Convergence acceleration for continued fractions 𝐾(𝑎_{𝑛}/1)

Abstract

A known method for convergence acceleration of limit periodic continued fractions K ( a n / 1 ) , a n → a K({a_n}/1),{a_n} \to a , is to replace the approximants S n ( 0 ) {S_n}(0) by "modified approximants" S n ( f ∗ ) {S_n}({f^{\ast }}) , where f ∗ = K ( a / 1 ) f^{\ast } = K(a/1) . The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be S n ( f ( n ) ′ ) {S_n}({f^{(n)’}}) , where K ( a n ′ / 1 ) K({a’_n}/1) is a converging continued fraction whose tails f ( n ) ′ {f^{(n)\prime }} are all known, and where a n − a n ′ → 0 {a_n} - a_n^\prime \to 0 . As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.