Optimal eighth-order Steffensen-type iterative family for multiple roots with applications to nonlinear models

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989100%3A27740%2F25%3A10258508" target="_blank" >RIV/61989100:27740/25:10258508 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.isr-publications.com/jmcs/articles-15574-optimal-eighth-order-steffensen-type-iterative-family-for-multiple-roots-with-applications-to-nonlinear-models" target="_blank" >https://www.isr-publications.com/jmcs/articles-15574-optimal-eighth-order-steffensen-type-iterative-family-for-multiple-roots-with-applications-to-nonlinear-models</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.22436/jmcs.040.04.03" target="_blank" >10.22436/jmcs.040.04.03</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Optimal eighth-order Steffensen-type iterative family for multiple roots with applications to nonlinear models

Popis výsledku v původním jazyce

Several advanced iterative techniques for finding multiple roots of higher order, along with the evaluation of derivatives, have been extensively studied and documented in the literature. However, the development of higher order methods without derivatives remains a challenging task, resulting in a scarcity of such techniques in existing research. Motivated by this observation, we propose a novel eighth-order iteration function of the Traub-Steffensen type. The suggested family employs the first-order divided difference and weight functions of one and three variables, optimizing performance for multiple roots with known multiplicity. The iterative scheme requires four functional evaluations per iteration achieving optimal eighth-order convergence in the sense of the Kung-Traub conjecture with an efficiency index of 1.6818. A comprehensive convergence analysis is conducted to confirm the optimality of the proposed method. Extensive numerical testing demonstrates the stability of the theoretical predictions and the favorable convergence behavior of the new scheme. To validate its practical utility, we explore various real-world nonlinear problems involving multiple roots, such as modeling energy distribution in a blackbody radiation, root clustering, and other applications. These comparisons reveal the effectiveness of the proposed scheme relative to other eighth-order iterative methods in terms of computational order of convergence, residual error, and the difference between successive iterations. Furthermore, the stable convergence behavior of the proposed method analyzed through graphical analysis using polynomial and transcendental functions. Basins of attraction are plotted for the designed eighth-order algorithm and compared with similar methods in the field. These graphical representations highlight the superior convergence speed and overall performance of the proposed algorithm, demonstrating its robust competitiveness in solving nonlinear problems with multiple roots.

Název v anglickém jazyce

Optimal eighth-order Steffensen-type iterative family for multiple roots with applications to nonlinear models

Popis výsledku anglicky

Several advanced iterative techniques for finding multiple roots of higher order, along with the evaluation of derivatives, have been extensively studied and documented in the literature. However, the development of higher order methods without derivatives remains a challenging task, resulting in a scarcity of such techniques in existing research. Motivated by this observation, we propose a novel eighth-order iteration function of the Traub-Steffensen type. The suggested family employs the first-order divided difference and weight functions of one and three variables, optimizing performance for multiple roots with known multiplicity. The iterative scheme requires four functional evaluations per iteration achieving optimal eighth-order convergence in the sense of the Kung-Traub conjecture with an efficiency index of 1.6818. A comprehensive convergence analysis is conducted to confirm the optimality of the proposed method. Extensive numerical testing demonstrates the stability of the theoretical predictions and the favorable convergence behavior of the new scheme. To validate its practical utility, we explore various real-world nonlinear problems involving multiple roots, such as modeling energy distribution in a blackbody radiation, root clustering, and other applications. These comparisons reveal the effectiveness of the proposed scheme relative to other eighth-order iterative methods in terms of computational order of convergence, residual error, and the difference between successive iterations. Furthermore, the stable convergence behavior of the proposed method analyzed through graphical analysis using polynomial and transcendental functions. Basins of attraction are plotted for the designed eighth-order algorithm and compared with similar methods in the field. These graphical representations highlight the superior convergence speed and overall performance of the proposed algorithm, demonstrating its robust competitiveness in solving nonlinear problems with multiple roots.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10100 - Mathematics

Projekt

—

Návaznosti

O - Projekt operacniho programu

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Mathematics and Computer Science

ISSN

2008-949X

e-ISSN

2008-949X

Svazek periodika

40

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

IR - Íránská islámská republika

Počet stran výsledku

20

Strana od-do

481-500

Kód UT WoS článku

001551738000001

EID výsledku v databázi Scopus

2-s2.0-105014743138