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The behaviours of convergents in θ-expansions: computational insights based on θ-expansions algorithm using the maple software


Citation

Muhammad, K. N. and Kamarulhaili, H. and Asbullah, M. A. and Sapar, S. H. (2024) The behaviours of convergents in θ-expansions: computational insights based on θ-expansions algorithm using the maple software. Mathematical Modeling and Computing, 11 (4). pp. 1141-1151. ISSN 2312-9794; eISSN: 2415-3788

Abstract

Continued fractions arise naturally in long division and the theory of the approximation to real numbers by rational numbers. This research considered the implementation on the convergent of θ-expansions of real numbers of x∈(0,θ) with 0<θ<1. The convergent of θ-expansions are also called as θ-convergent of continued fraction expansions. This study aimed to establish the properties for a family of θ-convergent in θ-expansions. The idea of discovering the behaviours of θ-convergent evolved from the concept of regular continued fraction (RCF) expansion and sequences involved in θ-expansions. The θ-expansions algorithm was used to compute the values of θ-convergent with the help of Maple software. Consequently, it proved to be an efficient method for fast computer implementation. The growth rate of θ-convergent was investigated to highlight the performance of θ-convergent. The analysis on θ-convergent revealed the convergent that gives a better approximation yielding to fewer convergence errors. This whole paper thoroughly derived the behaviours of θ-convergent, which measure how well a number x is approximated by its convergents for almost all irrational numbers.


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Additional Metadata

Item Type: Article
Divisions: Faculty of Science
Centre for Foundation Studies in Science of Universiti Putra Malaysia
DOI Number: https://doi.org/10.23939/mmc2024.04.1141
Publisher: Lviv Polytechnic National University
Keywords: Θ-convergent; Θ-expansions; Θ-expansions algorithm; Continued fraction; Con- vergence errors
Depositing User: Ms. Zaimah Saiful Yazan
Date Deposited: 26 Jun 2025 04:36
Last Modified: 26 Jun 2025 04:36
Altmetrics: http://www.altmetric.com/details.php?domain=psasir.upm.edu.my&doi=10.23939/mmc2024.04.1141
URI: http://psasir.upm.edu.my/id/eprint/118163
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