Some behaviour of convergents for θ-expansion and regular continued fraction (RCF) expansion
Khairun Nisak Muhammad, Hailiza Kamarulhaili · AIP conference proceedings · 2021
Motivated by problems in random continued fraction expansions, this research describes the design and implementation on θ-expansions of a real number, x in (0, θ) where 0 < θ < 1. Initially, for such a number θ, the convergent of θ-expansions is computed with comparison to the regular continued fraction (RCF)-convergent based on the samples of the approximated value, x. θ-expansions algorithm and continued fraction algorithm are applied in a Maple software to compute the θ-convergent and RCF-convergent. In this respect, the purpose of this research is to examine the growth rate of convergent for both expansions and provide the theoretical behaviours of θ-convergent in θ-expansions. Numerical results based on these samples show 3 conditions of convergence growth rate, which are C2 and C4 give the best decimal approximations in θ-expansions, whereas, Cn be the best approximations in RCF expansions. Therefore, this research reveals that various value of θ affect the performance on their growth rate of convergent for these two expansions. As θ approaching to 1, it reduces their convergence errors. The analysis of the experimental results reveals the RCF-convergent give a better performance compared to θ-convergent as its yield to less convergence errors. In addition, the value of convergent do converge to the approximated value, x and that is how the name of convergent comes.