: convergence acceleration
zbMATH Keywords: continued fractions
MaRDI profile type: Publication
full work available at URL: 0168-9274(83)90013-2
OpenAlex ID: W2088997256
cites work: Q3961720
cites work: Convergence Acceleration for Continued Fractions K(a n /1)
cites work: Further Results on Convergence Acceleration for Continued Fractions K(a n /1)
cites work: Convergence of Limit k -Periodic Continued Fractions K (an /bn ), and of Subsequences of Their Tails
cites work: Q3729308
cites work: Q3824804
cites work: Q3345221
cites work: An asymptotic property for tails of limit periodic continued fractions
cites work: Convergence of Continued Fractions
cites work: Accelerating convergence of limit periodic continued fractions \(K(a_n/1)\)
cites work: Truncation Error Bounds for Limit Periodic Continued Fractions
cites work: Tales About Tails
cites work: Q3797644
cites work: Derivatives of continued fractions with applications to hypergeometric functions
cites work: Linear approximations to continued fractions \(K(z_ n/1)\)
description: scientific article; zbMATH DE number 4048906