Page 131 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.503-516 (2025)
https://doi.org/10.36922/ijocta.8171

RESEARCH ARTICLE

A novel ninth-order root-finding algorithm for nonlinear equations
with implementations in various software tools

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3
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2
Srinivasarao Thota , Amir Naseem , Thumati Gopi , Kashireddy Sai NandanReddy , Padarthi Sai
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Kousik , Thulasi Bikku , and Shanmugasundaram Palanisamy 4*
1
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amaravati,
Andhra Pradesh, India
2
Department of Mathematics, University of Management and Technology, Lahore, Punjab, Pakistan
3
Department of Computer Science and Engineering, Amrita School of Computing, Amrita Vishwa
Vidyapeetham, Amaravati, Andhra Pradesh, India
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Department of Mathematics, College of Natural Computational Sciences, MizanTepi University,
Mizan-Aman, South West Ethiopia Peoples’ Region, Ethiopia
t srinivasarao@av.amrita.edu, amir.kasuri89@gmail.com, av.en.u4cse22043@av.students.amrita.edu,
av.en.u4cse22020@av.students.amrita.edu, av.en.u4cse22037@av.students.amrita.edu,
b thulasi@av.amrita.edu, psserode@mtu.edu.et
ARTICLE INFO ABSTRACT
Article History:
Received: December 24, 2024
Nonlinear phenomena are prevalent in numerous fields, including economics,
1st revised: March 19, 2025
engineering, and natural sciences. Computational science continues to advance
2nd revised: April 25, 2025
through the development of novel numerical schemes and the refinement of
3nd revised: May 9, 2025
existing ones. Ideally, these numerical systems should offer both high-order
Accepted: May 14, 2025
convergence and computational efficiency. This article introduces a new three-
Published Online: June 19, 2025
step algorithm for solving nonlinear scalar equations, aiming to meet these
Keywords: criteria. The proposed approach requires six function evaluations per iteration
Root-finding algorithm and achieves ninth-order convergence. To demonstrate the efficiency of the
Nonlinear equations technique, various numerical examples are shown. Implementations of the
Implementations method are available in both Maple and Python, and it can be readily adapted
Halley method for use in other computational environments.
Exponential method
AMS Classification:
65H05, 65Hxx

1. Introduction it computationally efficient. This novel approach
has the potential to be a significant breakthrough
Many scientific and engineering problems require
in computational science, offering accurate solu-
nonlinear root-finding techniques for their solu-
tions to nonlinear problems for various areas of
tions. The field of computational science is con-
knowledge.
tinually evolving, with ongoing development of
numerical methods that offer higher-order conver- The Newton–Raphson (NR) method is one
gence and more efficient numerical schemes. This of the most widely used approaches for solving
has led to the establishment of a new ninth-order nonlinear equations, mainly because of its sim-
nonlinear root-finding technique aimed at improv- plicity. As a foundational method, it serves as
ing the solution of nonlinear scalar equations. An- a basis for the development of more sophisti-
other advantage of this method is its high con- cated numerical techniques. However, achieving
verging power, which means fewer function eval- an optimal rate of convergence, as conjectured by
uations are needed per iteration, thereby making Kung–Traub, Traub 1,2 theorem may not always
*Corresponding author
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