A Class of Higher-Order Iterative Methods for Simultaneous Solutions of Nonlinear Equations with Applications to Biological Problems

Abstract

A new class of high-order simultaneous iterative methods is constructed for the simultaneous computation of all simple roots of a nonlinear equation. The objective is to develop efficient iterative schemes with improved convergence characteristics and computational performance. The schemes are constructed by applying suitable weight functions and attain a convergence order of $2p+1$, where p denotes the order of the predictor iterative method. Their performance is investigated through comprehensive numerical experiments on several nonlinear biological models, including the Population Model with Allee Effect and Density–Dependent Migration, the Epidemic Disease Model, and the Michaelis–Menten Enzyme Kinetics Model, to illustrate the accuracy, efficiency, and convergence behavior of the proposed schemes. Additionally, error graphs are plotted to compare the proposed methods with existing methods. The numerical results show that some of the proposed schemes achieve faster convergence and lower residual errors than existing schemes and have better computational performance. The results indicate that the proposed methods are competitive and reliable for solving nonlinear equations arising in biological and applied mathematical models.

1. Introduction

Many science, engineering, biology, and applied mathematical applications involve nonlinear equations [1,2]. Consider the nonlinear equation

$$\mathcal{F}\left(z\right)=0,$$

(1)

where $\mathcal{F}:D\subseteq \mathbb{C}\to \mathbb{C}$ is a sufficiently differentiable function. In many cases analytical solutions are not available and iterative techniques are very important in providing a numerical approximation to the roots. These methods could be categorized as “single root method”, “multiple root method”, and “simultaneous root finding method”. Simultaneous iterative methods are extensively used for finding all roots of an equation simultaneously. These methods are efficient, well suited to parallel implementation, and generally converge rapidly [3,4,5,6]. Recent developments in simultaneous and multipoint iterative methods can be found in [7,8,9,10].

Among the earliest and most popular methods is the Weierstrass method [11], which was later developed independently by Durand [12], Kerner [13], Dochev [14], and Presic [15]. This method exhibits second-order convergence for polynomials $P\left(z\right)=0$ of degree n and serves as the foundation for many later developments in simultaneous root-finding, and is given by

$$z_r^{(m+1)}=z_r^{\left(m\right)}-{\displaystyle \frac{P\left(z_r^{\left(m\right)}\right)}{{\displaystyle {\displaystyle \prod _{s=1,s\ne r}^n}(z_r^{\left(m\right)}-z_s^{\left(m\right)})}}},m=0,1,\dots$$

(2)

Another important method is the Ehrlich–Aberth method [16,17], which achieves third-order convergence for polynomial equations $P\left(z\right)=0$ of degree n and is as follows:

$$z_r^{(m+1)}=z_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{P^{\prime }\left(z_r^{\left(m\right)}\right)}{P\left(z_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{s=1,s\ne r}^n}\frac{1}{z_r^{\left(m\right)}-z_s^{\left(m\right)}}}}},m=0,1,\dots$$

(3)

The local convergence of the Ehrlich method has been studied in detail by Proinov [18], while further convergence analyses of simultaneous methods include the works of Proinov and Ivanov [10] and Proinov and Vasileva [9].

Motivated by these classical methods, many researchers have developed higher-order simultaneous schemes. Petković et al. [19,20] proposed several high-order methods for simultaneously approximating polynomial zeros. A comprehensive treatment of multipoint methods for nonlinear equations is given in [7,8]. Later, Mir et al. [21] introduced families of fourth- and sixth-order methods, while Shams et al. [22] extended iterative methods designed for single-root problems to the simultaneous approximation of all roots. Cordero et al. [23] presented a general technique for nonlinear equations that constructs a simultaneous method of convergence order $2p$, where p is the order of the predictor iterative method. The local convergence of higher-order iterative methods for nonlinear equations has also been investigated in [24].

The present work is motivated by the fourth-order simultaneous method proposed by Wang and Zheng [25] for approximating the zeros of polynomial equations $P\left(z\right)=0$:

$$\begin{array}{cc}\hfill z_r^{(m+1)}& =z_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}\hfill & {\displaystyle \frac{P^{\prime }\left(z_r^{\left(m\right)}\right)}{P\left(z_r^{\left(m\right)}\right)}-\frac{P^{″}\left(z_r^{\left(m\right)}\right)}{2P^{\prime }\left(z_r^{\left(m\right)}\right)}-}\hfill \\ \hfill & {\displaystyle \frac{P\left(z_r^{\left(m\right)}\right)}{2P^{\prime }\left(z_r^{\left(m\right)}\right)}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{z_r^{\left(m\right)}-z_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(z_r^{\left(m\right)}-z_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(4)

$r=1,2,\dots ,n$, where n is the degree of $P\left(z\right)$. The Wang–Zheng method achieves fourth-order convergence for polynomials by making use of second-derivative information. Recent work on Wang–Zheng-type methods further investigates their convergence properties, including extensions to multiple zeros [26].

Inspired by this idea, we extend the method to nonlinear scalar equations. To avoid the direct evaluation of the second derivative, we replace it with the finite-difference approximation, which approximates $P^{″}\left(z_r^{\left(m\right)}\right)$. By doing this, the number of functional evaluations decrease. Furthermore, suitable weight functions are incorporated to construct a new class of simultaneous iterative methods. We also introduce these weight functions to obtained a more generalized family of methods. The obtained family is presented in Equation (5). The convergence analysis shows that the resulting schemes attain a convergence order of $2p+1$, where p denotes the order of the predictor iterative method.

The remainder of the paper is organized as follows. Section 2 presents the construction of the proposed methods and their convergence analysis. Section 3 presents numerical experiments on several nonlinear biological models, along with error graphs, tables, and comparative analyses to evaluate the convergence behavior and performance of the proposed schemes. Finally, concluding remarks are given in Section 4.

2. Construction and Convergence Analysis of the Proposed Scheme

Consider the nonlinear equation $\mathcal{F}\left(z\right)=0,$ where $\mathcal{F}:\mathbb{C}\to \mathbb{C}$ is an analytic function possessing n distinct simple roots denoted by ${\lambda }_r$, for $r=1,2,\dots ,n$. Let $z_r^{\left(m\right)}$ denote the approximation to the root ${\lambda }_r$ at the $m^{th}$ iteration, where $m=0,1,2,\dots$. The proposed simultaneous iterative scheme is defined as follows:

$$\begin{array}{cc}\hfill w_r^{\left(m\right)}& =\Phi \left(z_r^{\left(m\right)}\right),\hfill \\ \hfill z_r^{(m+1)}& =w_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}\hfill & {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-}\hfill \\ \hfill & {\displaystyle \frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(5)

where $\Phi$ denotes the fixed-point function corresponding to the predictor method of order p, which is used to construct the proposed multipoint method. Here, ${\Gamma }_r^{\left(m\right)}={\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)-{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}{w_r^{\left(m\right)}-z_r^{\left(m\right)}}}$ represents an approximation to the second derivative, and the weight function $\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right):\mathbb{C}\to \mathbb{C}$ is a holomorphic mapping in a neighborhood of the origin 0. The variable ${\sigma }_r^{\left(m\right)}$ of the weight function is

$${\sigma }_r^{\left(m\right)}={\displaystyle \frac{\mathcal{F}\left(w_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}}.$$

The first step of the proposed scheme given in Equation (5) can be any iterative method having convergence order p. The second step is designed in such a way that the resulting method attains a higher convergence order. Theorem 1 establishes that the proposed method $K{M}_{{\Phi }_s}$ achieves a convergence order of at least $2p+1$, provided that the weight function $\mathcal{W}$ satisfies certain suitable conditions given in Theorem 1.

2.1. Convergence Analysis

The following result establishes the convergence analysis of the proposed simultaneous iterative scheme (5), denoted by $\left(K{M}_{{\Phi }_s}\right)$.

Theorem 1.

Let $\mathcal{F}:D\subseteq \mathbb{C}\to \mathbb{C}$ be an analytic function defined in a neighborhood D containing the simple roots ${\lambda }_r$, $r=1,2,\dots ,n$, such that $\mathcal{F}\left({\lambda }_r\right)=0{\mathcal{F}}^{\prime }\left({\lambda }_r\right)\ne 0,r=1,2,\dots ,n.$ Assume that $z_1^{\left(0\right)},z_2^{\left(0\right)},\dots ,z_n^{\left(0\right)}$ are sufficiently close initial approximations to the roots ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n,$ respectively. Further, suppose that the predictor step of (5) is generated by any iterative method of convergence order p that obtains simple roots non-simultaneously. Then the proposed scheme (5) attains at least $2p+1$ order of convergence provided that the weight function $\mathcal{W}$ is a not null function that satisfies the conditions

$$\mathcal{W}\left(0\right)=0,|{\mathcal{W}}^{\left(1\right)}\left(0\right)|<\infty .$$

(6)

Proof. 

Let $e_r^{\left(m\right)}=z_r^{\left(m\right)}-{\lambda }_r$ denote the error in the approximation of the $r^{th}$ root at the $m^{th}$ iteration. By expanding $\mathcal{F}\left(z_r^{\left(m\right)}\right)$ and ${\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)$ in a Taylor series about the simple root ${\lambda }_r$, $r=1,2,\dots ,n$, we obtain

$$\begin{array}{cc}\hfill \mathcal{F}\left(z_r^{\left(m\right)}\right)=& {\mathcal{F}}^{\prime }\left({\lambda }_r\right)[e_r^{\left(m\right)}+B_2{\left(e_r^{\left(m\right)}\right)}^2+B_3{\left(e_r^{\left(m\right)}\right)}^3+B_4{\left(e_r^{\left(m\right)}\right)}^4\hfill \\ \hfill & +B_5{\left(e_r^{\left(m\right)}\right)}^5+B_6{\left(e_r^{\left(m\right)}\right)}^6+O\left({\left(e_r^{\left(m\right)}\right)}^7\right)].\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill {\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)=& {\mathcal{F}}^{\prime }\left({\lambda }_r\right)[1+2B_2{e}_r^{\left(m\right)}+3B_3{\left(e_r^{\left(m\right)}\right)}^2\hfill \\ \hfill & +4B_4{\left(e_r^{\left(m\right)}\right)}^3+5B_5{\left(e_r^{\left(m\right)}\right)}^4+6B_6{\left(e_r^{\left(m\right)}\right)}^5+O\left({\left(e_r^{\left(m\right)}\right)}^6\right)],\hfill \end{array}$$

(8)

where $B_q={\displaystyle \frac{1}{q!}}{\displaystyle \frac{{\mathcal{F}}^{\left(q\right)}\left({\lambda }_r\right)}{{\mathcal{F}}^{\prime }\left({\lambda }_r\right)}}.$ On the other hand, let ${\widehat{e}}_{w,r}^{\left(m\right)}=w_r^{\left(m\right)}-{\lambda }_r$ denote the error in the predictor approximation $w_r^{\left(m\right)}$ to the $r^{\mathrm{th}}$ root ${\lambda }_r$. Taking into account that the predictor has convergence order p, its error can be expanded around ${\lambda }_r$ as

$$\begin{array}{ccc}\hfill {\widehat{e}}_{w,r}^{\left(m\right)}& =& w_r^{\left(m\right)}-{\lambda }_r\hfill \\ & =& M_1{\left(e_r^{\left(m\right)}\right)}^p+M_2{\left(e_r^{\left(m\right)}\right)}^{p+1}+M_3{\left(e_r^{\left(m\right)}\right)}^{p+2}\hfill \\ & & +M_4{\left(e_r^{\left(m\right)}\right)}^{p+3}+M_5{\left(e_r^{\left(m\right)}\right)}^{p+4}+M_6{\left(e_r^{\left(m\right)}\right)}^{p+5}+O\left({\left(e_r^{\left(m\right)}\right)}^{p+6}\right),\hfill \end{array}$$

(9)

where $M_1\ne 0$.

Using the Taylor series expansion of $\mathcal{F}\left(w_r^{\left(m\right)}\right)$ and ${\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)$ about the simple root ${\lambda }_r$, we obtain

$$\begin{array}{ccc}\hfill \mathcal{F}\left(w_r^{\left(m\right)}\right)& =& {\mathcal{F}}^{\prime }\left({\lambda }_r\right)\left[M_1{\left(e_r^{\left(m\right)}\right)}^p+M_2{\left(e_r^{\left(m\right)}\right)}^{p+1}+M_3{\left(e_r^{\left(m\right)}\right)}^{p+2}+M_4{\left(e_r^{\left(m\right)}\right)}^{p+3}\right.\hfill \\ & & +M_5{\left(e_r^{\left(m\right)}\right)}^{p+4}+M_6{\left(e_r^{\left(m\right)}\right)}^{p+5}+B_2{M}_1^2{\left(e_r^{\left(m\right)}\right)}^{2p}+2B_2{M}_1{M}_2{\left(e_r^{\left(m\right)}\right)}^{2p+1}\hfill \\ & & \left.+B_2{M}_2^2{\left(e_r^{\left(m\right)}\right)}^{p+2}+2B_2{M}_1{M}_3{\left(e_r^{\left(m\right)}\right)}^{2p+2}+B_3{M}_1^3{\left(e_r^{\left(m\right)}\right)}^{3p}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+1}\right)\right],\hfill \end{array}$$

(10)

and

$$\begin{array}{ccc}\hfill {\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)& =& {\mathcal{F}}^{\prime }\left({\lambda }_r\right)\left[1+2B_2{M}_1{\left(e_r^{\left(m\right)}\right)}^p+2B_2{M}_2{\left(e_r^{\left(m\right)}\right)}^{p+1}+2B_2{M}_3{\left(e_r^{\left(m\right)}\right)}^{p+2}\right.\hfill \\ & & +2B_2{M}_4{\left(e_r^{\left(m\right)}\right)}^{p+3}+2B_2{M}_5{\left(e_r^{\left(m\right)}\right)}^{p+4}+2B_2{M}_6{\left(e_r^{\left(m\right)}\right)}^{p+5}+3B_3{M}_1^2{\left(e_r^{\left(m\right)}\right)}^{2p}\hfill \\ & & \left.+6B_3{M}_1{M}_2{\left(e_r^{\left(m\right)}\right)}^{2p+1}+(3B_3{M}_2^2+6B_3{M}_1{M}_3){\left(e_r^{\left(m\right)}\right)}^{2p+2}+O\left({\left(e_r^{\left(m\right)}\right)}^{2p+3}\right)\right].\hfill \end{array}$$

(11)

The value of ${\sigma }_r^{\left(m\right)}$ is obtained from Equalities (10) and (11).

$$\begin{array}{ccc}\hfill {\sigma }_r^{\left(m\right)}& =& {\displaystyle \frac{\mathcal{F}\left(w_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}}\hfill \\ & =& M_1{\left(e_r^{\left(m\right)}\right)}^p+M_2{\left(e_r^{\left(m\right)}\right)}^{p+1}+M_3{\left(e_r^{\left(m\right)}\right)}^{p+2}+M_4{\left(e_r^{\left(m\right)}\right)}^{p+3}+M_5{\left(e_r^{\left(m\right)}\right)}^{p+4}\hfill \\ & & +M_6{\left(e_r^{\left(m\right)}\right)}^{p+5}-B_2{M}_1^2{\left(e_r^{\left(m\right)}\right)}^{2p}-2B_2{M}_1{M}_2{\left(e_r^{\left(m\right)}\right)}^{2p+1}\hfill \\ & & +(-B_2{M}_2^2-2B_2{M}_1{M}_3){\left(e_r^{\left(m\right)}\right)}^{2p+2}+(2B_2^2-2B_3)M_1^3{\left(e_r^{\left(m\right)}\right)}^{3p}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+1}\right).\hfill \end{array}$$

(12)

From Equation (12), it follows that ${\sigma }_r^{\left(m\right)}$ is of order $O\left({\left(e_r^{\left(m\right)}\right)}^p\right)$. Since the weight function $\mathcal{W}:\mathbb{C}\to \mathbb{C}$ is holomorphic in a neighborhood of the origin 0, as assumed above, it admits a convergent Taylor (Maclaurin) expansion about 0 as follows:

$$\begin{array}{ccc}\hfill \mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)& =& \mathcal{W}\left(0\right)+{\sigma }_r^{\left(m\right)}{\mathcal{W}}^{\left(1\right)}\left(0\right)+{\displaystyle \frac{1}{2!}}{\left({\sigma }_r^{\left(m\right)}\right)}^2{\mathcal{W}}^{\left(2\right)}\left(0\right)+O\left({\left({\sigma }_r^{\left(m\right)}\right)}^3\right)\hfill \\ & =& \mathcal{W}\left(0\right)+W^{\left(1\right)}\left(0\right)\left(M_1{\left(e_r^{\left(m\right)}\right)}^p+M_2{\left(e_r^{\left(m\right)}\right)}^{p+1}+M_3{\left(e_r^{\left(m\right)}\right)}^{p+2}+M_4{\left(e_r^{\left(m\right)}\right)}^{p+3}\right.\hfill \\ & & \left.+M_5{\left(e_r^{\left(m\right)}\right)}^{p+4}+M_6{\left(e_r^{\left(m\right)}\right)}^{p+5}\right)+\left(-B_2{W}^{\left(1\right)}\left(0\right)M_1^2+W^{\left(2\right)}\left(0\right)M_1^2\right){\left(e_r^{\left(m\right)}\right)}^{2p}\hfill \\ & & +\left(-2B_2{W}^{\left(1\right)}\left(0\right)M_1{M}_2+2W^{\left(2\right)}\left(0\right)M_1{M}_2\right){\left(e_r^{\left(m\right)}\right)}^{2p+1}\hfill \\ & & +\left(-B_2{W}^{\left(1\right)}\left(0\right)M_2^2+W^{\left(2\right)}\left(0\right)M_2^2-2B_2^2{W}^{\left(1\right)}\left(0\right)M_1{M}_3+2W^{\left(2\right)}\left(0\right)M_1{M}_3\right){\left(e_r^{\left(m\right)}\right)}^{2p+2}\hfill \\ & & +\left(2B_2^2{W}^{\left(1\right)}\left(0\right)-2B_3{W}^{\left(1\right)}\left(0\right)-2B_2{W}^{\left(2\right)}\left(0\right)\right)M_1^3{\left(e_r^{\left(m\right)}\right)}^{3p}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+1}\right).\hfill \end{array}$$

(13)

Further, using the expansions (8) and (11), we obtain

$$\begin{array}{ccc}\hfill {\Gamma }_r^{\left(m\right)}& =& {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)-{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}{w_r^{\left(m\right)}-z_r^{\left(m\right)}}}\hfill \\ & =& {\mathcal{F}}^{\prime }\left({\lambda }_r\right)\left[2B_2+3B_3\left(e_r^{\left(m\right)}\right)+4B_4{\left(e_r^{\left(m\right)}\right)}^2+3B_3{M}_1{\left(e_r^{\left(m\right)}\right)}^p\right.\hfill \\ & & +\left(4B_4{M}_1+3B_3{M}_2\right){\left(e_r^{\left(m\right)}\right)}^{p+1}+\left(4B_4{M}_2+3B_3{M}_3\right){\left(e_r^{\left(m\right)}\right)}^{p+2}\hfill \\ & & +\left(4B_4{M}_3+3B_3{M}_4\right){\left(e_r^{\left(m\right)}\right)}^{p+3}+\left(4B_4{M}_4+3B_3{M}_5\right){\left(e_r^{\left(m\right)}\right)}^{p+4}\hfill \\ & & +\left(4B_4{M}_5+3B_3{M}_6\right){\left(e_r^{\left(m\right)}\right)}^{p+5}+4B_4{M}_1^2{\left(e_r^{\left(m\right)}\right)}^{2p}\hfill \\ & & +\left(8B_4{M}_1{M}_2+3B_3{M}_2^2+6B_3{M}_1{M}_3\right){\left(e_r^{\left(m\right)}\right)}^{2p+1}\hfill \\ & & +\left(4B_4{M}_2^2+8B_4{M}_1{M}_3+6B_3{M}_2{M}_3+6B_3{M}_1{M}_4\right.\hfill \\ & & \left.-4B_2{M}_2{M}_4-4B_2{M}_1{M}_5\right){\left(e_r^{\left(m\right)}\right)}^{2p+2}\hfill \\ & & +\left(-6B_3{M}_1{M}_2^2-2B_2{M}_2^3-3B_3{M}_1^2{M}_3\right.\hfill \\ & & \left.\left.-12B_2{M}_1{M}_2{M}_3-6B_2{M}_1^2{M}_4\right){\left(e_r^{\left(m\right)}\right)}^{3p}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+1}\right)\right].\hfill \end{array}$$

Now, the second step of the proposed scheme (5) as $z_r^{(m+1)}=w_r^{\left(m\right)}-{\Delta }_r^{\left(m\right)}$, where ${\Delta }_r^{\left(m\right)}$ can be written as:

$${\Delta }_r^{\left(m\right)}={\displaystyle \frac{2\mathcal{F}\left(w_r^{\left(m\right)}\right){\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{2{\left({\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)\right)}^2-{\Gamma }_r^{\left(m\right)}F\left(w_r^{\left(m\right)}\right)-W\left({\sigma }_r^{\left(m\right)}\right)\mathcal{F}\left(w_r^{\left(m\right)}\right){\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)\Omega }},$$

(14)

where

$$\Omega =\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}{\displaystyle \frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}{\displaystyle \frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}}\right].$$

Therefore, using the values of (14) and (9) into the second step of proposed method (5)

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& {\widehat{e_w}}_r^{\left(m\right)}-{\Delta }_r^{\left(m\right)}\hfill \\ & =& -1/2W\left(0\right)M_1^2{\left(e_r^{\left(m\right)}\right)}^{2p}+\left(-(3/2)B_3{M}_1-W\left(0\right)M_2\right)M_1{\left(e_r^{\left(m\right)}\right)}^{2p+1}\hfill \\ & & +\left(-3B_3{M}_1{M}_2-1/2W\left(0\right)M_2^2-W\left(0\right)M_1{M}_3-2B_4{M}_1^2\right){\left(e_r^{\left(m\right)}\right)}^{2p+2}\hfill \\ & & +\left(B_2^2+1/2B_3-1/2{\mathcal{F}}^{\prime }\left({\lambda }_r\right)W^{\left(1\right)}\left(0\right)-1/4W{\left(0\right)}^2\right)M_1^3{\left(e_r^{\left(m\right)}\right)}^{3p}\hfill \\ & & +3B_2{B}_3{M}_1^3{\left(e_r^{\left(m\right)}\right)}^{3p+1}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+2}\right).\hfill \end{array}$$

(15)

To achieve convergence of order at least $2p+1$ using the proposed scheme defined in Equation (5), the coefficient of the term ${\left(e_r^{\left(m\right)}\right)}^{2p}$ must vanish. Thus, from Equation (15), the following condition involving the weight function is obtained:

$$\mathcal{W}\left(0\right)=0\mathrm{and}\left|W^{\left(1\right)}\left(0\right)\right|<+\infty .$$

(16)

Equation (16) provides the required restriction on the weight function $\mathcal{W}$. By substituting Equation (16) into Equation (15), the proposed scheme (5) satisfies the following error equation:

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& z_r^{(m+1)}-{\lambda }_r\hfill \\ & =& -{\displaystyle \frac{3}{2}}{B}_3{M}_1^2{\left(e_r^{\left(m\right)}\right)}^{2p+1}+\left(-2B_4{M}_1-3B_3{M}_2\right)M_1{\left(e_r^{\left(m\right)}\right)}^{2p+2}\hfill \\ & & +\left(B_2^2+{\displaystyle \frac{1}{2}}{B}_3-{\displaystyle \frac{1}{2}}{\mathcal{F}}^{\prime }\left({\lambda }_r\right)W^{\left(1\right)}\left(0\right)\right)M_1^3{\left(e_r^{\left(m\right)}\right)}^{3p}\hfill \\ & & +3B_2{B}_3{M}_1^3{\left(e_r^{\left(m\right)}\right)}^{3p+1}+O\left({\left(e_r^{\left(m\right)}\right)}^{3p+2}\right).\hfill \end{array}$$

(17)

Hence, the proposed simultaneous iterative scheme attains a convergence order of at least $2p+1$ when the first step is chosen as any existing iterative method of order p. This completes the proof.    □

2.2. Special Cases of the Proposed Class

In this subsection, several special cases of the proposed class of iterative schemes (5) are constructed by choosing different existing iterative methods in the first step $\Phi \left(z_r^{\left(m\right)}\right)$. For each selected scheme, a new modified variant is obtained and denoted by $K{M}_{{\Phi }_s}$, $s=1,2,3,4$.

(1)

By considering the first step as the damped Newton’s method of first-order convergence and then applying the second step of the proposed scheme, a third-order method is obtained, denoted by $K{M}_{{\Phi }_1}$, which is given as follows:

$$\begin{array}{ccc}\hfill w_r^{\left(m\right)}& =& z_r^{\left(m\right)}-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}& =& w_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}& {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-}\hfill \\ & {\displaystyle \frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(18)

In this case, $p=1$ and, therefore, $2p+1=3p=3$. So, its error equation is

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& \left[-{\displaystyle \frac{3}{2}}{B}_3{M}_1^2+\left(B_2^2+{\displaystyle \frac{1}{2}}{B}_3-{\displaystyle \frac{1}{2}}{\mathcal{F}}^{\prime }\left({\lambda }_r\right)W^{\left(1\right)}\left(0\right)\right)M_1^3\right]{\left(e_r^{\left(m\right)}\right)}^3+O\left({\left(e_r^{\left(m\right)}\right)}^4\right).\hfill \end{array}$$

(2)

Choosing the first step as the classical Newton’s method [1] of quadratic convergence and then applying the proposed correction step yields a fifth-order method, denoted by $K{M}_{{\Phi }_2}$, as follows:

$$\begin{array}{ccc}\hfill w_r^{\left(m\right)}& =& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}& =& w_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}& {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-}\hfill \\ & {\displaystyle \frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(19)

In this case, $p=2$ and $2p+1=5<3p=2p+2=6$. So, its error equation is

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& -3/2B_3{M}_1^2{\left(e_r^{\left(m\right)}\right)}^5+O\left({\left(e_r^{\left(m\right)}\right)}^6\right).\hfill \end{array}$$

(3)

Applying the proposed correction step to the modified Newton’s method [27] of cubic convergence results in a seventh-order method denoted by $K{M}_{{\Phi }_3}$, which is expressed as

$$\begin{array}{ccc}\hfill w_r^{\left(m\right)}& =& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}}\right)+\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}& =& w_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}& {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-}\hfill \\ & {\displaystyle \frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(20)

In this case, as $p=3$, $7=2p+1<3p$, and the resulting error equation is

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& -3/2B_3{M}_1^2{\left(e_r^{\left(m\right)}\right)}^7+O\left({\left(e_r^{\left(m\right)}\right)}^8\right).\hfill \end{array}$$

(4)

If two consecutive Newton’s steps are employed in the first stage, then the resulting predictor possesses fourth-order convergence ($p=4$). By applying the proposed correction step, a ninth-order method is obtained, denoted by $K{M}_{{\Phi }_4}$, and defined as follows:

$$\begin{array}{ccc}\hfill u_r^{\left(m\right)}& =& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill w_r^{\left(m\right)}& =& u_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(u_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(u_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}& =& w_r^{\left(m\right)}-{\displaystyle \frac{1}{\begin{array}{cc}& {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-}\hfill \\ & {\displaystyle \frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}\left[{\left({\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}\right)}^2+{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{{\left(w_r^{\left(m\right)}-w_s^{\left(m\right)}\right)}^2}\right]}\hfill \end{array}}}.\hfill \end{array}$$

(21)

Now, $9=2p+1<3p$, being it error equation

$$\begin{array}{ccc}\hfill e_r^{(m+1)}& =& -3/2B_3{M}_1^2{\left(e_r^{\left(m\right)}\right)}^9+O\left({\left(e_r^{\left(m\right)}\right)}^{10}\right).\hfill \end{array}$$

2.3. Some Special Cases of Weight Functions

This section presents different rational forms of the weight function $\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)$ used in the proposed method (5). For convenience, the methods corresponding to the first, second, and third weight functions are denoted by $K{M}_{{\Phi }_s}^{\left(1\right)},K{M}_{{\Phi }_s}^{\left(2\right)},K{M}_{{\Phi }_s}^{\left(3\right)},$ respectively, where $s=1,2,3,4$. The following weight functions are considered:

• For the first weight function, the corresponding methods are denoted by $K{M}_{{\Phi }_s}^{\left(1\right)}$:

  $$\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)={\sigma }_r^{\left(m\right)}.$$
• For the second weight function, the corresponding methods are denoted by $K{M}_{{\Phi }_s}^{\left(2\right)}$:

  $$\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)={\displaystyle \frac{{\sigma }_r^{\left(m\right)}}{{\sigma }_r^{\left(m\right)}-1}}.$$
• For the third weight function, the corresponding methods are denoted by $K{M}_{{\Phi }_s}^{\left(3\right)}$:

  $$\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)={\displaystyle \frac{{\sigma }_r^{\left(m\right)}\left({\left({\sigma }_r^{\left(m\right)}\right)}^2+5\right)}{1+6{\sigma }_r^{\left(m\right)}+3{\left({\sigma }_r^{\left(m\right)}\right)}^2+{\left({\sigma }_r^{\left(m\right)}\right)}^3}}.$$

3. Numerical Experiments and Computational Analysis

In this section, several nonlinear test problems are considered to investigate the convergence behavior, numerical stability, and computational efficiency of the proposed simultaneous iterative schemes $\left(K{M}_{{\Phi }_s}^{\left(i\right)}\right)$, $s=1,2,3,4$ and $i=1,2,3$, defined in Equation (5).

All numerical computations and graphical illustrations were performed using Mathematica Version 11.1 with variable-precision arithmetic at 10,000 significant digits. The implementation of the proposed $K{M}_{{\Phi }_s}^{\left(i\right)}$ method follows directly from the iterative scheme given in (5), and its computational steps are outlined in Algorithm 1. For each iteration, the values $w_r^{\left(m\right)}=\Phi \left(z_r^{\left(m\right)}\right)$ are obtained from the current approximations, and the new approximations $z_r^{(m+1)}$, $r=1,\dots ,n$, are then computed from (5) simultaneously. The compared methods were implemented in the same computational environment according to their respective iterative schemes. The iterations were terminated when

$${∥z^{\left(m\right)}-z^{(m-1)}∥}_2+{∥\mathcal{F}\left(z^{\left(m\right)}\right)∥}_2<{10}^{-500},$$

or when the prescribed maximum number of iterations was reached.

Algorithm 1 Computational procedure for the scheme $K{M}_{{\Phi }_s}^{\left(i\right)}$

Require:  $\mathcal{F}$, ${\mathcal{F}}^{\prime }$; fixed-point function $\Phi$; weight function $\mathcal{W}$; initial approximations $z^{\left(0\right)}=\left(z_1^{\left(0\right)},\dots ,z_n^{\left(0\right)}\right)$ with pairwise distinct components; tolerance $ϵ={10}^{-500}$; maximum number of iterations $m_{\max }$; working precision (10,000 digits).   1: $m\leftarrow 0$   2: repeat   3:     for $r=1,\dots ,n$ do   4:         $w_r^{\left(m\right)}\leftarrow \Phi \left(z_r^{\left(m\right)}\right)$; evaluate $\mathcal{F}\left(w_r^{\left(m\right)}\right)$ and ${\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)$   5:     end for   6:     for $r=1,\dots ,n$ do   7:         ${\Gamma }_r^{\left(m\right)}\leftarrow {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)-{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}{w_r^{\left(m\right)}-z_r^{\left(m\right)}}}$;    ${\sigma }_r^{\left(m\right)}\leftarrow {\displaystyle \frac{\mathcal{F}\left(w_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}}$   8:         $q_{rs}^{\left(m\right)}\leftarrow {\displaystyle \frac{1}{w_r^{\left(m\right)}-w_s^{\left(m\right)}}}$ for $s\ne r$;    ${\Omega }_r^{\left(m\right)}\leftarrow {\left({\sum }_{s\ne r}{q}_{rs}^{\left(m\right)}\right)}^2+{\sum }_{s\ne r}{\left(q_{rs}^{\left(m\right)}\right)}^2$   9:         $z_r^{(m+1)}\leftarrow w_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}-\frac{{\Gamma }_r^{\left(m\right)}}{2{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}-\frac{\mathcal{W}\left({\sigma }_r^{\left(m\right)}\right)}{2}{\Omega }_r^{\left(m\right)}}}}$  10:     end for  11:     $m\leftarrow m+1$; compute $\parallel z^{\left(m\right)}-z^{(m-1)}{\parallel }_2$ and $\parallel \mathcal{F}\left(z^{\left(m\right)}\right){\parallel }_2$  12: until  $\parallel z^{\left(m\right)}-z^{(m-1)}{\parallel }_2+{\parallel \mathcal{F}\left(z^{\left(m\right)}\right)\parallel }_2<ϵ$  or  $m=m_{\max }$ Ensure:  approximations $z_1^{\left(m\right)},\dots ,z_n^{\left(m\right)}$; the sequences $\parallel z^{\left(k\right)}-z^{(k-1)}{\parallel }_2$ and $\parallel \mathcal{F}\left(z^{\left(k\right)}\right){\parallel }_2$, $k=1,\dots ,m$; the ACOC computed from the last three iterates; the CPU time of the run.

For comparison purposes, the following iterative methods are employed:

• Ehrlich’s method, denoted by $\left(E\right)$ [16], possessing quadratic convergence for nonlinear transcendental equations and cubic convergence for polynomial equations:

  $$z_r^{(m+1)}=z_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}{\mathcal{F}\left(z_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{z_r^{\left(m\right)}-z_s^{\left(m\right)}}}}},m=1,2,\dots ,r=1,2,\dots ,n.$$

  (22)
• The method proposed by Cordero et al. [23], denoted by $\left(A{C}_1\right)$, having fourth-order convergence:

  $$\begin{array}{cc}\hfill w_r^{\left(m\right)}=& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}=& w_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-z_s^{\left(m\right)}}}}},m=1,2,\dots ,r=1,2,\dots ,n.\hfill \end{array}$$
• Another method proposed by Cordero et al. [23], denoted by $\left(A{C}_2\right)$, having sixth-order convergence:

  $$\begin{array}{cc}\hfill w_r^{\left(m\right)}=& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}}\right)+\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}=& w_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-z_s^{\left(m\right)}}}}},m=1,2,\dots ,r=1,2,\dots ,n.\hfill \end{array}$$
• The method proposed by Cordero et al. [23], denoted by $\left(A{C}_3\right)$, having eighth-order convergence:

  $$\begin{array}{cc}\hfill y_r^{\left(m\right)}=& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill w_r^{\left(m\right)}=& y_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(y_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(y_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}=& w_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\mathcal{F}\left(w_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{w_r^{\left(m\right)}-z_s^{\left(m\right)}}}}},m=1,2,\dots ,r=1,2,\dots ,n.\hfill \end{array}$$
• The fifth-order simultaneous iterative method proposed by Shams et al., denoted by $\left(S_5\right)$ [22]:

  $$\begin{array}{cc}\hfill w_r^{\left(m\right)}=& z_r^{\left(m\right)}-{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill t_r^{\left(m\right)}=& w_r^{\left(m\right)}-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)-{\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)}{\alpha {\mathcal{F}}^{\prime }\left(w_r^{\left(m\right)}\right)+(2-\alpha ){\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}}{\displaystyle \frac{\mathcal{F}\left(z_r^{\left(m\right)}\right)}{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}},\hfill \\ \hfill z_r^{(m+1)}=& z_r^{\left(m\right)}-{\displaystyle \frac{1}{{\displaystyle {\displaystyle \frac{{\mathcal{F}}^{\prime }\left(z_r^{\left(m\right)}\right)}{\mathcal{F}\left(z_r^{\left(m\right)}\right)}}-{\displaystyle \sum _{\begin{array}{c}s=1\\ s\ne r\end{array}}^n}\frac{1}{z_r^{\left(m\right)}-t_s^{\left(m\right)}}}}},m=1,2,\dots ,r=1,2,\dots ,n,\alpha \in \mathbb{R}.\hfill \end{array}$$
  For the numerical computations, the parameter is chosen as $\alpha ={\displaystyle \frac{1}{10}}$.

The numerical results presented in Table 1, Table 2 and Table 3 provide detailed information regarding the convergence characteristics and computational performance of the considered methods. The following information is given:

Table 1. Comparison of numerical results for Example 1 obtained using various iterative methods.

Method	$\left|\right|{\mathit{z}}^{\left(2\right)}-{\mathit{z}}^{\left(1\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(3\right)}-{\mathit{z}}^{\left(2\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(4\right)}-{\mathit{z}}^{\left(3\right)}\left|\right|$	$\left|\right|\mathcal{F}\left({\mathit{z}}^{\left(4\right)}\right)\left|\right|$	ACOC	CPU Time
$K{M}_{{\Phi }_1}^{\left(1\right)}$	$2.7(-8)$	$3.2(-24)$	$5.2(-72)$	$2.8(-71)$	$3.0000$	$1.641$
$K{M}_{{\Phi }_2}^{\left(1\right)}$	$5.0(-10)$	$4.2(-47)$	$1.7(-232)$	$1.1(-231)$	$5.0000$	$1.453$
$K{M}_{{\Phi }_3}^{\left(1\right)}$	$2.5(-6)$	$2.8(-39)$	$6.8(-270)$	$4.6(-269)$	$7.0000$	$1.718$
$K{M}_{{\Phi }_4}^{\left(1\right)}$	$1.3(-40)$	$1.5(-359)$	$4.0(-3230)$	$2.7(-3229)$	$9.0000$	$1.467$
$K{M}_{{\Phi }_1}^{\left(2\right)}$	$2.7(-7)$	$6.7(-21)$	$1.0(-61)$	$5.6(-61)$	$3.0000$	$1.641$
$K{M}_{{\Phi }_2}^{\left(2\right)}$	$9.6(-11)$	$1.0(-50)$	$1.7(-250)$	$1.1(-249)$	$5.0000$	$1.327$
$K{M}_{{\Phi }_3}^{\left(2\right)}$	$4.6(-6)$	$2.0(-37)$	$6.6(-257)$	$4.5(-256)$	$7.0000$	$1.765$
$K{M}_{{\Phi }_4}^{\left(2\right)}$	$1.1(-40)$	$2.1(-360)$	$8.6(-3228)$	$5.8(-3227)$	$9.0000$	$1.531$
$K{M}_{{\Phi }_1}^{\left(3\right)}$	$1.6(-6)$	$8.8(-19)$	$1.4(-55)$	$7.6(-55)$	$3.0000$	$0.624$
$K{M}_{{\Phi }_2}^{\left(3\right)}$	$1.2(-9)$	$3.6(-45)$	$8.4(-233)$	$5.7(-222)$	$5.0000$	$1.407$
$K{M}_{{\Phi }_3}^{\left(3\right)}$	$9.4(-6)$	$2.8(-35)$	$5.8(-242)$	$4.0(-241)$	$7.0000$	$1.704$
$K{M}_{{\Phi }_4}^{\left(3\right)}$	$2.0(-40)$	$4.9(-358)$	$1.3(-3216)$	$9.4(-3216)$	$9.0000$	$1.360$
E	$5.4(-2)$	$4.3(-3)$	$2.8(-5)$	$1.9(-4)$	$1.9997$	$0.891$
$A{C}_1$	$5.8(-7)$	$1.4(-25)$	$6.1(-100)$	$4.1(-99)$	$4.0000$	$0.906$
$A{C}_2$	$2.2(-1)$	$7.9(-6)$	$9.9(-31)$	$6.7(-30)$	$6.0000$	$1.313$
$A{C}_3$	$3.1(-30)$	$7.0(-237)$	$5.1(-1890)$	$3.5(-1889)$	$8.0000$	$0.828$
$S_5$	$5.3(-2)$	$4.2(-3)$	$2.7(-5)$	$1.8(-4)$	$1.9997$	$2.906$

Table 2. Comparison of numerical results for Example 2 obtained using various iterative methods.

Method	$\left|\right|{\mathit{z}}^{\left(3\right)}-{\mathit{z}}^{\left(2\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(4\right)}-{\mathit{z}}^{\left(3\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(5\right)}-{\mathit{z}}^{\left(4\right)}\left|\right|$	$\left|\right|\mathcal{F}\left({\mathit{z}}^{\left(5\right)}\right)\left|\right|$	ACOC	CPU Time
$K{M}_{{\Phi }_1}^{\left(1\right)}$	$2.4(-9)$	$4.0(-28)$	$1.8(-84)$	$1.3(-83)$	$3.0000$	$5.343$
$K{M}_{{\Phi }_2}^{\left(1\right)}$	$2.0(-7)$	$7.2(-36)$	$4.0(-178)$	$3.9(-177)$	$5.0000$	$4.485$
$K{M}_{{\Phi }_3}^{\left(1\right)}$	$6.1(-6)$	$2.0(-39)$	$1.0(-273)$	$1.0(-272)$	$7.0000$	$5.751$
$K{M}_{{\Phi }_4}^{\left(1\right)}$	$8.3(-44)$	$2.1(-392)$	$1.0(-3529)$	$1.0(-3528)$	$9.0000$	$4.859$
$K{M}_{{\Phi }_1}^{\left(2\right)}$	$1.2(-7)$	$1.0(-22)$	$5.0(-68)$	$5.3(-67)$	$3.0000$	$5.188$
$K{M}_{{\Phi }_2}^{\left(2\right)}$	$1.0(-15)$	$2.9(-77)$	$4.3(-385)$	$4.2(-384)$	$5.0000$	$4.640$
$K{M}_{{\Phi }_3}^{\left(2\right)}$	$3.0(-3)$	$1.4(-20)$	$8.4(-142)$	$8.2(-141)$	$7.0000$	$5.781$
$K{M}_{{\Phi }_4}^{\left(2\right)}$	$5.8(-44)$	$8.6(-394)$	$3.2(-3542)$	$3.1(-3541)$	$9.0000$	$4.984$
$K{M}_{{\Phi }_1}^{\left(3\right)}$	$2.6(-4)$	$3.5(-13)$	$2.4(-40)$	$1.9(-39)$	$3.0000$	$6.203$
$K{M}_{{\Phi }_2}^{\left(3\right)}$	$2.5(-7)$	$2.3(-35)$	$1.3(-175)$	$1.3(-179)$	$5.0000$	$4.890$
$K{M}_{{\Phi }_3}^{\left(3\right)}$	$4.8(-2)$	$5.6(-12)$	$1.1(-81)$	$1.0(-80)$	$7.0000$	$6.015$
$K{M}_{{\Phi }_4}^{\left(3\right)}$	$1.5(-43)$	$4.4(-390)$	$8.2(-3509)$	$8.0(-3508)$	$9.0000$	$5.375$
E	$1.3(-1)$	$7.3(-3)$	$3.0(-5)$	$3.1(-4)$	$2.0057$	$2.813$
$A{C}_1$	$2.4(-6)$	$1.1(-24)$	$4.4(-98)$	$4.3(-97)$	$4.0000$	$2.719$
$A{C}_2$	dvgt	dvgt	dvgt	dvgt	dvgt	dvgt
$A{C}_3$	$5.4(-35)$	$1.1(-278)$	$6.8(-2228)$	$6.6(-2227)$	$8.0000$	$3.063$
$S_5$	$1.4(-1)$	$1.2(-2)$	$8.6(-5)$	$9.0(-4)$	$2.0010$	$10.343$

dvgt denotes that the iterative scheme diverges.

Table 3. Comparison of numerical results for Example 3 obtained using various iterative methods.

Method	$\left|\right|{\mathit{z}}^{\left(3\right)}-{\mathit{z}}^{\left(2\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(4\right)}-{\mathit{z}}^{\left(3\right)}\left|\right|$	$\left|\right|{\mathit{z}}^{\left(5\right)}-{\mathit{z}}^{\left(4\right)}\left|\right|$	$\left|\right|\mathcal{F}\left({\mathit{z}}^{\left(5\right)}\right)\left|\right|$	ACOC	CPU Time
$K{M}_{{\Phi }_1}^{\left(1\right)}$	$3.1(-8)$	$3.0(-25)$	$5.8(-76)$	$3.0(-76)$	$3.0000$	$1.672$
$K{M}_{{\Phi }_2}^{\left(1\right)}$	$3.5(-14)$	$2.6(-69)$	$5.6(-345)$	$3.8(-345)$	$5.0000$	$1.453$
$K{M}_{{\Phi }_3}^{\left(1\right)}$	$8.4(-23)$	$3.0(-156)$	$2.6(-1090)$	$1.8(-1090)$	$7.0000$	$1.749$
$K{M}_{{\Phi }_4}^{\left(1\right)}$	$6.2(-40)$	$1.8(-355)$	$3.9(-3195)$	$2.6(-3195)$	$9.0000$	$1.609$
$K{M}_{{\Phi }_1}^{\left(2\right)}$	$2.1(-5)$	$5.7(-16)$	$1.0(-47)$	$7.1(-48)$	$3.0000$	$1.875$
$K{M}_{{\Phi }_2}^{\left(2\right)}$	$1.4(-11)$	$2.8(-56)$	$9.0(-280)$	$6.1(-280)$	$5.0000$	$1.422$
$K{M}_{{\Phi }_3}^{\left(2\right)}$	$5.2(-21)$	$1.0(-143)$	$1.1(-1002)$	$7.5(-1003)$	$7.0000$	$1.781$
$K{M}_{{\Phi }_4}^{\left(2\right)}$	$2.9(-39)$	$2.0(-349)$	$7.9(-3141)$	$5.3(-3141)$	$9.0000$	$1.610$
$K{M}_{{\Phi }_1}^{\left(3\right)}$	$5.2(-5)$	$1.7(-15)$	$6.7(-47)$	$3.4(-47)$	$3.0000$	$1.859$
$K{M}_{{\Phi }_2}^{\left(3\right)}$	$8.0(-16)$	$1.6(-77)$	$5.6(-386)$	$3.8(-386)$	$5.0000$	$1.421$
$K{M}_{{\Phi }_3}^{\left(3\right)}$	$2.5(-30)$	$7.8(-209)$	$1.0(-1458)$	$6.7(-1459)$	$7.0000$	$2.531$
$K{M}_{{\Phi }_4}^{\left(3\right)}$	$1.9(-41)$	$5.6(-369)$	$7.7(-3317)$	$5.2(-3317)$	$9.0000$	$1.500$
$A{C}_1$	$9.3(-12)$	$1.0(-45)$	$1.4(-181)$	$9.7(-182)$	$4.0000$	$0.843$
$A{C}_2$	$9.7(-23)$	$2.4(-133)$	$6.3(-797)$	$4.3(-797)$	$6.0000$	$1.094$
$A{C}_3$	$1.0(-43)$	$6.0(-346)$	$6.5(-2764)$	$4.4(-2764)$	$8.0000$	$0.922$
$S_5$	$6.4(-2)$	$6.5(-4)$	$6.9(-8)$	$6.3(-8)$	$2.0392$	$2.906$

(i)

The notation $m(\pm n)$ denotes $m\times {10}^{\pm n}$.

(ii)

The quantity $\parallel z^{(m+1)}-z^{\left(m\right)}{\parallel }_2$ represents the Euclidean norm of the difference between two consecutive iterates.

(iii)

CPU denotes the computational time measured in seconds. Note that the time for CPU executions was measured for a mean of 10 computation runs for each test problem. The CPU times should be considered as indicative measures of the computational effectiveness of the considered methods.

(iv)

The residual norm at the ${\left(m\right)}^{th}$ iteration is represented by $\parallel \mathcal{F}\left(z^{\left(m\right)}\right){\parallel }_2,$ where

$$\mathcal{F}\left(z^{\left(m\right)}\right)=\left(f\left(z_1^{\left(m\right)}\right),f\left(z_2^{\left(m\right)}\right),\dots ,f\left(z_n^{\left(m\right)}\right)\right),$$

(v)

The approximate computational order of convergence (ACOC), [28] which estimates the theoretical convergence order $\rho$, is computed by

$$ACOC\approx {\displaystyle \frac{\ln \left({\displaystyle \frac{\parallel z^{(m+2)}-z^{(m+1)}{\parallel }_2}{\parallel z^{(m+1)}-z^{\left(m\right)}{\parallel }_2}}\right)}{\ln \left({\displaystyle \frac{\parallel z^{(m+1)}-z^{\left(m\right)}{\parallel }_2}{\parallel z^{\left(m\right)}-z^{(m-1)}{\parallel }_2}}\right)}},$$

The computational experiments were performed on a desktop computer with the following specifications:

• 11th Gen Intel(R) Core(TM) i3-1125G4 processor, 2.00 GHz (64-bit architecture);
• 8 GB RAM and Intel(R) UHD Graphics;
• Microsoft Windows 10 operating system.

Further, the computational analysis in terms of iteration process and residual error reduction for Examples 1–3 is illustrated graphically in Figure 1, Figure 2 and Figure 3. In these figures, ${\lambda }_r$ denotes the $r^{th}$ root of the nonlinear equation under consideration. The solid curves correspond to the proposed methods, whereas the dotted curves represent the existing iterative methods used for comparison. Since all three weight functions exhibit similar convergence behavior, only the graphs corresponding to the third weight function are presented.

Figure 1. Error analysis of $K{M}_{{\Phi }_1}^{\left(3\right)}$ and E corresponding to the roots ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$ for Example 1.

Figure 2. Error analysis of $K{M}_{{\Phi }_2}^{\left(3\right)}$, $A{C}_1$, and $S_5$ corresponding to the roots ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_8$ for Example 2.

Figure 3. Error analysis of $K{M}_{{\Phi }_3}^{\left(3\right)}$ and $A{C}_2$ corresponding to the roots ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$ for Example 3.

Example 1

(Population Model with Allee Effect and Density-Dependent Migration). To predict the growth, persistence, and extinction of a biological species in response to changes in environmental conditions, it is necessary to understand the population dynamics of the species [29,30]. If the population density drops below a critical threshold, for many species, the reduction in growth rate is connected with the inability to reproduce, cooperate, and defend themselves from predators. This effect is referred to as a strong Allee effect. Moreover, migration behavior can be influenced by population density and environmental fluctuations. We consider the following nonlinear model of a population to see these effects.

$${\displaystyle \frac{dz}{dt}}=rz\left(1-{\displaystyle \frac{z}{K}}\right)\left({\displaystyle \frac{z}{A}}-1\right)+B\left(1+\sin \left(\pi z\right)\right),$$

(23)

in which $z\left(t\right)$ is the population density at time t, r the intrinsic growth rate, K the carrying capacity, A the Allee threshold and B the strength of the migration. The first term is logistic growth with a strong Allee effect, and the second term is density dependent migration with environmental variability. When the population is in equilibrium, the density of the population is unchanged, so that:

$${\displaystyle \frac{dz}{dt}}=0.$$

(24)

Consequently, the equilibrium condition yields the nonlinear transcendental equation

$$f\left(z\right)=rz\left(1-{\displaystyle \frac{z}{K}}\right)\left({\displaystyle \frac{z}{A}}-1\right)+B\left(1+\sin \left(\pi z\right)\right)=0.$$

(25)

For the numerical examples, the parameter values are chosen as

$$r=2,K=10,A=3,B=-2.$$

Substituting these parameter values into Equation (25) yields

$$f\left(z\right)=2z\left(1-{\displaystyle \frac{z}{10}}\right)\left({\displaystyle \frac{z}{3}}-1\right)-2\left(1+\sin \left(\pi z\right)\right)=0.$$

(26)

Equation (26) is a nonlinear transcendental equation possessing multiple real roots that correspond to different equilibrium states of the biological population system. For the selected parameter values, the computed roots are $-0.254\dots ,3.288\dots ,3.910\dots ,9.806\dots .$ The initial approximations employed for the simultaneous iterative schemes are chosen as $-0.5,3,4,10.$ There is no biological meaning to the negative equilibrium since the population densities are restricted to being nonnegative. Three positive roots are located in the vicinity of the Allee threshold; they represent critical transition states that are very sensitive to environmental perturbations and migration effects. The largest positive root, which lies near the carrying capacity $K=10$, represents a stable population persistence state resulting from the combined effects of intrinsic growth, the Allee effect, and density-dependent migration.

Example 2

(Epidemic Disease Model). In many problems in epidemiology, population biology and infectious disease dynamics, a nonlinear epidemic disease model is encountered [31,32]. Many biological systems have infectious diseases that are spread nonlinearly between susceptible and infected populations. In addition, the system is subject to environmental fluctuations, seasonal influences, and nonlinear transmission mechanisms that add further complexities to the dynamic behavior. These biological factors can interact in a nonlinear way, resulting in a transcendental equation with several equilibrium points.

In the present study, a modified epidemic disease model is considered and represented by the following nonlinear equation:

$$f\left(z\right)=z+8e^{-z}-12\sin \left(z\right)-30=0,$$

where z denotes the dimensionless infected population density, $8e^{-z}$ represents the nonlinear disease transmission or infection interaction term, $\sin \left(z\right)$ models seasonal or environmental fluctuations affecting disease spread, and the constant term 30 corresponds to the external recovery or control parameter.

The equilibrium states of the epidemic system are obtained by solving $f\left(z\right)=0.$ The approximate numerical roots of the equation are found as $-0.968\dots ,22.650\dots ,24.672\dots ,28.407\dots ,$ $31.545\dots ,34.199\dots ,38.484\dots ,39.874\dots .$ The corresponding initial approximations are taken as $-1,22,24,28,31,34,38,40.$

The negative root corresponds to a mathematically admissible but biologically unrealistic equilibrium state since negative population densities are not physically meaningful in epidemiological systems. This equilibrium is generally associated with unstable or nonphysical disease dynamics generated by the strong nonlinear behavior of the model. The positive roots represent biologically feasible equilibrium states where infection transmission, recovery mechanisms, and environmental effects remain balanced. The equilibrium states near $22.650\dots ,24.672\dots ,28.407\dots$ correspond to comparatively stable endemic states associated with controlled disease spread and balanced epidemic behavior. The higher states of equilibrium near $31.545\dots ,34.199\dots ,38.484\dots ,39.874\dots$ associations with increased disease transmission and unstable or critical epidemic areas. This higher equilibrium can be associated with outbreaks of epidemics, fast disease progression, or unstable populations. Simultaneous root-finding methods are especially significant in finding all the possible roots of a nonlinear transcendental equation in one run, because the model has several different equilibrium solutions. To investigate epidemic stability, disease persistence and biologically stable operating conditions, the computation of all the roots is indispensable. Hence, the present model serves as a useful test problem for the study of higher order simultaneous iterative schemes, analysis of convergence and the stability behavior of numerical algorithms.

Example 3

(Michaelis–Menten Enzyme Kinetics Model). Michaelis–Menten enzyme kinetics models frequently arise in biochemistry, pharmacokinetics, molecular biology, and metabolic reaction analysis [33]. In biological systems, enzymes catalyze biochemical reactions by interacting with substrate molecules. The reaction rate initially increases rapidly with substrate concentration; however, after a certain level, enzyme saturation occurs and the reaction rate approaches a limiting value. This saturation phenomenon introduces strong nonlinear behavior into the biochemical system.

In many biological processes, environmental fluctuations, regulatory mechanisms, and nonlinear metabolic effects further complicate the system dynamics, leading to nonlinear transcendental equations possessing multiple equilibrium states.

In the present study, a modified Michaelis–Menten type biological model is considered and represented by the following nonlinear equation:

$$f\left(z\right)={\displaystyle \frac{15z}{2+z}}-0.5z+\cos \left(z\right)-5=0,$$

where z denotes the dimensionless substrate concentration, ${\displaystyle \frac{15z}{2+z}}$ represents the nonlinear enzyme saturation kinetics, $-0.5z$ models degradation or removal of substrate concentration, $\cos \left(z\right)$ represents oscillatory biological or environmental effects, and the constant term 5 corresponds to external biochemical regulation.

The equation is nonlinear and non-polynomial because it simultaneously contains rational and trigonometric nonlinearities. Owing to the transcendental nature of the equation, exact analytical determination of equilibrium states becomes difficult. Therefore, efficient numerical iterative methods are required for accurate computation of all feasible roots.

The equilibrium states of the biological system are obtained by solving $f\left(z\right)=0.$ The approximate numerical roots of the equation are found as $0.980\dots ,14.977\dots ,17.628\dots ,19.096\dots .$ The seed values corresponding to the above are considered as $1,14,17,20.$ The smaller equilibrium point is near $0.980\dots$ Low substrate operating condition which is biologically stable and maintains a balance between substrate use and enzyme activity.The higher lying equilibrium states near $14.976\dots ,17.628\dots ,19.095\dots$ indicate high levels of substrate concentration, when enzyme saturation occurs and biochemical activity is high, in the case of nonlinear enzyme saturation behaviour. They are equilibrium states that could represent very high metabolic activity or enzyme reaction saturation. Since the model possesses multiple equilibrium solutions, simultaneous root-finding methods are important for computing all feasible roots efficiently. Therefore, the present model serves as an effective benchmark problem for testing higher-order simultaneous iterative schemes and convergence behavior for nonlinear biological systems.

Analysis of Numerical Results

Table 1, Table 2 and Table 3 illustrate that the proposed methods consistently converge and validate the theoretical findings presented in Section 2 and Section 3. The numerical experiments are performed on three nonlinear problems, namely, Examples 1–3. The obtained numerical outcomes show the effectiveness, stability, and higher convergence behavior of some of the proposed schemes. Furthermore, the graphical analysis presented in Figure 1, Figure 2 and Figure 3 corresponds to the third weight-function-based methods, namely, $K{M}_{{\Phi }_1}^{\left(3\right)}$, $K{M}_{{\Phi }_2}^{\left(3\right)}$, and $K{M}_{{\Phi }_3}^{\left(3\right)}$, respectively. Since all weight functions exhibit similar convergence behavior, the graphical study is carried out only for one representative method in each example and compared with the corresponding existing methods to avoid redundancy while preserving the essential convergence characteristics.

• Table 1 presents the numerical results obtained for the Population Model with Allee Effect and Density-Dependent Migration. It is observed that the proposed method $K{M}_{{\Phi }_4}$, with three different weight functions consistently achieve smaller absolute errors and faster convergence toward the required roots as compared to the existing methods. The errors decrease rapidly at each iteration, demonstrating the high computational efficiency and robustness of the proposed scheme. Figure 1 presents the error analysis for Example 1, where the proposed method $K{M}_{{\Phi }_1}^{\left(3\right)}$ is compared with the existing method E for the roots ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$, since both methods possess the same order of convergence.
• Table 2 corresponds to the Epidemic Disease Model. The numerical outcomes indicate that the proposed method $K{M}_{{\Phi }_4}$, with three different weight functions, get better results than the existing iterative schemes in terms of convergence speed, accuracy, and residual reduction. In particular, these proposed methods reduce the errors more efficiently in successive iterations, thereby requiring fewer computational steps to obtain high-precision approximations of the equilibrium states of the epidemic system. It is also observed that the method $A{C}_2$ diverges for this problem, demonstrating its lack of robustness compared with the proposed schemes. The error analysis presented in Figure 2 corresponds to Example 2 and compares the proposed method $K{M}_{{\Phi }_2}^{\left(3\right)}$ with the existing methods $A{C}_1$ and $S_5$ for the roots ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_8$. The results show a faster decrease in the errors for the proposed method, highlighting its convergence behavior and computational efficiency.
• Table 3 shows the Michaelis–Menten Enzyme Kinetics Model. The numerical results clearly prove that the proposed method $K{M}_{{\Phi }_4}$, with three different weight functions achieve faster convergence, smaller residual errors and better computational accuracy than the available iterative schemes. The errors are seen to become more significant in the successive iterations for the proposed methods, resulting in reaching to the equilibrium enzyme states in fewer iterations. Furthermore, the error analysis presented in Figure 3 corresponds to Example 3 and compares the proposed method $K{M}_{{\Phi }_3}^{\left(3\right)}$ with the existing method $A{C}_2$ for the roots ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$. The graphical results further confirm the convergence behavior, stability, and computational efficiency of the proposed method.
• Importantly, the proposed algorithms and Cordero’s schemes ($A{C}_1$, $A{C}_2$, and $A{C}_3$) require exactly the same number of function evaluations of the algorithm per iteration. The methods proposed have a convergence order of $2p+1$ while Cordero’s methods have an order of $2p$. With this, the proposed schemes provide greater computational efficiency, in terms of functional evaluations.
• Furthermore, the numerical experiments shows that method $S_5$ is only second order computational convergent even though it is a fifth order method, theoretically. On the other hand, the proposed methods retain their theoretical convergence orders and consistently show high-order convergence for all the nonlinear problems that were tested, with stable results. Using the computational order of convergence described by ACOC, we calculate their Ostrowski’s efficiency index [34], defined by $EI=ACO{C}^{1/d}$, where d is the number of functional evaluations per iteration. Using this definition, the $EI$ of all the used methods for a function of n zeros, appears in Table 4. In it, $K{M}_{{\varphi }_i}$, $i=1,2,3,4$ denotes the proposed methods with any of the $j=1,2,3$ weight functions (none of them have functional evaluations). In Figure 4, we observe that the most efficient scheme is $K{M}_{{\varphi }_2}$, for all n between 1 and 10. It has been checked that this performance is held for bigger values of n.
  Table 4. Efficiency indices for the different methods.

  Method	d	$\mathbf{EI}$
  $K{M}_{{\varphi }_1}$	4n	$3^{1/\left(4n\right)}$
  $K{M}_{{\varphi }_2}$	4n	$5^{1/\left(4n\right)}$
  $K{M}_{{\varphi }_3}$	5n	$7^{1/\left(5n\right)}$
  $K{M}_{{\varphi }_4}$	6n	$9^{1/\left(6n\right)}$
  E	2n	$2^{1/\left(2n\right)}$
  $A{C}_1$	4n	$4^{1/\left(4n\right)}$
  $A{C}_2$	5n	$6^{1/\left(5n\right)}$
  $A{C}_3$	6n	$8^{1/\left(6n\right)}$
  $S_5$	3n	$2^{1/\left(3n\right)}$

  

  Figure 4. Efficiency indices for new and known methods.

In summary, the numerical investigations confirm that some of the proposed iterative techniques can be very efficient and they are competitive with the already available iterative methods.

4. Conclusions

In this paper, a new class of higher-order simultaneous iterative methods has been presented for the simultaneous computation of the roots of nonlinear equations. The proposed schemes incorporate different weight functions into a predictor–corrector framework and achieve a convergence order of $2p+1$, where p denotes the order of the predictor iterative method. The theoretical convergence analysis establishes the order of convergence and confirms the validity of the proposed schemes.

Several nonlinear biological models, including population and epidemic disease models, have been considered to demonstrate the applicability and computational performance of the proposed methods. The numerical results show that the proposed schemes converge rapidly to the desired roots with high accuracy and small residual errors and error estimations. The error graphs further illustrate the effective reduction in errors over successive iterations. With the same number of iterations, some of the proposed methods achieve smaller residual and estimated errors than existing methods for the considered test problems, while maintaining comparable function-evaluation costs.

The comparative numerical results demonstrate that the proposed simultaneous iterative methods provide reliable convergence behavior, small residual errors, and improved computational performance. Therefore, the developed methods constitute effective and competitive computational tools for the simultaneous solution of nonlinear equations arising in applied mathematics, biological sciences, engineering, and other scientific applications.