Page 132 - IJOCTA-15-3

Srinivasarao Thota et al. / IJOCTA, Vol.15, No.3, pp.503-516 (2025)
be possible. Consequently, researchers have fo- threshold in less computational time. Further-
cused on developing algorithms with better con- more, it is more efficient than other methods in
vergence rates while minimizing the number of terms of its efficiency index, demonstrating that
function evaluations per iteration. The efficiency it can effectively solve scalar and vector nonlin-
index E = p 1/c introduced as a critical measure, ear equations. This paper is organized as follows:
estimates the convergence order p relative to the Section 2 presents an overview of various existing
number of function evaluations per step necessary schemes, along with the formulation and conver-
for a numerical scheme. The aim is to maximize gence analysis for the proposed ninth-order algo-
this index as larger values indicate improved ef- rithm. Theoretical comparisons and stability as-
ficiency. Some notable recently developed non- sessments are also discussed. Section 3 explores
linear methods include the homotopy perturba- real-life applications and demonstrates the effec-
tion approach, 3–5 the Adomian decomposition tiveness of the proposed scheme through numeri-
6
method, the variational iteration method, 7,8 and cal experiments. Finally, Section 4 discusses key
methods based on quadrature rules. 9,10 findings and highlights areas for future research
development. Numerous hybrid root-finding al-
This paper is based on prior work and de-
gorithms, using different techniques, are available
signs a new three-step numerical method with 26–37
in the literature and references therein.
ninth-order convergence. This method combines
an exponential series-based scheme (ExpM) 11–14 In this article, we present a new three-step it-
with the classical third-order Halley method erative method with a single parameter and mem-
(HM) 12,15–18 to achieve higher performance mea- ory. It is designed in a modular way, making it
sures than existing methods. Specifically, this easier to use for solving nonlinear equations effi-
ciently.
method minimizes the number of function evalu-
ations while maintaining high accuracy and com-
putational efficiency. Noor et al. 19 developed a 1.1. Previous materials and methods
novel two-step scheme with order p 1 p 2 by com-
bining two schemes of orders p 1 and p 2 . At each We reviewed various numerical techniques applied
iteration, the process adds new function evalu- to nonlinear models. Given the difficulty in ob-
ations, but these have a higher order of con- taining exact analytical solutions for such mod-
vergence. However, Noor et al. 20 presented two els, we employed numerical methods that offer
1,2
schemes with third- and fourth-order convergence approximations. The classical NR method is
that required five and eight function evaluations typically the first to come to mind when solv-
per iteration, respectively. Shah et al. 21 devel- ing nonlinear equations numerically. NR has a
oped a three-step scheme with third-order conver- second-order convergence, which is considered op-
gence at the cost of five function evaluations per timal . In simple terms, this entails one evalua-
iteration. Abro and Shaikh 21 developed a three- tion per iteration for both the function and its
step sixth-order convergent technique employing first derivative:
seven function evaluations per iteration. Even
though these methods converge at high orders, x n+1 = x n − f(x n ) , n = 0, 1, 2, . . . (1)
′
they incur extra computational costs in terms of f (x n )
central processing unit time. A similar method Halley’s third-order numerical scheme, Hal-
was applied in a recently published work 23 to pro- ley3, is a precise method 16,24 that calls for three
duce an effective three-step numerical technique. different function evaluations per iteration as fol-
The primary goal of this study is to design a novel lows: the function itself, its first derivative, and
hybrid three-step numerical approach for solving its second-order derivative.
scalar and vector nonlinear equations. Inspired by
these methods, we combined the classical third- 2f(x n)f (x n)
′
order Halley scheme 24 with a modified third-order x n+1 = x n − 2f (x n) −f(x n)f (x n) , n = 0, 1, 2, . . .
′′
′
2
Newton scheme 25 to create a novel three-step nu- (2)
merical scheme. This led to the development of a In a recent study, a pioneering numeri-
ninth-order method that requires just six function cal scheme called the modified Halley method
evaluations per iteration: three functions, two (MHM) was presented. As an interpolative tech-
first-order derivatives, and one second-order de- nique, it combines NR and HM as a correc-
rivative. Several advantages are associated with tion, while also employing a finite difference ap-
the proposed scheme over previous methods. It proximation for estimating the second derivative.
provides improved accuracy such that fewer iter- Remarkably, MHM 17 achieves fifth-order conver-
ations are necessary to reach a defined accuracy gence and requires four function evaluations per
504