A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

Abstract

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

1. Introduction

Scalar nonlinear equations of a single variable $\gamma$,

$$f\left(\gamma \right)=0,$$

(1)

play a pivotal role in advancing scientific understanding and engineering [1,2]. Various scientific disciplines, including physics, chemistry, biology, and economics, utilize nonlinear equations to describe complex correlations and interactions between variables. These equations enable scientists to more accurately characterize chaotic systems, fluid dynamics, and population dynamics compared to linear models. In engineering, nonlinear equations are crucial in areas such as control systems [3], structural analysis [4], and electrical circuits [5]. Engineers use these equations to model and predict real-world behaviors by accounting for nonlinearities in materials and systems. Nonlinear optimization techniques are essential for solving engineering problems including parameter estimation [6], optimal control [7], and system design [8]. The significance of nonlinear equations extends to emerging fields like artificial intelligence and machine learning [9], where they are used for complex data processing and pattern recognition [10]. Overall, nonlinear equations and their associated systems are indispensable tools for scientists and engineers striving to understand and manage complex systems, thereby fostering the advancement of knowledge and technology.

Solving nonlinear equations analytically can be challenging, and often impossible, due to the intrinsic complexity of nonlinear interactions. Nonlinear equations include terms that are not simply proportional to the variable of interest, and their solutions may not be expressible in closed-form expressions or simple algebraic equations [11,12,13,14]. Therefore, we turn to numerical iterative schemes. Iterative numerical methods are effective in solving nonlinear equations and systems, making them invaluable tools for researchers across various fields [15,16,17,18]. These numerical iterative techniques are classified into three types: single root-finding schemes with local convergence behavior, simultaneous methods for finding all roots of (1) with global convergence behavior, and schemes that find all solutions to nonlinear systems of equations (i.e., vectorial problems). Iterative techniques for solving nonlinear systems of equations, such as gradient descent [19] or evolutionary algorithms that search for roots simultaneously across multiple dimensions in the solution space [20], exhibit local convergence behavior.

The simplest and most efficient method is the classical Newton’s method [21] for solving (1), given by

$${\sigma }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}};k=0,1,...,$$

(2)

The method (2) exhibits local quadratic convergence. To reduce the computational cost of (2), Steffensen [22] proposed the following modified version:

$${\sigma }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\gamma }^{\left[k\right]};f\right]}},$$

(3)

where ${\theta }_{11}^{\left[k\right]}={\gamma }^{\left[k\right]}+f\left({\gamma }^{\left[k\right]}\right)$, and $\left[{\theta }_{11}^{\left[k\right]},{\gamma }^{\left[k\right]};f\right]:\Delta \subset \mathbb{R}\to \mathbb{R}$ is the first-order forward difference on $\Delta$, i.e., $\left[{\theta }_{11}^{\left[k\right]},{\gamma }^{\left[k\right]};f\right]={\displaystyle \frac{f\left({\theta }_{11}^{\left[k\right]}\right)-f\left({\gamma }^{\left[k\right]}\right)}{{\theta }_{11}^{\left[k\right]}-{\gamma }^{\left[k\right]}}}$ [23]. In high-precision computing, the divided difference is replaced with a first-order central difference on $\Delta$ as follows:

$${\sigma }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]}},$$

(4)

where ${\theta }_{12}^{\left[k\right]}={\gamma }^{\left[k\right]}-f\left({\gamma }^{\left[k\right]}\right)$ and $\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]$ is the first-order central difference operator. The two-step modified Newton’s method [24] of the third-order convergence has the form

$${\varsigma }^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{f\left({\tau }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},$$

(5)

where ${\tau }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}}.$

Higher-order schemes offer considerable advantages over lower-order schemes for solving nonlinear equations due to improved accuracy and efficiency. They achieve higher accuracy per iteration step, reducing truncation errors and requiring fewer iterations to obtain the desired precision. The order of convergence can increase up to three, four, and so on, such as the well-known Ostrowski’s method [25].

$$\left\{\begin{array}{c}{\tau }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ {\varsigma }^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f\left({\gamma }^{\left[k\right]}\right)-2f\left({\tau }^{\left[k\right]}\right)}}{\displaystyle \frac{f\left({\tau }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}}.\end{array}\right.$$

(6)

Similarly, Kou et al. [26] developed sixth-order methods using the weight function technique, written as

$$\left\{\begin{array}{c}{\tau }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ z^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f\left({\gamma }^{\left[k\right]}\right)-2f\left({\tau }^{\left[k\right]}\right)}}{\displaystyle \frac{f\left({\tau }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ {\gamma }^{[k+1]}=z^{\left[k\right]}-{\displaystyle \frac{f\left(z^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}}\left[{\left({\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)-f\left({\tau }^{\left[k\right]}\right)}{f\left({\gamma }^{\left[k\right]}\right)-2f\left({\tau }^{\left[k\right]}\right)}}\right)}^2+{\displaystyle \frac{f\left(z^{\left[k\right]}\right)}{f\left({\tau }^{\left[k\right]}\right)-\alpha f\left(z^{\left[k\right]}\right)}}\right].\end{array}\right.$$

(7)

Liu et al. [27] presented the following eighth-order methods using the weight function technique:

$$\left\{\begin{array}{c}{\tau }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ z^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f\left({\gamma }^{\left[k\right]}\right)-2f\left({\tau }^{\left[k\right]}\right)}}{\displaystyle \frac{f\left({\tau }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ {\gamma }^{[k+1]}=z^{\left[k\right]}-{\displaystyle \frac{f\left(z^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}}\left[\begin{array}{c}{\left({\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)-f\left({\tau }^{\left[k\right]}\right)}{f\left({\gamma }^{\left[k\right]}\right)-2f\left({\tau }^{\left[k\right]}\right)}}\right)}^2+\\ {\displaystyle \frac{f\left(z^{\left[k\right]}\right)}{f\left({\tau }^{\left[k\right]}\right)-{\alpha }_{1}f\left(z^{\left[k\right]}\right)}}+{\displaystyle \frac{4f\left(z^{\left[k\right]}\right)}{f\left({\tau }^{\left[k\right]}\right)-{\alpha }_{2}f\left(z^{\left[k\right]}\right)}}\end{array}\right],\end{array}\right.$$

(8)

where ${\alpha }_1,{\alpha }_2\in \mathbb{R}.$

Numerous single- and multi-step methods exist for solving (1), and some of these methods can be applied to solving systems of nonlinear equations with local convergence behavior [28]. Noot et al. [29], Darvishi et al. [30], Babajee et al. [31], Ortega et al. [32], and others (see e.g., [33,34] and references therein) have employed (2) as a predictor step to construct multi-step approaches to solving nonlinear equation systems. Iterative methods for finding a single root of nonlinear equations, though widely used, have certain inherent limitations that researchers must consider. One primary concern is convergence; these methods may fail to find a solution if the initial guess is not close enough to a root or if the function has abrupt changes. The dependence on initial guesses poses a significant challenge, as inaccurate or poorly chosen starting points can result in a slow convergence or divergence [35,36]. Furthermore, iterative methods usually provide only local solutions, with no guarantee of accurately identifying all roots, particularly multiple roots. The computational cost can be significant, especially for complex functions or high-dimensional systems, and ill-conditioned problems can lead to numerical instability. Additionally, these methods generally provide root values without information about their multiplicity, and their applicability may be limited in the presence of discontinuities or non-smooth features [37]. Due to these limitations, when using iterative methods, we may need to investigate alternative approaches based on the specific characteristics of the nonlinear equations under investigation. Therefore, we turn to simultaneous methods, which are more stable, consistent, and reliable, and can also be applied to parallel computing (see e.g., [38,39]).

In 1891, Weierstrass [40] introduced the generalized form of (2) by incorporating the Weierstrass correction, which was later explored by Presic [41], Durand [42], Dochev [43], and Kerner [44]. In 2015, Proinov et al. [45] proposed the local convergence theorem for the double Weierstrass technique. In 2016, Nedzibove created a modified version of the Weierstrass technique [46] and presented its local convergence [47] in 2018. In 1973, Aberth [48] developed a third-order convergent simultaneous method with derivatives, which was then accelerated by Nourein [49] to the fourth order in 1977, by Petković [50] to the sixth order in 2020, and by Mir et al. [51] to the tenth order using various single root-finding methods as corrections. Cholakov [52,53] and Marcheva et al. [54] proposed the local convergence of multi-step simultaneous methods for determining all roots of (1). In 2023, Shams et al. [55,56] described the global convergence behavior of simultaneous algorithms using random initial gauge values, along with contributions from many others.

Among derivative-free simultaneous methods, the Weierstrass–Dochive method [57] (abbreviated as BM) is the most attractive. It is given by

$${\gamma }_t^{[k+1]}={\gamma }_t^{\left[k\right]}-{\Delta }_t^{\ast }\left({\gamma }_t^{\left[k\right]}\right),$$

(9)

where

$${\Delta }_t^{\ast }\left({\gamma }_t^{\left[k\right]}\right)={\displaystyle \frac{f\left({\gamma }_t^{\left[k\right]}\right)}{{\prod }_{\begin{array}{c}j=1j\ne t\end{array}}^{\mathfrak{m}}({\gamma }_t^{\left[k\right]}-{\gamma }_j^{\left[k\right]})}},(t,j=1,2,...,m),$$

(10)

is the Weierstrass correction. The method (10) has local quadratic convergence.

In 1977, Ehrlich [58] presented the following convergent simultaneous method (abbreviated as EM) of the third order:

$${\gamma }_t^{[k+1]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N_t\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\gamma }_t^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}\right)}},$$

(11)

where ${\gamma }_j^{\left[k\right]}=u_j^{\left[k\right]}$ is used as a correction. Petkovic et al. [59] accelerated the convergence order of (11) from three to six:

$${\gamma }_t^{[k+1]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N_t\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\gamma }_t^{\left[k\right]}-u_j^{\left[k\right]}\right)}\right)}},$$

(12)

where $u_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left(v_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}{2f\left(v_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}}{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}}$ and $v_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}}.$

Petkovic et al. [60] accelerated the convergence order of (11) from three to ten, as shown in the following method (abbreviated as PM^ϵ):

$${\gamma }_t^{[k+1]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N_t\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\gamma }_t^{\left[k\right]}-Z_j^{\left[k\right]}\right)}\right)}},$$

(13)

where $z_j^{\left[k\right]}=u_j^{\left[k\right]}-{\displaystyle \frac{\left({\tau }_j^{\left[k\right]}-u_j^{\left[k\right]})f\left(u_j^{\left[k\right]}\right)\right)\left(\frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}\right)}{{\left(f\left({\gamma }_j^{\left[k\right]}\right)-f\left(u_j^{\left[k\right]}\right)\right)}^2}}\left[f\left({\tau }_j^{\left[k\right]}\right)-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f\left({\tau }_j^{\left[k\right]}\right)-f\left(u_j^{\left[k\right]}\right)}}\right]$, $u_j^{\left[k\right]}={\tau }_j^{\left[k\right]}-$${\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)f\left({\tau }_j^{\left[k\right]}\right)\left(\frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}\right)}{{\left(f\left({\gamma }_j^{\left[k\right]}\right)-f\left({\tau }_j^{\left[k\right]}\right)\right)}^2}},{\tau }_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}}.$

Shams et al. [61] proposed a three-step simultaneous scheme for finding all polynomial roots (abbreviated as MM^ϵ):

$${\gamma }_t^{[k+1]}=z_t^{\left[k\right]}-{\displaystyle \frac{f\left(z_t^{\left[k\right]}\right)}{\underset{\stackrel{j=1}{j\ne t}}{\stackrel{\mathfrak{m}}{\Pi }}(z_t^{\left[k\right]}-z_j^{\left[k\right]})}},$$

(14)

where $z_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\tau }_t^{\left[k\right]}\right)}{\underset{\stackrel{j=1}{j\ne t}}{\stackrel{\mathfrak{m}}{\Pi }}({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]})}},{\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_t^{\left[k\right]}\right)}{\underset{\stackrel{j=1}{j\ne t}}{\stackrel{\mathfrak{m}}{\Pi }}({\gamma }_t^{\left[k\right]}-v_j^{\left[k\right]})}}$ and $v_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{\alpha {\left(f\left({\gamma }_k^{\left[k\right]}\right)\right)}^2}{f({\gamma }_j^{\left[k\right]}+\alpha f\left({\gamma }_j^{\left[k\right]}\right))-f\left({\gamma }_j^{\left[k\right]}\right)}}$; $\alpha \in \mathbb{R}.$ The numerical scheme (14) exhibits a convergence order of twelve.

A review of the existing literature reveals the following:

• Most iterative methods used for solving nonlinear equations and systems are highly effective at converging to solutions when the initial guess is close to a root.
• These iterative techniques are particularly sensitive to initial guesses and may fail to converge if the initial values are not chosen precisely.
• Local convergence algorithms may lack stability and consistency in many cases.
• Iterative methods are susceptible to rounding errors and may fail to converge when the problem is poorly conditioned.
• Nonlinear equations and systems can have multiple solutions, and achieving convergence to the desired solution based on initial estimates can be challenging.

Hirstov et al. [62] proposed the generalized Weierstrass method for solving systems of nonlinear equations (abbreviated as BM^ϵ):

$${\gamma }_{t.,r}^{[k+1]}={\gamma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{f_t\left({\gamma }_{t,r}^{\left[k\right]}\right)}{\underset{l\ne s}{\stackrel{\mathfrak{m}}{\Pi }}({\gamma }_{t,l}^{\left[k\right]}-{\gamma }_{t,s}^{\left[k\right]})}}.$$

(15)

Chinesta et al. [63] proposed the generalized method (11) for nonlinear systems (abbreviated as EM^ϵ):

$${\gamma }_{t.,r}^{[k+1]}={\gamma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{f_t\left({\gamma }_{t,r}^{\left[k\right]}\right)}{\left[{\theta }_{31}^{\left[k\right]},{\gamma }^{\left[k\right]};f_t\right]-f_t\left({\gamma }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{l\ne s}^{\mathfrak{m}}}\left(\frac{1}{\left({\gamma }_{t,l}^{\left[k\right]}-{\gamma }_{t,s}^{\left[k\right]}\right)}\right)}},$$

(16)

where ${\theta }_{31}^{\left[k\right]}={\gamma }^{\left[k\right]}+{\beta }^{[\ast ]}f\left({\gamma }^{\left[k\right]}\right)$, and ${\beta }^{[\ast ]}\in \mathbb{R}.$ The order of convergence of EM^ϵ is 2. Motivated by prior work, the main objective of this study is to develop a novel family of efficient, higher-order simultaneous schemes. These schemes aim not only to compute all roots of nonlinear equations simultaneously but also to solve nonlinear systems of equations comprehensively, thereby addressing the limitations outlined earlier. The structure of the paper is as follows: after the introduction, Section 2 introduces and analyzes a new family of two-step vectorial simultaneous algorithms. Section 3 is dedicated to discussing computational efficiencies, while in Section 4, we present and discuss the numerical results obtained from our proposed schemes. Finally, the concluding section summarizes the key findings, contributions of this research, and directions for future work.

2. Constructing a Family of Simultaneous Methods for Distinct Roots

Consider the two-step Newton’s method [64] for finding the simple root of (1):

$$\left\{\begin{array}{c}{\tau }^{\left[k\right]}={\gamma }^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }^{\left[k\right]}\right)}},\\ {\vartheta }^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{f\left({\tau }^{\left[k\right]}\right)}{f^{\prime }\left({\tau }^{\left[k\right]}\right)}}.\end{array}\right.$$

(17)

The methods in (17) exhibit third-order convergence if $\zeta$ is a simple root of (1), $ϵ={\gamma }^{\left[k\right]}-\zeta$, and the error equation is described by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\vartheta }^{\left[k\right]}-\zeta }{{\left({\gamma }^{\left[k\right]}-\zeta \right)}^4}}& \to & -C_2^2\left(C_3{C}_2^5+2C_3^2{C}_2^3-C_3{C}_4{C}_2^2\right);\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill C_{\omega }\left(\varsigma \right)& =& {\displaystyle \frac{f^{\left(\omega \right)}\left(\zeta \right)}{\omega !f^{\prime }\left(\zeta \right)}},\omega =2,3,...\hfill \end{array}$$

(19)

or

$${\vartheta }^{\left[k\right]}-\zeta =O\left({ϵ}^4\right).$$

(20)

Suppose (1) has m distinct roots. Then, $f\left(\gamma \right)$ and $f^{\prime }\left(\gamma \right)$ can be approximated as

$$f\left(\gamma \right)={\displaystyle \prod _{j=1}^{\mathfrak{m}}}\left(\gamma -{\gamma }_j\right)\mathrm{and}{f}^{\prime }\left(\gamma \right)={\displaystyle \sum _{t=1}^n}\underset{\underset{j=1}{j\ne t}}{\prod ^{\mathfrak{m}}}\left(\gamma -{\gamma }_j\right).$$

(21)

This implies that

$${\displaystyle \frac{f^{\prime }\left(\gamma \right)}{f\left(\gamma \right)}}={\displaystyle \sum _{j=1}^{\mathfrak{m}}}\left({\displaystyle \frac{1}{(\gamma -{\gamma }_j)}}\right)={\displaystyle \frac{1}{\frac{1}{\gamma -{\gamma }_k}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{(\gamma -{\gamma }_j)}\right)}}.$$

(22)

By using (22) in (11), we developed a new simultaneous scheme (abbreviated as BD^ϵ):

$${\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N\left({\tau }_k^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]})}\right)}},$$

(23)

where ${\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]})}\right)}},$ with $N\left({\gamma }_t^{\left[k\right]}\right)={\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_j^{\left[k\right]}\right)}}$ and ${\vartheta }_j^{\left[k\right]}={\gamma }_t^{\left[k\right]}-$${\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)\left(b_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}{f\left(b_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}},$ where $b_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{2\left(\frac{f\left(a_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}{\left(a_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}\right)-\frac{f\left({\gamma }_j^{\left[k\right]}\right)}{\left(a_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}}}$ and $a_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-$${\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f\left[{\gamma }_j^{\left[k\right]},{\gamma }_j^{\left[k\right]}+f\left({\gamma }_j^{\left[k\right]}\right)\right]}}.$ The method (23) for multiple roots can also be expressed as (abbreviated as BM^ϵ)

$${\varsigma }^{\left[k\right]}={\tau }^{\left[k\right]}-{\displaystyle \frac{{\wp }_i}{\frac{{\wp }_i}{N\left({\tau }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\tau }_t^{\left[k\right]}-{\tau }_{\mathfrak{j}}^{\left[k\right]})}\right)}},$$

(24)

where ${\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_i}{\frac{{\wp }_i}{N\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]})}\right)}}.$ To develop a derivative-free approach, we replace $f^{\prime }\left({\gamma }_i^{\left[i\right]}\right)$ with the central difference operator $\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]$, resulting in (abbreviated as DF^ϵ)

$${\phi }_t^{\left[k\right]}={\varsigma }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_t}{{\wp }_t\frac{\left[{\theta }_{21}^{\left[k\right]},{\theta }_{22}^{\left[k\right]};f\right]}{f\left({\varsigma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\varsigma }_t^{\left[k\right]}-{\varsigma }_{\mathfrak{j}}^{\left[k\right]})}\right)}},$$

(25)

where ${\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_t}{{\wp }_t\frac{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]}{f\left({\tau }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]})}\right)}}$, ${\theta }_{11}^{\left[k\right]}={\tau }_t^{\left[k\right]}+f\left({\tau }_t^{\left[k\right]}\right),{\theta }_{12}^{\left[k\right]}={\tau }_t^{\left[k\right]}+f\left({\tau }_t^{\left[k\right]}\right),{\theta }_{21}^{\left[k\right]}={\varsigma }_t^{\left[k\right]}+f\left({\varsigma }_t^{\left[k\right]}\right),$ and ${\theta }_{22}^{\left[k\right]}={\varsigma }_t^{\left[k\right]}+f\left({\varsigma }_t^{\left[k\right]}\right).$

Assume the system

$$F\left(\gamma \right)=0,$$

(26)

has m solutions ${\xi }_t=\left({\xi }_{t_1},{\xi }_{t_2},...{\xi }_{t_n}\right)$ for $t=1,...,m$, where $F:\Delta \subset R^m⟶R^m$ is defined over an open convex domain $\Delta \subset R^m$. The primary goal of this research study is to develop a numerical scheme that can find all solutions of the nonlinear system of equations (26) simultaneously. To find all solutions, we assume a set of m initial guesses ${\gamma }_t^{\left[0\right]}=\left({\gamma }_{t_1}^{\left[0\right]},\dots ,{\gamma }_{t_n}^{\left[0\right]}\right),$ for $t=1,\dots ,m$, and define ${⅁}_t^{\mathrm{sum}}\left({\gamma }^{\left[k\right]}\right)=\left({⅁}_{t_1}^{\mathrm{sum}}\left({\gamma }^{\left[k\right]}\right),\dots ,{⅁}_{t_n}^{\mathrm{sum}}\left({\gamma }^{\left[k\right]}\right)\right)$ [65],

where

$${⅁}_{t,r}^{\mathrm{sum}}\left({\gamma }^{\left[k\right]}\right)={\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left({\displaystyle \frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}}\right),$$

(27)

and

$${⅁}_{t,r}^{\mathrm{sum}\ast }\left({\varsigma }^{\left[k\right]}\right)={\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left({\displaystyle \frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}}\right).$$

(28)

Using (27) and (28) in (25), we develop a new simultaneous scheme (abbreviated as DS^ϵ) as follows:

$${\phi }_{t,r}^{\left[k\right]}={\varsigma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{F\left({\varsigma }_{t,r}^{\left[k\right]}\right)}{\left[{⌀}_{21}^{\left[k\right]},{⌀}_{22}^{\left[k\right]};F\right]-F\left({\varsigma }_{t,r}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}}\left({\varsigma }_{t,r}^{\left[k\right]}\right)}},$$

(29)

where ${\varsigma }_{t,r}^{\left[k\right]}={\tau }_{t,r}^{\left[k\right]}-{\displaystyle \frac{F\left({\tau }_{t,r}^{\left[k\right]}\right)}{\left[{⌀}_{11}^{\left[k\right]},{⌀}_{12}^{\left[k\right]};F\right]-F\left({\tau }_{t,r}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}}\left({\tau }_{t,r}^{\left[k\right]}\right)}}$, ${⌀}_{11}^{\left[k\right]}={\tau }_{t,r}^{\left[k\right]}+F\left({\tau }_{t,r}^{\left[k\right]}\right),{⌀}_{12}^{\left[k\right]}={\tau }_{t,r}^{\left[k\right]}-F\left({\tau }_{t,r}^{\left[k\right]}\right),{⌀}_{21}^{\left[k\right]}={\varsigma }_{t,r}^{\left[k\right]}+F\left({\varsigma }_{t,r}^{\left[k\right]}\right),{⌀}_{22}^{\left[k\right]}={\varsigma }_{t,r}^{\left[k\right]}-F\left({\varsigma }_{t,r}^{\left[k\right]}\right),$ and $\left[{⌀}_{11}^{\left[k\right]},{⌀}_{12}^{\left[k\right]};F\right]:\Delta \subset R^m\to R^m$ is the first-central forward difference on $\Delta$.

The theoretical order of convergence of the parallel scheme BD^ϵ − BM^ϵ for approximating the roots of nonlinear equations is demonstrated in Theorem 1. For the case of multiplicity unity, we observe the convergence of BD^ϵ.

Theorem 1. 

Let ${\xi }_1,...,{\xi }_m$ be simple roots of (1) with multiplicity ℘. If the initial guesses ${\gamma }^{\left[0\right]},\dots ,$ ${\gamma }_m^{\left[0\right]}$ are sufficiently close to the actual roots, then the BM^ϵ method has a convergence order of ten.

Proof. 

Let ${ϵ}_t={\gamma }_t^{\left[k\right]}-{\xi }_t$, ${ϵ}_t^{\prime }={\tau }_t^{\left[k\right]}-{\xi }_t$ and ${ϵ}_t^{″}={\varsigma }_t^{\left[k\right]}-{\xi }_t$ represent the errors in ${\gamma }_t^{\left[k\right]},$${\tau }^{\left[k\right]}$ and ${\varsigma }_t^{\left[k\right]}$, respectively. Considering the first step of BM^ϵ, we have

$${\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{1}{({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]})}\right)}},$$

(30)

where $N\left({\gamma }_t^{\left[k\right]}\right)=\left({\displaystyle \frac{f\left({\gamma }_t^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_t^{\left[k\right]}\right)}}\right).$ For distinct roots, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{N\left({\gamma }_t^{\left[k\right]}\right)}}& =& \left({\displaystyle \frac{f^{\prime }\left({\gamma }_t^{\left[k\right]}\right)}{f\left({\gamma }_t^{\left[k\right]}\right)}}\right)={\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1}{({\gamma }_t^{\left[k\right]}-{\xi }_j)}}\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& \hspace{1em}\hspace{1em}=& {\displaystyle \frac{1}{({\gamma }_t^{\left[k\right]}-{\xi }_t)}}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left({\displaystyle \frac{1}{{\gamma }_t^{\left[k\right]}-{\xi }_j)}}\right).\hfill \end{array}$$

(32)

For multiple roots, the BD^ϵ method can be expressed as

$${\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_k}{\frac{{\wp }_k}{({\gamma }_t^{\left[k\right]}-{\xi }_t)}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^n}\left(\frac{{\wp }_j}{({\gamma }_t^{\left[k\right]}-{\xi }_j)}\right)-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{{\wp }_j}{({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]})}\right)}},$$

(33)

$${\tau }_t^{\left[k\right]}-{\xi }_t={\gamma }_t^{\left[k\right]}-{\xi }_t-{\displaystyle \frac{{\sigma }_k}{\frac{{\sigma }_t}{({\gamma }_t^{\left[k\right]}-{\xi }_t)}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{{\wp }_j\left({\gamma }_t^{\left[k\right])}-{\vartheta }_j^{\left[k\right]}-{\gamma }_t^{\left[k\right]}+{\xi }_j\right)}{\left(\left({\gamma }_t^{\left[k\right]}-{\xi }_j\right)\left({\gamma }_k^{\left[k\right]}-{\vartheta }_j^{\left[k\right]}\right)\right)}\right)}},$$

(34)

$${ϵ}_t^{\prime }={ϵ}_t-{\displaystyle \frac{{\wp }_t}{\frac{{\wp }_t}{{ϵ}_t}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{-{\wp }_j\left({\vartheta }_j^{\left[k\right]}-{\xi }_j\right)}{\left({\gamma }_t^{\left[k\right]}-{\xi }_j\right)\left({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]}\right)}\right)}},$$

(35)

$$\begin{array}{ccc}\hfill {ϵ}_t^{\prime }& =& {ϵ}_t-{\displaystyle \frac{{\wp }_t{ϵ}_t}{{\wp }_t+{ϵ}_t{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{-{\wp }_j\left({\vartheta }_j^{\left[k\right]}-{\xi }_{\mathfrak{j}}\right)}{\left({\gamma }_t^{\left[k\right]}-{\xi }_j\right)\left({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]}\right)}\right)}},\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& =& {ϵ}_t-{\displaystyle \frac{{\wp }_t{ϵ}_t}{{\wp }_t+{ϵ}_t{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(U_t{ϵ}_j^4\right)}},\hfill \end{array}$$

(37)

where ${\varsigma }_j^{\left[k\right]}-{\xi }_j={ϵ}_j^3$ as per [24]. Using Equation (37) and $U_t=\left({\displaystyle \frac{-{\wp }_j}{\left({\gamma }_t^{\left[k\right]}-{\xi }_j\right)\left({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]}\right)}}\right)$, we obtain

$${ϵ}_k^{\prime }={\displaystyle \frac{{ϵ}_t^2{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(U_t{ϵ}_j^3\right)}{{\sigma }_t+{ϵ}_t{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(U_t{ϵ}_j^3\right)}}.$$

(38)

Assuming $\left|{ϵ}_t\right|=\left|{ϵ}_j\right|=O\left|ϵ\right|$, from Equation (38), we have

$${ϵ}_t^{\prime }=\mathrm{O}{\left({ϵ}_t\right)}^5.$$

(39)

Now, consider the second step of BM^ϵ:

$${\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_t}{\frac{{\wp }_t}{\left({\tau }_t^{\left[k\right]}-{\xi }_k\right)}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{{\wp }_j}{\left({\tau }_t^{\left[k\right]}-{\xi }_j\right)}\right)-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^n}\left(\frac{{\wp }_j}{{\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}}\right)}},$$

(40)

$${\varsigma }_t^{\left[k\right]}-{\xi }_t={\tau }_t^{\left[k\right]}-{\xi }_t-{\displaystyle \frac{{\wp }_t}{\frac{{\wp }_t}{\left({\tau }_t^{\left[k\right]}-{\xi }_t\right)}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^n}\left(\frac{{\wp }_j\left({\gamma }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}-{\tau }_t^{\left[k\right]}+{\xi }_j\right)}{\left({\tau }_t^{\left[k\right]}-{\xi }_j\right)\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},$$

(41)

$${ϵ}_t^{″}={ϵ}_t^{\prime }-{\displaystyle \frac{{\wp }_t}{\frac{{\wp }_t}{{ϵ}_t}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^m}\left(\frac{-{\wp }_{\mathfrak{j}}\left({\tau }_j^{\left[k\right]}-{\xi }_j\right)}{\left({\tau }_t^{\left[k\right]}-{\xi }_j\right)\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},$$

(42)

$$\begin{array}{ccc}\hfill {ϵ}_t^{″}& =& {ϵ}_t^{\prime }-{\displaystyle \frac{{\wp }_k}{\frac{{\sigma }_t}{{ϵ}_t}+{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^m}\left(\frac{-{\wp }_j\left({\tau }_j^{\left[k\right]}-{\xi }_j\right)}{\left({\tau }_t^{\left[k\right]}-{\xi }_j\right)\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& =& {ϵ}_t^{\prime }-{\displaystyle \frac{{\wp }_k}{\frac{{\wp }_t}{{ϵ}_t}+{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^n}\left(U_t^{\ast }{ϵ}_j^{\prime }\right)}},\hfill \end{array}$$

(44)

Using Equation (44) and the definition of $U_t^{\ast }=\left({\displaystyle \frac{-{\wp }_j\left({\tau }_j^{\left[k\right]}-{\xi }_j\right)}{\left({\tau }_t^{\left[k\right]}-{\xi }_j\right)\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}}\right)$, we have

$${ϵ}_t^{″}={\displaystyle \frac{{ϵ}_t^{\prime }{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left({ϵ}_j^{\prime }{U}_t^{\ast }\right)}{{\wp }_t+{ϵ}_t^{\prime }{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(U_t^{\ast }{ϵ}_j^{\prime }\right)}}.$$

(45)

Assuming $\left|{ϵ}_t^{\prime }\right|=\left|{ϵ}_j^{\prime }\right|=O\left|{ϵ}^{\prime }\right|$, from Equation (45), we obtain

$${ϵ}_t^{\prime }=O{\left({ϵ}^{\prime }\right)}^2=O{\left({ϵ}^5\right)}^2=O{\left(ϵ\right)}^{10}.$$

(46)

Hence, the theorem is proved.    □

The theoretical order of convergence of the derivative-free parallel scheme DF^ϵ to approximate all the roots of nonlinear equations is demonstrated in Theorem 2.

Theorem 2. 

Let ${\xi }_1,...,{\xi }_m$ be the simple roots of Equation (1). If the initial guesses ${\gamma }_1^{\left[k\right]},\dots ,{\gamma }_m^{\left[k\right]}$ are sufficiently close to these roots, then the DF^ϵ scheme achieves eighth-order convergence.

Proof. 

Let ${ϵ}_{\tau }={\tau }_t^{\left[k\right]}-{\xi }_t$, ${ϵ}_{\varsigma }={\varsigma }_t^{\left[k\right]}-{\xi }_t$, and ${ϵ}_{\phi }={\phi }_t^{\left[k\right]}-{\xi }_t$ be the errors in ${\tau }_t^{\left[k\right]}$, ${\varsigma }^{\left[k\right]}$, and ${\phi }_t^{\left[k\right]}$, respectively. Expanding $f\left({\tau }_t^{\left[k\right]}\right)$ using the Taylor series around $\xi$, we have

$$f\left({\tau }_t^{\left[k\right]}\right)=f^{\prime }\left(\xi \right)\left({ϵ}_{\tau }+c_2{\left({ϵ}_{\tau }\right)}^2+c_3{\left({ϵ}_{\tau }\right)}^3+\dots \right),$$

where $c_2={\displaystyle \frac{f^i\left(\xi \right)}{i!f^{\prime }\left(\xi \right)}}$ for $i\ge 2$. Thus,

$$\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]=f^{\prime }\left(\xi \right)\left(1+2c_2\left({ϵ}_{\tau }\right)+\left(\begin{array}{c}3c_3+\\ c_3{\left(f^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{\tau }\right)}^2+\dots \right).$$

Considering the first step of DF^ϵ, we have

$${\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\tau }_t^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]-f\left({\tau }_t^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},$$

(47)

$${\varsigma }_t^{\left[k\right]}-{\xi }_t={\tau }_t^{\left[k\right]}-{\xi }_t-{\displaystyle \frac{f\left({\tau }_t^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]-f\left({\tau }_t^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},$$

(48)

$${ϵ}_{\varsigma }={ϵ}_{\tau }-{\displaystyle \frac{\left[{ϵ}_{\tau }+c_2{\left({ϵ}_{\tau }\right)}^2+c_3{\left({ϵ}_{\tau }\right)}^3+...\right]}{\left[\begin{array}{c}1+2c_2\left({ϵ}_{\tau }\right)+\left(\begin{array}{c}3c_3+\\ c_3{\left(f^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{\tau }\right)}^2...\\ -{ϵ}_{\tau }{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)+...\end{array}\right]}},$$

(49)

$${ϵ}_{\varsigma }={ϵ}_{\tau }-{\displaystyle \frac{\left[{ϵ}_{\tau }+c_2{\left({ϵ}_{\tau }\right)}^2+c_3{\left({ϵ}_{\tau }\right)}^3+\dots \right]}{\left[1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)\right){ϵ}_{\tau }+O\left({ϵ}_{\tau }^2\right)\right]}},$$

(50)

$${ϵ}_{\varsigma }={\displaystyle \frac{\left[\begin{array}{c}{ϵ}_{\tau }+\left(2c_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{\tau }\right)}^2+\dots \\ -{ϵ}_{\tau }-c_2{\left({ϵ}_{\tau }\right)}^2-c_3{\left({ϵ}_{\tau }\right)}^3-\dots \end{array}\right]}{1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)\right){ϵ}_{\tau }+O\left({ϵ}_{\tau }^2\right)}},$$

(51)

$${ϵ}_{\varsigma }={\displaystyle \frac{\left(2c_2-c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{\tau }\right)}^2+...}{1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)\right){ϵ}_{\tau }+O\left({ϵ}_{\tau }^2\right)}}$$

(52)

$${ϵ}_{\varsigma }=O\left({ϵ}_{\tau }^2\right).$$

(53)

Now, considering the second step of DF^ϵ, we have

$${\phi }_t^{\left[k\right]}={\varsigma }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\varsigma }_t^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]-f\left({\varsigma }_t^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)}},$$

(54)

$${\phi }_t^{\left[k\right]}-{\xi }_t={\varsigma }_t^{\left[k\right]}-{\xi }_t-{\displaystyle \frac{f\left({\varsigma }_t^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]-f\left({\varsigma }_t^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)}},$$

(55)

$${ϵ}_{\phi }={ϵ}_{\varsigma }-{\displaystyle \frac{\left[{ϵ}_{\varsigma }+c_2{\left({ϵ}_{\varsigma }\right)}^2+c_3{\left({ϵ}_{\varsigma }\right)}^3+\dots \right]}{\left[\begin{array}{c}1+2c_2\left({ϵ}_{\varsigma }\right)+\left(\begin{array}{c}3c_3+\\ c_3{\left(f^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{\varsigma }\right)}^2\dots \\ -{ϵ}_{\varsigma }{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)+\dots \end{array}\right]}},$$

(56)

$${ϵ}_{\phi }={ϵ}_{\varsigma }-{\displaystyle \frac{{ϵ}_{\varsigma }+c_2{\left({ϵ}_{\varsigma }\right)}^2+c_3{\left({ϵ}_{\varsigma }\right)}^3+...}{1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{\tau }+O\left({ϵ}_{\tau }^2\right)}},$$

(57)

$${ϵ}_{\phi }={\displaystyle \frac{\left[\begin{array}{c}{ϵ}_{\varsigma }+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{\varsigma }\right)}^2+...\\ -{ϵ}_{\varsigma }-c_2{\left({ϵ}_{\varsigma }\right)}^2-c_3{\left({ϵ}_{\varsigma }\right)}^3-...\end{array}\right]}{1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)\right){ϵ}_{\varsigma }+O\left({ϵ}_{\varsigma }^2\right)}},$$

(58)

$${ϵ}_{\phi }={\displaystyle \frac{\left(2c_2-c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{\varsigma }\right)}^2+...}{1+\left(2c_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_t^{\left[k\right]}-{\varsigma }_j^{\left[k\right]}\right)}\right)\right){ϵ}_{\varsigma }+O\left({ϵ}_{\varsigma }^2\right)}},$$

(59)

$${ϵ}_{\phi }=O\left({ϵ}_{\varsigma }^2\right).$$

(60)

Since ${ϵ}_{\tau }=O\left({ϵ}^2\right)$ (as shown in [66]), then ${ϵ}_{\tau }=O\left({ϵ}_{\tau }^2\right)=O\left({\left({ϵ}^2\right)}^2\right)=O\left({ϵ}^4\right)$, and

$${ϵ}_{\phi }=O{\left({ϵ}^4\right)}^2=O\left({ϵ}^8\right).$$

(61)

Hence, the theorem is proved.    □

The theoretical order of convergence of the derivative-free parallel scheme DS^ϵ to approximate all solutions of the vectorial problems simultaneously is demonstrated in Theorem 3.

Theorem 3. 

Consider a sufficiently differentiable function $F:{\mathbb{R}}^n\to {\mathbb{R}}^d$ defined on a convex neighborhood of ${\xi }_{k,r}$ denoted by $\Delta \subset {\mathbb{R}}^n$, satisfying $F\left({\xi }_1\right)=0$, for $t=1,\dots ,m$. If $F^{\prime }\left({\xi }_1\right)\ne 0$, then there exists ${\gamma }_t^{\left[0\right]}=\left({\gamma }_{t1}^{\left[0\right]},\dots ,{\gamma }_{t_n}^{\left[0\right]}\right)\in {\mathbb{R}}^n$ close enough to ${\xi }_{1.r}\forall r=1,\dots ,n$, such that the iterative sequence DS^ϵ converges to the exact solution of (26) with order-eight convergence.

Proof. 

Let

$${ϵ}_{t,k_{j_1}}^{\left[\tau \right]}={\tau }_{t_{j_1}}^{\left[k\right]}-{\xi }_{t_{j_1}},{ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}={\varsigma }_{t_{j_1}}^{\left[k\right]}-{\xi }_{t_{j_1}}$$

and ${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}={\phi }_{t_{j_1}}^{\left[k\right]}-{\xi }_{t_{j_1}}$ be the errors in the estimates ${\tau }_{k,r}^{\left[k\right]},{\varsigma }_{k,r}^{\left[k\right]}$, and${\phi }_{k,r}^{\left[k\right]}$, respectively. Consider $F=\left(F_1,\dots ,F_d\right),$ where $F_p:{\mathbb{R}}^n\to \mathbb{R}$ and the coordinate of $F_p\left({\tau }_t^{\left[k\right]}\right)$ is $p=1,\dots ,d$. Using Taylor series expansion around ${\xi }_t$, we have

$$F_p\left({\tau }_t^{\left[k\right]}\right)=\left[\begin{array}{c}{\displaystyle \sum _{j_1=1}^n}{\displaystyle \frac{\partial F_p\left({\xi }_t\right)}{\partial {\gamma }_{j_1}}}{ϵ}_{k,k_{j_1}}\left({\gamma }_k^{\left[k\right]}\right)+\\ {\displaystyle \sum _{j_1=1}^n}{\displaystyle \sum _{j_2=1}^n}{\displaystyle \frac{{\partial }^2{F}_p\left({\xi }_t\right)}{\partial {\gamma }_{j_1}\partial {\gamma }_{j_2}}}{ϵ}_{t,k_{j_1}}\left({\gamma }_t^{\left[k\right]}\right){ϵ}_{t,k_{j_2}}\left({\gamma }_t^{\left[k\right]}\right)+O_3\left({ϵ}_{t,k}\right),\end{array}\right]$$

where ${ϵ}_{t,k_{j_1}}^{\left[\tau \right]}={\tau }_{k_{j_1}}^{\left[k\right]}-{\xi }_t$ for all $t=\left\{1,\dots ,m\right\}$ and $j_1=\left\{1,\dots ,n\right\}$. The residual term $O_3\left({ϵ}_{t,k}\right)$ contains the higher-order terms of the Taylor series, where the sum of exponents of ${ϵ}_{t,k_{j_1}}^q\left({\gamma }_k^{\left[k\right]}\right)$ for $j_1\in \left\{1,\dots ,n\right\}$ satisfies $q\ge 3$. We have

$$F_p\left({\tau }_t^{\left[k\right]}\right){⅁}_{t,r}^{\mathrm{sum}}\left({\tau }^{\left[k\right]}\right)=\left({\displaystyle \sum _{j_1=1}^n}{\displaystyle \frac{\partial F_p\left({\xi }_t\right)}{\partial {\tau }_{j_1}}}{ϵ}_{t,k_{j_1}}\right){⅁}_{t,r}^{\mathrm{sum}}\left({\tau }^{\left[k\right]}\right)+O_2\left({ϵ}_{t,k}\right).$$

Then

$$F_p\left({\tau }_{t,t}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}}\left({\tau }^{\left[k\right]}\right)=G_{p,r}\left({ϵ}_{t,k}^{\left[\tau \right]}\right)+O_2\left({ϵ}_{t,k}^{\left[\tau \right]}\right),\mathrm{and}$$

(62)

$$F_p\left({\varsigma }_{t,t}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}\ast }\left({\varsigma }^{\left[k\right]}\right)=G_{p,r}\left({ϵ}_{t,k}^{\left[\varsigma \right]}\right)+O_2\left({ϵ}_{t,k}^{\left[\varsigma \right]}\right),$$

(63)

for $p=1,\dots ,d$ and $k=1,\dots ,n$. Therefore, $F_p\left({\tau }_{k,t}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}}\left({\tau }_{t,r}^{\left[k\right]}\right)$ is replaced by $A{ϵ}_{t,k}^{\left[\tau \right]}+O_2\left({ϵ}_{t,k}^{\left[\tau \right]}\right)$ and $F_p\left({\varsigma }_{t,r}^{\left[k\right]}\right){⅁}_{k,r}^{\mathrm{sum}\ast }\left({\varsigma }_{t,r}^{\left[k\right]}\right)$ by $A^{[\ast ]}{ϵ}_{t,k}^{\left[\varsigma \right]}+O_2\left({ϵ}_{t,k}^{\left[\varsigma \right]}\right)$. Now, expanding $F\left({\gamma }_t^{\left[k\right]}\right),F^{\prime }\left({\gamma }_t^{\left[k\right]}\right)$, and $F^{″}\left({\gamma }_{t,r}^{\left[k\right]}\right)$ around ${\xi }_{k,t}$, we have

$$F\left({\tau }_{t,r}^{\left[k\right]}\right)=F^{\prime }\left({\xi }_{t,k_{j_1}}\right)\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+O_3\left({ϵ}_{t,k_{j_1}}\right)\right),$$

(64)

$$F^{\prime }\left({\tau }_{t,r}^{\left[k\right]}\right)=F^{\prime }\left({\xi }_{t,k_{j_1}}\right)\left(I+2C_2\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)+O_2\left({ϵ}_{t,k_{j_1}}\right)\right),$$

(65)

$$F^{″}\left({\tau }_{t,r}^{\left[k\right]}\right)=F^{\prime }\left({\xi }_{t,k_{j_1}}\right)\left(2C_2+O_1\left({ϵ}_{t,k_{j_1}}\right)\right).$$

(66)

Therefore,

$$\left[{⌀}_{21}^{\left[k\right]},{⌀}_{22}^{\left[k\right]};F\right]=F^{\prime }\left({\xi }_{t,k_{j_1}}\right)\left(\begin{array}{c}1+2C_2\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)\\ +\left(\begin{array}{c}3C_3+\\ C_3{\left(F^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+...O_3\left({ϵ}_{t,k_{j_1}}\right)\end{array}\right),$$

where $C_{\omega }\left(\varsigma \right)={\displaystyle \frac{F^{\left(\omega \right)}\left({\xi }_{t,k_{j_1}}\right)}{\omega !F^{\prime }\left({\xi }_{t,k_{j_1}}\right)}},\omega \ge 2.$ Considering the first step of DS^ϵ, we have

$${\varsigma }_{t,r}^{\left[k\right]}={\tau }_{t,r}^{\left[k\right]}-{\displaystyle \frac{F\left({\tau }_{t,r}^{\left[k\right]}\right)}{\left[{⌀}_{11}^{\left[k\right]},{⌀}_{12}^{\left[k\right]};F\right]-F\left({\tau }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]}\right)}\right)}},$$

(67)

$${\varsigma }_{t,r}^{\left[k\right]}-{\xi }_{t,r}={\tau }_{t,r}^{\left[k\right]}-{\xi }_{t,r}-{\displaystyle \frac{F{\tau }_{t,r}^{\left[k\right]})}{\left[{⌀}_{11}^{\left[k\right]},{⌀}_{12}^{\left[k\right]};F\right]-F\left({\tau }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)}},$$

(68)

$${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}={ϵ}_{t,k_{j_1}}^{\left[\tau \right]}-{\displaystyle \frac{{ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+C_3{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^3+...}{\left[\begin{array}{c}1+2c_2\left(C_{t,k_{j_1}}^{\left[\tau \right]}\right)+\\ \left(\begin{array}{c}3C_3+\\ C_3{\left(F^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2...-{ϵ}_{\tau }{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)+...\end{array}\right]}},$$

(69)

$${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}={ϵ}_{t,k_{j_1}}^{\left[\tau \right]}-{\displaystyle \frac{{ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+C_3{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^3+...}{1+\left(2C_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2}},$$

(70)

$${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}={\displaystyle \frac{\left[\begin{array}{c}{ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+\left(2C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+...\\ -{ϵ}_{t,k_{j_1}}^{\left[\tau \right]}-C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2-C_3{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^3-..\end{array}\right].}{1+\left(2C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2}},$$

(71)

$${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}={\displaystyle \frac{\left(2C_2-C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2+...}{1+\left(2C_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\tau \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2}},$$

(72)

$${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}=O{\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2.$$

(73)

Now, considering the second step of DS^ϵ, we have

$${\phi }_{t,r}^{\left[k\right]}={\varsigma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{F\left({\varsigma }_{t,r}^{\left[k\right]}\right)}{\left[{⌀}_{21}^{\left[k\right]},{⌀}_{22}^{\left[k\right]};f\right]-F\left({\varsigma }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)}},$$

(74)

$${\phi }_{t,r}^{\left[k\right]}-{\xi }_{t,r}={\varsigma }_{t,r}^{\left[k\right]}-{\xi }_{t,r}-{\displaystyle \frac{F\left({\varsigma }_{t,r}^{\left[k\right]}\right)}{\left[{⌀}_{21}^{\left[k\right]},{⌀}_{22}^{\left[k\right]};F\right]-F\left({\varsigma }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)}},$$

(75)

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}={ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}-{\displaystyle \frac{{ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}+C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2+c_3{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^3+...}{\left[\begin{array}{c}1+2C_2\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)+\\ \left(\begin{array}{c}3C_3+\\ C_3{\left(F^{\prime }\left(\xi \right)\right)}^2\end{array}\right){\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2\dots \\ -{ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)+\dots \end{array}\right]}},$$

(76)

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}={ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}-{\displaystyle \frac{{ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}+C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2+C_3{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^3+\dots }{1+\left(2C_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2}},$$

(77)

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}={\displaystyle \frac{\left[\begin{array}{c}{ϵ}_{t,k_{j_1}}^{\left[\phi \right]}+\left(2C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{t,k_{j_1}}^{\left[\phi \right]}\right)}^2+\dots \\ -{ϵ}_{t,k_{j_1}}^{\left[\phi \right]}-C_2{\left({ϵ}_{t,k_{j_1}}^{\left[\phi \right]}\right)}^2-C_3{\left({ϵ}_{t,k_{j_1}}^{\left[\phi \right]}\right)}^3-\dots \end{array}\right]}{1+\left(2C_2-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2}},$$

(78)

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}={\displaystyle \frac{\left(2C_2-C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){\left({ϵ}_{t,k_{j_1}}^{\left[\phi \right]}\right)}^2+...}{1+\left(2C_2-{\displaystyle \sum _{\stackrel{t=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{\left({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]}\right)}\right)\right){ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}+O{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2}},$$

(79)

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}=O{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2.$$

(80)

As ${ϵ}_{t,k_{j_1}}^{\left[\tau \right]}=O\left({\left({ϵ}_{t,k_{j_1}}\right)}^2\right)$ [67], we have ${ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}=O\left({\left({ϵ}_{t,k_{j_1}}^{\left[\tau \right]}\right)}^2\right)=O\left({\left({\left({ϵ}_{t,k_{j_1}}\right)}^2\right)}^2\right)=O\left({\left({ϵ}_{t,k_{j_1}}\right)}^4\right)$, and

$${ϵ}_{t,k_{j_1}}^{\left[\phi \right]}=O{\left({ϵ}_{t,k_{j_1}}^{\left[\varsigma \right]}\right)}^2=O{\left({\left({ϵ}_{t,k_{j_1}}\right)}^4\right)}^2=O\left({\left({ϵ}_{t,k_{j_1}}\right)}^8\right).$$

(81)

Hence the theorem is proved.    □

3. Computational Analysis of Simultaneous Methods

The computational efficiency of iterative methods for finding all roots of nonlinear equations is a topic of significant importance in numerical analysis. Iterative methods, both sequential and parallel numerical schemes, often demonstrate favorable computational efficiency due to their ability to iteratively refine estimates for each root. The iterative process of root-finding methods allows them to adapt effectively to complex and nonlinear functions. However, their efficiency can be influenced by factors such as the choice of initial guesses, the characteristics of the function being solved, and potential convergence issues. In cases of rapid convergence, iterative methods offer a computationally efficient approach to simultaneously find all roots. Conversely, slow convergence, particularly with high-degree polynomials or ill-conditioned problems, can negatively impact computational efficiency. Evaluating the computational efficiency of iterative methods involves assessing their convergence rates, sensitivity to initial conditions, and suitability for various types of nonlinear equations, contributing to a nuanced understanding of their performance across different scenarios. For further details on computational efficiency, refer to [68].

Figure 1a–d presents the percentage computational efficiency ratios of MM^ϵ, with respect to PM^ϵ, BD^ϵand, respectively. Meanwhile, Figure 2a–e illustrate the computational efficiency of MM^ϵ, BD^ϵ, EM^ϵ, BM^ϵ, and DF^ϵ relative to the PM^ϵ technique. The computational efficiency curves clearly demonstrate that the newly developed method is more efficient and consistent than the PM^ϵ method.

$${▿}^{[\ast ]}\left[m\right]={\displaystyle \frac{log\left[\mathbf{r}\right]}{{⅁}_{11}^{[\ast ]}+{⅁}_{12}^{[\ast ]}+{⅁}_{13}^{[\ast ]}}},$$

where ${⅁}_{11}^{[\ast ]}=w_{as}\ast A{S}_m,{⅁}_{12}^{[\ast ]}=w_m{M}_m,$ and ${⅁}_{13}^{[\ast ]}=w_d{D}_m$ and $A{S}_m,M{M}_m,D{D}_m$ represent the number of arithmetic operations of addition and subtraction, and multiplications and division, respectively [69]. Using the data provided in

Table 1. Computational cost and basic operations for simultaneous schemes.

Methods	PMϵ	MMϵ	BDϵ, EMϵ, BMϵ, DFϵ
ASm	22 × $\left(m^2\right)$ + O $\left[m\right]$	17 × $\left(m^2\right)$ + O $\left[m\right]$	19 ×  $\left(m^2\right)$ + O $\left[m\right]$
MMm	12 × $\left(m^2\right)$ + O $\left[m\right]$	6 × $\left(m^2\right)$ + O $\left[m\right]$	6 × $\left(m^2\right)$ + O $\left[m\right]$
DDm	2 × $\left(m^2\right)$ + O $\left[m\right]$	2 × $\left(m^2\right)$ + O $\left[m\right]$	2 × $\left(m^2\right)$ + O $\left[m\right]$

, we have

$${▿}_1^{[\ast ]}[m,n]=\left({\displaystyle \frac{{▿}^{[\ast ]}\left[m\right]-{▿}^{[\ast ]}\left[n\right]}{{▿}^{[\ast ]}\left[n\right]}}\times 100\right).$$

(82)

Remark 1. 

Figure 1a–e and Figure 2a–e graphically illustrate these percentage ratios. It is evident from Figure 1a,b that the newly constructed simultaneous methods BD^ϵ, DF^ϵ are more efficient compared to PM^ϵ and MM^ϵ, respectively.

4. Numerical Outcomes

To evaluate the performance and efficiency of the newly designed vectorial scheme, we solved nonlinear vectorial problems arising in science and engineering. We terminated the computer program using the following criteria:

$$e_t^{\left[k\right]}={∥{\gamma }_t^{[k+1]}-{\gamma }_t^{\left[k\right]}∥}_2<\in ={10}^{-15},$$

(83)

where $e_t^{\left[k\right]}$ represents the absolute error using the Euclidean norm-2, i.e., ${∥ ∥}_2$ [70,71]. In the numerical calculations, we utilized vectors v1–v3 from Appendix A 

Table A1. Random set of initial vectors used in Example 2 for solving nonlinear systems of equations.

$\mathit{\zeta }$	${\mathit{\zeta }}_1,$	${\mathit{\zeta }}_2,$	${\mathit{\zeta }}_3,$	${\mathit{\zeta }}_4$
v	$\left[{\zeta }_1\right]$	$\left[{\zeta }_2\right]$	$\left[{\zeta }_3\right]$	$\left[{\zeta }_4\right]$
v1 = $\left[v_1\right]$	$\left[0.32\right]$	$\left[0.12\right]$	$\left[0.12\right]$	$\left[0.91\right]$
v2 = $\left[v_2\right]$	$\left[0.11\right]$	$\left[0.65\right]$	$\left[0.41\right]$	$\left[0.12\right]$
v3 = $\left[v_3\right]$	$\left[0.02\right]$	$\left[0.93\right]$	$\left[0.14\right]$	$\left[0.31\right]$
⋮	⋮,	⋮,	⋮,	⋮

,

Table A2. Random set of initial vectors used in Example 3 for solving nonlinear systems of equations.

$\mathit{\zeta }$	${\mathit{\zeta }}_1,$	${\mathit{\zeta }}_2,$	${\mathit{\zeta }}_3,$	${\mathit{\zeta }}_4$
v	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$
v1 = $\left[v_{11},v_{12}\right]$	$\left[0.12,0.72\right]$	$\left[0.14,0.13\right]$	$\left[0.32,0.36\right]$	$\left[0.01,0.65\right]$
v2 = $\left[v_{11},v_{12}\right]$	$\left[0.61,0.72\right]$	$\left[0.43,0.54\right]$	$\left[0.01,0.42\right]$	$\left[0.92,0.45\right]$
v3 = $\left[v_{11},v_{12}\right]$	$\left[0.52,0.32\right]$	$\left[0.13,0.23\right]$	$\left[0.14,0.11\right]$	$\left[0.35,0.55\right]$
⋮	⋮,	⋮,	⋮,	⋮

and

Table A3. Random set of initial vectors used in Example 4 for solving nonlinear systems of equations.

$\mathit{\zeta }$	${\mathit{\zeta }}_1,$	${\mathit{\zeta }}_2,$	${\mathit{\zeta }}_3,$	${\mathit{\zeta }}_4$
v	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$
v1 = $\left[v_{11},v_{12}\right]$	$\left[0.21,0.12\right]$	$\left[0.13,0.54\right]$	$\left[0.01,0.13\right]$	$\left[0.02,0.05\right]$
v2 = $\left[v_{11},v_{12}\right]$	$\left[0.02,0.32\right]$	$\left[0.12,0.13\right]$	$\left[0.02,0.76\right]$	$\left[0.01,0.60\right]$
v3 = $\left[v_{11},v_{12}\right]$	$\left[0.01,0.12\right]$	$\left[0.54,0.94\right]$	$\left[0.01,0.43\right]$	$\left[0.10,0.40\right]$
⋮	⋮,	⋮,	⋮,	⋮

for (29) and abbreviated as ${\mathrm{DS}}^{{ϵ}_1}$, ${\mathrm{DS}}^{{ϵ}_2}$, and ${\mathrm{DS}}^{{ϵ}_3}$, respectively. The numerical outcomes considered the following points to analyze the simultaneous schemes PM^ϵ, MM^ϵ, BD^ϵ, BM^ϵ, DF^ϵ, and DS^ϵ:

• Computational CPU time (CPU-time);
• Residual error computed for all roots using Algorithms 1–3;
• Efficiency of the simultaneous schemes PM^ϵ, MM^ϵ, BD^ϵ, BM^ϵ, DF^ϵ, and DS^ϵ;
• Consistency and stability analysis;
• Global convergence of the simultaneous schemes analyzed using a random set of initial vectors provided in Appendix A Table A1, Table A2, Table A3,

  Table A4. Random set of initial vectors used in Example 5 for solving nonlinear systems of equations.

  $\mathit{\zeta }$	${\mathit{\zeta }}_1,$	${\mathit{\zeta }}_2,$	${\mathit{\zeta }}_3,$	${\mathit{\zeta }}_4$
  v	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$	$\left[{\zeta }_{11},{\zeta }_{12}\right]$	$\left[{\zeta }_{21},{\zeta }_{22}\right]$
  v1 = $\left[v_{11},v_{12}\right]$	$\left[0.02,0.32\right]$	$\left[0.21,0.13\right]$	$\left[0.19,0.99\right]$	$\left[0.91,0.65\right]$
  v2 = $\left[v_{11},v_{12}\right]$	$\left[0.02,0.32\right]$	$\left[0.93,0.03\right]$	$\left[0.04,0.11\right]$	$\left[0.31,0.02\right]$
  v3 = $\left[v_{11},v_{12}\right]$	$\left[0.82,0.32\right]$	$\left[0.93,0.03\right]$	$\left[0.14,0.01\right]$	$\left[0.31,0.22\right]$
  ⋮	[$⋮$,	⋮,	⋮,	$⋮]$

  and

  Table A5. Random set of initial vectors used in Example 6 for solving nonlinear systems of equations.

  $\mathit{\zeta }$	${\mathit{\zeta }}_1,$	${\mathit{\zeta }}_2,$	${\mathit{\zeta }}_3,$	${\mathit{\zeta }}_4$	⋯	${\mathit{\zeta }}_{\mathit{n}}$
  v	$\left[{\zeta }_{11},...,{\zeta }_{12}\right]$	$\left[{\zeta }_{21},...,{\zeta }_{22}\right]$	$\left[{\zeta }_{11},...,{\zeta }_{12}\right]$	$\left[{\zeta }_{21},...,{\zeta }_{22}\right]$	⋯	$\left[{\zeta }_{n1},...,{\zeta }_{n2}\right]$
  v1 = $\left[v_{11},...,v_{12}\right]$	$\left[0.02,...,0.42\right]$	$\left[0.02,...,0.03\right]$	$\left[0.12,...,0.46\right]$	$\left[0.91,...,0.65\right]$	⋯	$\left[0.01,...,0.43\right]$
  v2 = $\left[v_{11},...,v_{12}\right]$	$\left[0.01,...,0.62\right]$	$\left[0.05,...,0.04\right]$	$\left[0.41,...,0.43\right]$	$\left[0.12,...,0.45\right]$	⋯	$\left[0.04,...,0.02\right]$
  v3 = $\left[v_{11},...,v_{12}\right]$	$\left[0.02,...,0.32\right]$	$\left[0.03,...,0.03\right]$	$\left[0.14,...,0.11\right]$	$\left[0.31,...,0.22\right]$	⋯	$\left[0.05,...,0.09\right]$
  ⋮	⋮,	⋮,	⋮,	⋮	⋮,	⋮

  .

Algorithm 1: Method for finding all distinct and multiple roots of (1).

$$\begin{array}{c}\hfill \left[\begin{array}{c}\mathrm{First}\mathrm{compute}{\gamma }_i^{\left(0\right)}(tt=1,\dots ,n),\mathrm{tolerance}\in >0\mathrm{and}\mathrm{set}jj=0\mathrm{for}\mathrm{iterations}tt\\ \left[\begin{array}{c}\mathrm{Calculate}\left\{\begin{array}{c}{a}_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{f\left[{\gamma }_j^{\left[k\right]},{\gamma }_j^{\left[k\right]}+f\left({\gamma }_j^{\left[k\right]}\right)\right]}},\\ b_j^{\left[k\right]}={\gamma }_j^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)}{2\left(\frac{f\left(a_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}{\left(a_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}\right)-\frac{f\left({\gamma }_j^{\left[k\right]}\right)}{\left(a_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}}},\\ {\vartheta }_j^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_j^{\left[k\right]}\right)\left(b_j^{\left[k\right]}-{\gamma }_j^{\left[k\right]}\right)}{f\left(b_j^{\left[k\right]}\right)-f\left({\gamma }_j^{\left[k\right]}\right)}}.\end{array}\right.\\ \mathrm{Update}\left\{\begin{array}{c}{\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N\left({\gamma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{({\gamma }_t^{\left[k\right]}-{\vartheta }_j^{\left[k\right]})}\right)}},\\ {\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{1}{\frac{1}{N\left({\tau }_k^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{1}{({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]})}\right)}},\end{array}\right.\\ \mathrm{where}N\left({\gamma }_t^{\left[k\right]}\right)={\displaystyle \frac{f\left({\gamma }_t^{\left[k\right]}\right)}{f^{\prime }\left({\gamma }_t^{\left[k\right]}\right)}}.\\ {\varsigma }_{\mathfrak{t}}^{[k+1]}={\varsigma }_t^{\left[k\right]}(tt=1,...,n).\\ \mathrm{if}{e}_t^{\left[k\right]}=\left|{\gamma }_t^{[k+1]}-{\gamma }_t^{\left[k\right]}\right|<\in ={10}^{-30}\mathrm{or}\sigma >tt,\mathrm{then}\mathrm{stop}.\end{array}\right.\\ \mathrm{Set}jj=jj+1\mathrm{and}\mathrm{go}\mathrm{to}{1}^{st}-\mathrm{iteration}.\\ \mathrm{End}\mathrm{do}.\end{array}\right.\end{array}$$

Algorithm 2: Derivative-free method for finding all roots of (1).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left[\begin{array}{c}\mathrm{First}\mathrm{compute}{\gamma }_i^{\left(0\right)}(tt=1,\dots ,n),\mathrm{tolerance}\in >0\mathrm{and}\mathrm{set}jj=0\mathrm{for}\mathrm{iterations}tt\\ \left[\begin{array}{c}\mathrm{Calculate}{\tau }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}-{\displaystyle \frac{f\left({\gamma }_t^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\gamma }_t^{\left[k\right]};f\right]}}.\\ \mathrm{Update}\left\{\begin{array}{c}{\varsigma }_t^{\left[k\right]}={\tau }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_t}{{\wp }_t\frac{\left[{\theta }_{11}^{\left[k\right]},{\theta }_{12}^{\left[k\right]};f\right]}{f\left({\tau }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\tau }_t^{\left[k\right]}-{\tau }_j^{\left[k\right]})}\right)}},\\ {\phi }_t^{\left[k\right]}={\varsigma }_t^{\left[k\right]}-{\displaystyle \frac{{\wp }_t}{{\wp }_t\frac{\left[{\theta }_{21}^{\left[k\right]},{\theta }_{22}^{\left[k\right]};f\right]}{f\left({\varsigma }_t^{\left[k\right]}\right)}-{\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\varsigma }_t^{\left[k\right]}-{\varsigma }_{\mathfrak{j}}^{\left[k\right]})}\right)}},\end{array}\right.\\ \\ {\phi }_t^{\left[k\right]}={\gamma }_t^{\left[k\right]}(tt=1,...,n).\\ \mathrm{if}{e}_t^{\left[k\right]}=\left|{\gamma }_t^{[k+1]}-{\gamma }_t^{\left[k\right]}\right|<\in ={10}^{-30}\mathrm{or}\sigma >tt,\mathrm{then}\mathrm{stop}.\end{array}\right.\\ \mathrm{Set}jj=jj+1\mathrm{and}\mathrm{go}\mathrm{to}{1}^{st}-\mathrm{iteration}.\\ \mathrm{End}\mathrm{do}.\end{array}\right.\end{array}\end{array}$$

Algorithm 3: For finding all solution of (26).

$$\begin{array}{c}\hfill \left[\begin{array}{c}\mathrm{First}\mathrm{compute}{\gamma }_i^{\left(0\right)}(tt=1,..,n),\mathrm{tolerance}\in >0\mathrm{and}\mathrm{set}jj=0\mathrm{for}\mathrm{iterations}tt\\ \left[\begin{array}{c}\mathrm{Calculate}{\sigma }_{t,r}^{\left[k\right]}={\gamma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{F\left({\gamma }_{t,r}^{\left[k\right]}\right)}{\left[{\theta }_{11}^{\left[k\right]},{\gamma }_{t,r}^{\left[k\right]};F\right]}}.\\ \mathrm{Update}\left\{\begin{array}{c}{\varsigma }_{t,r}^{\left[k\right]}={\tau }_{t,r}^{\left[k\right]}-{\displaystyle \frac{{\wp }_{t}F\left({\tau }_{t,r}^{\left[k\right]}\right)}{{\wp }_t\left[{⌀}_{11}^{\left[k\right]},{⌀}_{12}^{\left[k\right]};F\right]-F\left({\tau }_{t,r}^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]})}\right)}},\\ {\phi }_{t,r}^{\left[k\right]}={\varsigma }_{t,r}^{\left[k\right]}-{\displaystyle \frac{{\wp }_{t}F\left({\varsigma }_t^{\left[k\right]}\right)}{{\wp }_t\left[{⌀}_{21}^{\left[k\right]},{⌀}_{22}^{\left[k\right]};F\right]-F\left({\varsigma }_t^{\left[k\right]}\right){\displaystyle \sum _{\stackrel{j=1}{j\ne t}}^{\mathfrak{m}}}\left(\frac{{\wp }_j}{({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]})}\right)}},\end{array}\right.\\ \mathrm{where}{⅁}_{k,r}^{\mathrm{sum}\ast }\left({\tau }_{t,r}^{\left[k\right]}\right)={\displaystyle \sum _{\stackrel{\mathfrak{t}=1}{j\ne t}}^{\mathfrak{m}}}\left({\displaystyle \frac{1}{({\tau }_{t,r}^{\left[k\right]}-{\tau }_{j,r}^{\left[k\right]})}}\right),\mathrm{and}\\ {⅁}_{k,r}^{\mathrm{sum}\ast }\left({\varsigma }_{t,r}^{\left[k\right]}\right)={\displaystyle \sum _{\stackrel{\mathfrak{t}=1}{j\ne t}}^{\mathfrak{m}}}\left({\displaystyle \frac{1}{({\varsigma }_{t,r}^{\left[k\right]}-{\varsigma }_{j,r}^{\left[k\right]})}}\right).\\ {\varsigma }_{t,r}^{\left[k\right]}={\gamma }_{t,r}^{\left[k\right]}(tt=1,...,n).\\ \mathrm{if}{e}_t^{\left[k\right]}=\left|{\gamma }_t^{[k+1]}-{\gamma }_t^{\left[k\right]}\right|<\in ={10}^{-30}\mathrm{or}\sigma >tt,\mathrm{then}\mathrm{stop}.\end{array}\right.\\ \mathrm{Set}jj=jj+1\mathrm{and}\mathrm{go}\mathrm{to}{1}^{st}-\mathrm{iteration}.\\ \mathrm{End}\mathrm{do}.\end{array}\right.\end{array}$$

Example 1: Quarter car suspension model

The shock absorber in the suspension system regulates the transient behavior of both the vehicle and suspension mass [72,73]. The nonlinear behavior of the suspension system makes it one of its most complex components. Nonetheless, the damping force of the dampers is characterized by an asymmetric nonlinear hysteresis loop. Automobile engineers utilize a quarter car suspension model, which is a simplified representation, to examine the vertical dynamics of a vehicle’s single wheel and its interaction with the road. This model is a component of the broader field of vehicle dynamics and suspension design. The quarter car model divides the vehicle into two primary parts: the sprung mass and the unsprung mass.

Automobile Structure Sprung Weight: The vehicle body mass includes the chassis, occupants, and other components directly supported by the suspension. The majority of the sprung mass is typically concentrated around the vehicle’s center of gravity.

The Unsprung Weight and Suspension of the Wheels: The unsprung mass includes the wheel, tire, and any suspension components directly linked to the wheel. These components are not supported by the suspension springs.

The suspension system, comprising a spring and a damper, regulates the interaction between sprung and unsprung masses. The spring represents the suspension’s elasticity, while the damper simulates the shock absorber’s damping effect. Using the quarter car suspension model, engineers can analyze how a vehicle responds to potholes and other road irregularities. This model allows for the calculation of dynamic quantities such as suspension deflection, wheel displacement, and vehicle body forces. Understanding these fundamental dynamics and characteristics of suspension systems aids in designing and optimizing suspension systems for improved ride comfort, handling, and stability. Despite the availability of more advanced models, such as half-car or full-car models, the quarter car model remains a critical tool in vehicle dynamics studies. The equations for mass motion are as follows:

$$\left\{\begin{array}{c}{m}_s{\gamma }_s^{″}+k_s({\gamma }_s-{\gamma }_u)+F=0,\\ m_u{\gamma }_u^{″}-k_s({\gamma }_s-{\gamma }_u)-k_{\sigma }({\gamma }_r-{\gamma }_u)-F=0,\end{array}\right.$$

(84)

where $m_s$ represents the mass above the spring, $m_u$ denotes the mass below the spring, ${\gamma }_s$ signifies the displacement of the sprung mass, ${\gamma }_u$ indicates the displacement of the unsprung mass, ${\gamma }_r$ represents disturbances from road bumps, $k_s$ corresponds to the spring stiffness, and $k_{\sigma }$ pertains to the tire stiffness. To accurately model the damper force F, one can use the polynomial presented by Barethiye [74] in Equation (64):

$$f\left(\gamma \right)=-77.14{\gamma }^4+23.14{\gamma }^3+342.7{\gamma }^2+956.7\gamma +124.5.$$

(85)

The exact roots of Equation (65) are

$$\begin{array}{ccc}\hfill {\zeta }_1& =& 3.090556803,{\zeta }_2=-1.326919946+1.434668028i,{\zeta }_3=-0.1367428388,\hfill \\ \hfill {\zeta }_4& =& -1.326919946-1.434668028i.\hfill \end{array}$$

Initial guesses: $\stackrel{\left[0\right]}{{\gamma }_1}=0.1,$$\stackrel{\left[0\right]}{{\gamma }_2}=-1+1i,$${\stackrel{\left[0\right]}{\gamma }}_3=-0.1,{\gamma }_4^{\left[0\right]}=-1-1i.$ The numerical outcomes for the initial guesses, which are sufficiently close to the exact values, are presented in

Table 2. Residual error with initial values near exact root.

Methods	PMϵ	MMϵ	BM	EM	DFϵ	BDϵ
Error it	05	05	05	04	05	05
CPU	0.015231	0.013467	0.01235	0.001245	0.00123	0.00123
$e_1^{\left[k\right]}$	9.6754 × 10−40	9.3215 × 10−65	7.6543 × 10−65	6.5634 × 10−40	0.0	0.0
$e_2^{\left[k\right]}$	7.4532 × 10−43	0.8753 × 10−64	9.7865 × 10−67	8.7453 × 10−95	2.4642 × 10−71	1.0042 × 10−91
$e_3^{\left[k\right]}$	6.8750 × 10−43	6.9873 × 10−85	3.4534 × 10−80	6.4534 × 10−86	7.3432 × 10−87	7.545 × 10−107
$e_4^{\left[k\right]}$	8.9765 × 10−43	3.3421 × 10−95	7.3453 × 10−73	0.0	0.0	0.0
${\rho }_t^{[\sigma -1]}$	9.1221512	10.031452	10.914556	10.2323	9.9654	9.9865

.

The results of Table 2 clearly show that BF^ϵ and BD^ϵ are superior to PM^ϵ, MM^ϵ, BM, and EM in terms of computational order of convergence, CPU-time, residual error, and error iteration (Error it) for solving (85).

The initial vectors provided in Table A1 [75] are used to verify the global convergence of PM^ϵ, MM^ϵ, BM, EM, DF^ϵ, and BD^ϵ.

The numerical outcomes of the simultaneous vectorial method for solving (85) in terms of residual error, CPU time, local computational order of convergence, and iterations are shown in

Table 3. Residual error using a random set of initial-guess vectors.

Methods	${\mathit{e}}_1^{\left[\mathit{k}\right]}$	${\mathit{e}}_2^{\left[\mathit{k}\right]}$	${\mathit{e}}_3^{\left[\mathit{k}\right]}$	${\mathit{e}}_4^{\left[\mathit{k}\right]}$	${\mathit{\rho }}_{\mathit{\varsigma }}^{[\mathit{k}-1]}$
Error on random initial-guess vector v1
PMϵ	2.12 × 10−12	9.73 × 10−23	2.34 × 10−45	7.34 × 10−34	7.63563774
MMϵ	3.22 × 10−32	0.56 × 10−43	8.76 × 10−34	0.98 × 10−54	9.43565765
BM	4.33 × 10−12	0.76 × 10−23	1.54 × 10−54	9.99 × 10−34	9.13456683
EM	9.10 × 10−12	3.43 × 10−44	9.76 × 10−34	0.23 × 10−24	8.56575532
DFϵ	0.12 × 10−31	8.65 × 10−33	0.98 × 10−54	8.72 × 10−34	8.13424556
BDϵ	0.16 × 10−20	0.05 × 10−32	0.11 × 10−57	8.02 × 10−32	9.13424556
Error on random initial-guess vector v2
PMϵ	9.23 × 10−23	8.65 × 10−43	3.43 × 10−23	1.34 × 10−26	7.63563774
MMϵ	8.23 × 10−12	7.65 × 10−45	7.64 × 10−34	6.43 × 10−36	9.43565765
BM	2.34 × 10−34	0.86 × 10−34	8.97 × 10−46	7.89 × 10−35	9.13456683
EM	4.54 × 10−21	0.04 × 10−23	7.34 × 10−36	9.80 × 10−46	8.56575532
DFϵ	0.12 × 10−31	8.65 × 10−33	0.98 × 10−54	8.72 × 10−34	8.13424556
BDϵ	5.43 × 10−15	0.04 × 10−26	2.23 × 10−26	0.98 × 10−36	8.13424556
Error on random initial-guess vector v3
PMϵ	5.65 × 10−16	3.21 × 10−23	7.56 × 10−37	8.65 × 10−35	7.63563774
MMϵ	6.54 × 10−27	5.32 × 10−34	1.23 × 10−36	8.65 × 10−34	9.43565765
BM	7.54 × 10−36	5.32 × 10−45	8.23 × 10−38	4.66 × 10−46	9.13456683
EM	7.34 × 10−35	4.23 × 10−23	6.23 × 10−28	0.36 × 10−26	8.56575532
DFϵ	0.72 × 10−31	1.65 × 10−33	7.98 × 10−54	8.79 × 10−34	7.13424556
BDϵ	8.94 × 10−18	6.43 × 10−44	4.32 × 10−47	0.08 × 10−27	8.13424556

,

Table 4. CPU time using a random set of starting vectors v1–v3 taken from Appendix A Table A1.

Methods	${\mathit{e}}_1^{\left[\mathit{k}\right]}$	${\mathit{e}}_2^{\left[\mathit{k}\right]}$	${\mathit{e}}_3^{\left[\mathit{k}\right]}$	${\mathit{e}}_4^{\left[\mathit{k}\right]}$
PMϵ	2.5487	3.4533	2.4533	0.3543
MMϵ	1.0576	0.7657	0.6754	0.3421
BM	0.9342	3.6598	1.2343	0.7342
EM	0.7654	1.8765	1.9342	0.0142
DFϵ	0.4354	1.5005	1.7302	0.0142
BDϵ	0.0042	0.9842	0.3234	0.0342

and

Table 5. Iterations using random initial guess vectors v1–v3 from Appendix A Table A1.

Methods	${\mathit{e}}_1^{\left[\mathit{k}\right]}$	${\mathit{e}}_2^{\left[\mathit{k}\right]}$	${\mathit{e}}_3^{\left[\mathit{k}\right]}$	${\mathit{e}}_4^{\left[\mathit{k}\right]}$	${\mathit{\rho }}_{\mathit{\varsigma }\mathit{k}}^{[\mathit{k}-1]}$
Random initial guess vectors v1–v3
taken from Appendix A Table A1
PMϵ	08	08	08	08	07
MMϵ	08	08	07	07	08
BM	05	05	05	05	02
EM	06	06	06	07	02
DFϵ	04	04	05	07	07
BDϵ	03	03	03	03	08

and Figure 3. Table 3, Table 4 and Table 5 clearly illustrate that DF^ϵ outperforms MM^ϵ, BM, EM, PM^ϵ, and BD^ϵ in terms of global convergence, as it converges faster and utilizes less CPU time and fewer iterations than the other methods.

The numerical results from iterative methods using random initial vectors, as presented in Table 3, Table 4, Table 5 and

Table 6. Consistency of the numerical scheme for solving (85).

Methods	Max-${\mathit{e}}_{\mathit{i}}^{\left[\mathit{k}\right]}$	Avg-CPU	Avg-Iterations	Avg-COC
Random initial guess vectors v1–v3 taken from Appendix A Table A1
PMϵ	0.2144 × 10−10	1.9453	03	7.3454
MMϵ	4.874 × 10−20	0.8664	03	9.2215
BM	2.984 × 10−21	0.5616	05	1.9432
EM	2.983 × 10−21	0.5615	04	1.9987
DFϵ	2.982 × 10−21	0.5617	03	7.8758
BDϵ	3.126 × 10−21	0.1146	03	6.8796

, demonstrate that the newly developed schemes BD^ϵ and DF^ϵ outperform existing methods such as PM^ϵ, MM^ϵ, BM, and EM, achieving a significantly higher accuracy with maximum errors of 0.11 × 10⁻⁵⁷, 0.98 × 10⁻⁵⁴, and 7.98 × 10⁻⁵⁴ for the three sets of initial vectors v1–v3 (see Table 3 and Figure 3). These techniques also exhibit superior performance compared to DM, DM1, and DM3 in terms of average CPU time (Avg-CPU) and average number of iterations (Avg-Iterations). Table 6 provides an overall assessment of the simultaneous schemes, confirming that DF^ϵ shows greater stability and consistency compared to MM^ϵ, BD^ϵ, BM^ϵ, PM^ϵ, and DS^ϵ, respectively.

Example 2: Solving a non-differentiable system [76]

Consider the non-differentiable system:

$$⅁\left(\gamma \right)=\left\{\begin{array}{c}{\gamma }_1{\gamma }_2-\left|{\gamma }_1\right|=0,\\ {\gamma }_1{\gamma }_2-\left|{\gamma }_2\right|=0.\end{array}\right.$$

(86)

The exact solutions to $⅁\left(\gamma \right)=0$ are ${\xi }_1=\left({\xi }_{1,1},{\xi }_{1,2}\right)=\left(-1,-1\right);{\xi }_2=\left({\xi }_{2,1},{\xi }_{2,2}\right)=\left(1,1\right)$, and the trivial solution is $(0,0)$. We chose the following initial guesses: ${\gamma }_1^{\left[0\right]}=\left[{\gamma }_{1,1}^{\left[0\right]},{\gamma }_{1,2}^{\left[0\right]}\right]=\left[-2.5,-3.1\right]$ and ${\gamma }_2^{\left[0\right]}=\left[{\gamma }_{2,1}^{\left[0\right]},{\gamma }_{2,2}^{\left[0\right]}\right]=\left[2.5,3.1\right]$. The numerical results are presented in

Table 7. Simultaneous determination of ${\xi }_1=\left({\xi }_{1,1},{\xi }_{1,2}\right)$ of (86) using parallel schemes.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
CPU	0.0321	0.0321	0.0212	0.0213	0.0232	0.0342
$e_{1,1}^{\left[k\right]}$	0.103 × 10−34	0.123 × 10−34	0.433 × 10−45	0.343 × 10−34	9.876 × 10−32	0.876 × 10−54
$e_{1,2}^{\left[k\right]}$	4.325 × 10−33	1.325 × 10−33	0.643 × 10−41	0.875 × 10−23	0.875 × 10−44	7.876 × 10−47
${\rho }_{\varsigma }^{[k-1]}$	8.9765	8.9765	9.9678	7.9854	9.9876	7.0987
Simultaneous determination of ${\xi }_2=\left({\xi }_{2,1},{\xi }_{2,2}\right)$ of (86) using parallel scheme
CPU	0.08201	0.093	0.07542	0.093	0.0232	0.0002
$e_{1,1}^{\left[k\right]}$	5.435 × 10−37	0.324 × 10−34	6.543 × 10−65	4.324 × 10−34	8.654 × 10−43	2.324 × 10−51
$e_{1,2}^{\left[k\right]}$	7.534 × 10−45	5.004 × 10−34	7.654 × 10−34	5.424 × 10−34	0.985 × 10−43	6.534 × 10−54
${\rho }_{\varsigma }^{[k-1]}$	8.4365	7.9984	9.0978	7.9984	9.9006	7.0007

.

The results of Table 7 clearly demonstrate that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM in terms of computational order of convergence, CPU-time, and residual error for solving (86).

To check the global convergence behavior, we utilized the following starting set of vectors presented in Appendix A Table A2.

In terms of residual error, CPU time, local computational order of convergence, and iterations, the numerical results of the simultaneous vectorial method for solving (86) are presented in

Table 8. Residual error using a random set of initial-guess vectors.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Error on random initial-guess vector v1
$e_{1,1}^{\left[k\right]}$	0.12 × 10−12	0.705 × 10−13	4.212 × 10−7	9.872 × 10−1	0.246 × 10−13	4.076 × 10−17
$e_{1,2}^{\left[k\right]}$	9.123 × 10−13	1.300 × 10−13	0.765 × 10−13	9.8721 × 10−23	7.531 × 10−14	2.098 × 10−16
$e_{2,1}^{\left[k\right]}$	8.313 × 10−13	5.432 × 10−14	1.324 × 10−11	0.872 × 10−17	9.736 × 10−15	4.875 × 10−18
$e_{2,2}^{\left[k\right]}$	0.983 × 10−14	9.762 × 10−15	8.312 × 10−13	1.323 × 10−12	0.943 × 10−16	8.765 × 10−18
Error on random initial-guess vectors v2–v3
$e_{1,1}^{\left[k\right]}$	0.765 × 10−13	0.765 × 10−13	8.762 × 10−24	9.8762 × 10−14	0.002 × 10−29	0.125 × 10−19
$e_{1,2}^{\left[k\right]}$	1.334 × 10−12	3.334 × 10−11	0.762 × 10−13	0.9872 × 10−25	0.139 × 10−28	7.654 × 10−29
$e_{2,1}^{\left[k\right]}$	7.312 × 10−13	8.382 × 10−10	3.312 × 10−16	0.987 × 10−17	0.983 × 10−16	3.654 × 10−27
$e_{2,2}^{\left[k\right]}$	8.762 × 10−14	6.762 × 10−17	9.723 × 10−25	0.9872 × 10−17	0.765 × 10−25	0.965 × 10−15

,

Table 9. CPU time using a random set of starting vectors v1–v3 from Appendix A Table A2.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
CPU time using vector
$e_{1,1}^{\left[k\right]}$	0.3423	3.2817	2.4693	1.0982	3.231	2.3215
$e_{1,2}^{\left[k\right]}$	2.4324	9.3213	2.4365	3.2981	3.4321	4.3093
$e_{2,1}^{\left[k\right]}$	4.3253	2.4365	3.3132	3.2817	2.4365	1.3241
$e_{2,2}^{\left[k\right]}$	4.3432	3.231	3.2462	9.3213	3.2314	1.3424

and

Table 10. Iterations using a random set of starting vectors v1–v3 from Appendix A Table A2.

Methods	MMϵ	EMϵ1	BMϵ	DSϵ1	DSϵ2	DSϵ3
$e_{1,1}^{\left[k\right]}$	06	04	04	05	06	03
$e_{1,2}^{\left[k\right]}$	06	04	04	05	06	03
$e_{2,1}^{\left[k\right]}$	06	04	04	05	06	03
$e_{2,2}^{\left[k\right]}$	06	04	04	05	06	03

and Figure 4. These tables clearly illustrate that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ in terms of global convergence, converging faster, and utilizing less CPU time and fewer iterations than other methods.

The numerical results of the iterative method using random initial vectors presented in Table 8, Table 9 and Table 10 show that the newly developed scheme ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_3}$ is more efficient than existing methods MM^ϵ, EM^ϵ, BM^ϵ because it achieves a much higher accuracy, with errors of 4.8756 × 10⁻¹⁸, 7.651 × 10⁻²⁹, and 7.654 × 10⁻²⁹ for the three sets of initial vectors v1–v3, respectively. Additionally, it consumes less CPU time and requires fewer iterations.

Table 11. Consistency of the numerical scheme for solving (86) using vectors v1–v3 taken from Appendix A Table A2.

Methods	Max-${\mathit{e}}_{\mathit{i}}^{\left[\mathit{k}\right]}$	Avg-CPU	Avg-It	Avg-COC
MMϵ	7.876 × 10−31	2.2343	04	6.765
EMϵ	4.435 × 10−26	2.742	05	6.465
BMϵ	8.675 × 10−23	2.1234	05	5.897
DSϵ1	7.654 × 10−22	2.7654	05	4.765
DSϵ2	3.124 × 10−36	3.3427	04	4.765
DSϵ3	4.435 × 10−26	2.7654	03	3.765

depicts the overall behavior of the simultaneous schemes and demonstrates that ${\mathrm{DS}}^{{ϵ}_1}$ is more stable and consistent than ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ.

Example 3: Computing the steady state of the epidemic model [77]

Consider the following system of nonlinear equations:

$$⅁\left(\gamma \right)=\left\{\begin{array}{l}{\gamma }_1-\left(1-R\right)\left({\displaystyle \frac{D}{10\left(1+{\beta }_1\right)}}-{\gamma }_1\right)e^{\left({\displaystyle \frac{10{\gamma }_1}{1+\frac{10{\gamma }_1}{\omega }}}\right)}=0,\\ {\gamma }_1-\left(1-{\beta }_2\right){\gamma }_2+\left(1-R\right)\left({\displaystyle \frac{D}{10}}-{\beta }_1{\gamma }_1-\left(1+{\beta }_1\right){\gamma }_2\right)e^{\left({\displaystyle \frac{10{\gamma }_2}{1+\frac{10{\gamma }_2}{\omega }}}\right)}=0,\end{array}\right.$$

(87)

where $D=22$, ${\beta }_1={\beta }_1=2,R=0.935$ and $\omega =1000$, although different values of R may be considered. The exact solutions of $⅁\left(\gamma \right)=0$ are ${\xi }_1=\left({\xi }_{1,1},{\xi }_{1,2}\right)=\left(0.00752614,0.0589059\right)$ and ${\xi }_2=\left({\xi }_{2,1},{\xi }_{2,2}\right)=\left(1,1\right)$. We choose the initial guesses as ${\gamma }_1^{\left[0\right]}=\left[{\gamma }_{1,1}^{\left[0\right]},{\gamma }_{1,2}^{\left[0\right]}\right]=\left[-2.5,-3.1\right]$ and ${\gamma }_2^{\left[0\right]}=\left[{\gamma }_{2,1}^{\left[0\right]},{\gamma }_{2,2}^{\left[0\right]}\right]=\left[2.5,3.1\right]$. The numerical results are presented in

Table 12. Simultaneous finding of ${\xi }_1=\left({\xi }_{1,1},{\xi }_{1,2}\right)$s of (87) using parallel schemes.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
CPU-time	0.0321	0.0232	0.0212	0.0213	0.0232	0.0342
$e_{1,1}^{\left[k\right]}$	0.873 × 10−24	7.298 × 10−14	0.561 × 10−45	0.343 × 10−44	9.298 × 10−24	0.346 × 10−42
$e_{1,2}^{\left[k\right]}$	1.365 × 10−23	1.103 × 10−44	0.343 × 10−41	0.875 × 10−33	0.113 × 10−34	1.056 × 10−13
${\rho }_{\varsigma }^{[k-1]}$	8.9765	9.0076	9.8768	7.9854	9.0076	7.0987
Simultaneous finding of ${\xi }_2=\left({\xi }_{2,1},{\xi }_{2,2}\right)$s of (87) using parallel sachems.
CPU-time	0.08451	0.093	0.07542	0.093	0.0201	0.0232
$e_{1,1}^{\left[k\right]}$	6.345 × 10−14	1.134 × 10−24	6.554 × 10−55	4.334 × 10−24	8.994 × 10−33	8.345 × 10−42
$e_{1,2}^{\left[k\right]}$	9.834 × 10−15	5.004 × 10−27	7.674 × 10−34	5.884 × 10−24	0.115 × 10−34	3.534 × 10−47
${\rho }_{\varsigma }^{[k-1]}$	8.1165	8.9984	8.0458	8.9984	9.8306	8.0237

.

The results of Table 12 clearly show that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ in terms of computational order of convergence, CPU-time, and residual error for solving (87).

To evaluate the global convergence behavior, we utilize the initial vector set presented in Appendix A Table A3.

In terms of residual error, CPU time, local computational order of convergence, and iterations, the numerical outcomes of the simultaneous vectorial method for solving (87) are shown in

Table 13. Residual error using a random set of starting vectors.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Error on random initial-guess vector v1
$e_{1,1}^{\left[k\right]}$	7.12 × 10−12	9.072 × 10−1	3.871 × 10−7	9.072 × 10−1	0.126 × 10−13	4.002 × 10−17
$e_{1,2}^{\left[k\right]}$	1.653 × 10−23	7.232 × 10−23	1.465 × 10−13	7.212 × 10−23	7.531 × 10−14	2.098 × 10−16
$e_{2,1}^{\left[k\right]}$	4.093 × 10−23	3.832 × 10−17	8.624 × 10−11	0.872 × 10−17	9.736 × 10−15	4.875 × 10−18
$e_{2,2}^{\left[k\right]}$	0.983 × 10−34	1.333 × 10−12	8.312 × 10−13	1.323 × 10−12	0.943 × 10−16	8.765 × 10−18
Error on random initial-guess vectors v2–v3
$e_{1,1}^{\left[k\right]}$	2.453 × 10−13	9.072 × 10−11	1.542 × 10−24	0.1242 × 10−13	9.802 × 10−26	2.135 × 10−12
$e_{1,2}^{\left[k\right]}$	1.334 × 10−12	4.212 × 10−23	0.762 × 10−13	0.9872 × 10−24	0.139 × 10−25	2.354 × 10−23
$e_{2,1}^{\left[k\right]}$	7.312 × 10−13	0.822 × 10−17	3.312 × 10−16	0.9873 × 10−15	0.983 × 10−14	1.124 × 10−26
$e_{2,2}^{\left[k\right]}$	8.762 × 10−14	2.223 × 10−12	9.723 × 10−25	0.9872 × 10−16	0.765 × 10−22	0.345 × 10−14

,

Table 14. CPU time using a random set of starting vectors v1–v3 taken from Appendix A Table A3.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
$e_{1,1}^{\left[k\right]}$	0.7765	0.0573	0.7833	0.0573	0.7634	0.0573
$e_{1,2}^{\left[k\right]}$	0.0573	0.7642	0.0742	0.9342	0.0742	0.7642
$e_{2,1}^{\left[k\right]}$	0.9342	0.9342	0.2342	0.7765	0.0042	0.4211
$e_{2,2}^{\left[k\right]}$	0.7765	0.7765	0.0042	0.9842	0.3234	0.0342

and

Table 15. Iterations using a random set of starting vectors v1–v3 from Appendix A Table A3.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
$e_{1,1}^{\left[k\right]}$	05	04	04	05	06	03
$e_{1,2}^{\left[k\right]}$	05	04	04	05	06	03
$e_{2,1}^{\left[k\right]}$	05	04	04	05	06	03
$e_{2,2}^{\left[k\right]}$	05	04	04	05	06	03

and Figure 5. Table 13, Table 14 and Table 15 clearly illustrate that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM in terms of global convergence, converging faster and consuming less CPU time, and requiring fewer iterations than other methods.

With errors of 8.75 × 10⁻¹⁸, 1.124 × 10⁻²⁴, and 1.121 × 10⁻²⁴ for the three sets of initial vectors v1–v3, the newly developed scheme ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_3}$ achieved a significantly higher accuracy compared to the existing methods MM^ϵ, EM^ϵ, and BM^ϵ. It also consumed less CPU time and required fewer iterations, as evidenced by the numerical results of the iterative method using random initial vectors presented in Table 3, Table 4, Table 5 and Table 6.

Table 16. Consistency of the numerical scheme for solving (87) using vectors v1–v3 from Appendix A Table A3.

Methods	Max-${\mathit{e}}_{\mathit{i}}^{\left[\mathit{k}\right]}$	Avg-CPU	Avg-It	Avg-COC
MMϵ	3.622 × 10−20	1.1133	05	7.435
EMϵ	2.656 × 10−21	1.5433	04	7.435
BMϵ	6.87 × 10−33	2.4534	06	5.917
DSϵ1	0.854 × 10−32	2.7764	03	2.795
DSϵ2	6.54 × 10−36	3.3548	04	5.705
DSϵ3	4.435 × 10−26	2.7223	03	2.975

depicts the overall behavior of the simultaneous schemes and demonstrates that DS${\mathrm{DS}}^{{ϵ}_3}$ is more stable and consistent than ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ.

Example 4: Searching for the equilibrium point of the N-Body system [78]

Consider the nonlinear system of equations describing how to find the equilibrium solution in an N-body system as

$$Ņ\left(\gamma \right)=\left\{\begin{array}{c}\left(\sqrt{3}{\gamma }_1-{\gamma }_2\right)\left(1-{\displaystyle \frac{1}{{\left({\gamma }_1^2+{\gamma }_2^2\right)}^{\frac{3}{2}}}}\right)+{\Lambda }_1\left(\sqrt{3}\left({\gamma }_1-1\right)-{\gamma }_2\right)\left(1-{\displaystyle \frac{1}{{\left({\left({\gamma }_1-1\right)}^2+{\gamma }_2^2\right)}^{\frac{3}{2}}}}\right)=0,\\ 2{\gamma }_2\left(1-{\displaystyle \frac{1}{{\left({\gamma }_1^2+{\gamma }_2^2\right)}^{\frac{3}{2}}}}\right)+{\Lambda }_2\left(\sqrt{3}\left({\gamma }_1-1\right)-{\gamma }_2\right)\left(1-{\displaystyle \frac{1}{{\left({\left({\gamma }_1-\frac{1}{2}\right)}^2+{\left({\gamma }_2-\frac{\sqrt{3}}{2}\right)}^2\right)}^{\frac{3}{2}}}}\right)=0.\end{array}\right.$$

(88)

The nonlinear system of equations Ņ$\left(\gamma \right)$ has more than one solution depending on the parameter values. For instance, if we choose ${\Lambda }_1=0.3$ and ${\Lambda }_1=0.4$, Ņ$\left(\gamma \right)$ has five solutions. Initial starting values are chosen as ${\gamma }_1^{\left[0\right]}=\left[{\gamma }_{1,1}^{\left[0\right]},{\gamma }_{1,2}^{\left[0\right]}\right]=\left[-2.5,-3.1\right]$ and ${\gamma }_2^{\left[0\right]}=\left[{\gamma }_{2,1}^{\left[0\right]},{\gamma }_{2,2}^{\left[0\right]}\right]=\left[2.5,3.1\right]$. The numerical outcomes are presented in Table 7.

The results in

Table 17. Determination of ${\xi }_1=\left({\xi }_{1,1},{\xi }_{1,2}\right)$ of (88) using parallel schemes.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
CPU	0.0321	0.0223	0.0212	0.0213	0.0232	0.0342
$e_{1,1}^{\left[k\right]}$	0.003 × 10−34	0.343 × 10−14	0.411 × 10−45	0.343 × 10−14	5.876 × 10−44	0.876 × 10−42
$e_{1,2}^{\left[k\right]}$	1.325 × 10−33	0.115 × 10−23	0.512 × 10−41	0.115 × 10−23	0.875 × 10−34	7.876 × 10−43
${\rho }_{\varsigma }^{[k-1]}$	8.9765	7.9874	8.3248	7.9874	9.0076	7.0181
Simultaneous determination of ${\xi }_2=\left({\xi }_{2,1},{\xi }_{2,2}\right)$ of (88) using parallel schemes.
CPU	0.08201	0.0234	0.07542	0.093	0.0232	0.0002
$e_{1,1}^{\left[k\right]}$	5.435 × 10−54	0.343 × 10−14	6.040 × 10−55	7.324 × 10−12	0.654 × 10−34	2.324 × 10−43
$e_{1,2}^{\left[k\right]}$	7.534 × 10−45	0.115 × 10−23	7.654 × 10−34	3.424 × 10−54	4.3475 × 10−24	6.534 × 10−44
${\rho }_{\varsigma }^{[k-1]}$	8.423455	7.9874	9.0324	7.9654	9.4306	7.0127

clearly demonstrate that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ in terms of computational order of convergence, CPU-time, and residual error for solving (88).

To assess the global convergence behavior, we utilize the following initial set of vectors presented in Appendix A Table A4.

In terms of residual error, CPU time, local computational order of convergence, and iterations, the numerical outcomes of the simultaneous vectorial method for solving (88) are shown in

Table 18. Residual errors using a random set of starting vectors.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Error using random initial-guess vector v1
$e_{1,1}^{\left[k\right]}$	0.12 × 10−12	0.12 × 10−12	0.765 × 10−25	0.943 × 10−16	0.872 × 10−17	9.736 × 10−15
$e_{1,2}^{\left[k\right]}$	2.122 × 10−20	0.139 × 10−28	0.765 × 10−23	9.8721 × 10−23	7.531 × 10−14	2.098 × 10−16
$e_{2,1}^{\left[k\right]}$	1.003 × 10−11	0.983 × 10−16	1.324 × 10−21	0.872 × 10−17	9.736 × 10−15	4.875 × 10−18
$e_{2,2}^{\left[k\right]}$	3.765 × 10−21	0.765 × 10−25	8.312 × 10−23	1.323 × 10−12	0.943 × 10−16	8.765 × 10−18
Error using random initial-guess vectors v2–v3
$e_{1,1}^{\left[k\right]}$	0.765 × 10−13	0.12 × 10−12	1.324 × 10−21	0.872 × 10−17	9.736 × 10−15	0.125 × 10−19
$e_{1,2}^{\left[k\right]}$	1.404 × 10−11	0.139 × 10−28	8.312 × 10−23	1.323 × 10−12	0.943 × 10−16	2.098 × 10−16
$e_{2,1}^{\left[k\right]}$	4.343 × 10−10	0.983 × 10−16	0.943 × 10−16	0.872 × 10−17	9.736 × 10−15	4.875 × 10−18
$e_{2,2}^{\left[k\right]}$	8.762 × 10−14	0.765 × 10−25	9.723 × 10−25	0.9872 × 10−17	0.765 × 10−25	0.965 × 10−15

,

Table 19. CPU time using a random set of starting vectors v1–v3 taken from Appendix A Table A4.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
$e_{1,1}^{\left[k\right]}$	4.3077	4.2324	1.7611	3.2987	2.0011	4.2324
$e_{1,2}^{\left[k\right]}$	4.4234	3.2987	2.3432	2.4276	0.2024	2.4276
$e_{2,1}^{\left[k\right]}$	2.0011	2.4276	1.3424	2.3432	4.3077	1.7611
$e_{2,2}^{\left[k\right]}$	0.2024	2.3432	4.3077	1.7611	1.6524	4.3232

and

Table 20. Iterations using a random set of starting vectors v1–v3 taken from Appendix A Table A4.

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
$e_{1,1}^{\left[k\right]}$	06	04	04	05	06	03
$e_{1,2}^{\left[k\right]}$	06	04	04	05	06	03
$e_{2,1}^{\left[k\right]}$	06	04	04	05	06	03
$e_{2,2}^{\left[k\right]}$	06	04	04	05	06	03

and Figure 6. These tables clearly illustrate that ${\mathrm{DS}}^{{ϵ}_3}$ outperforms ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ in terms of global convergence, again converging faster and requiring less CPU time and iterations than other methods.

The newly developed schemes ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_3}$ achieved a substantially higher accuracy than the existing methods, with errors of 4.876 × 10⁻¹⁸, 4.875 × 10⁻¹⁸, and 4.34 × 10⁻¹⁸ for the three sets of initial vectors v1–v3. The numerical results of the iterative method using random initial vectors reported in Table 18, Table 19 and Table 20 also indicate that ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_3}$ required less CPU time and fewer iterations

Table 21. Consistency of the numerical scheme for solving (88) using vectors v1–v3 from Appendix Table A4.

Methods	Max-${\mathit{e}}_{\mathit{i}}^{\left[\mathit{k}\right]}$	Avg-CPU	Avg-It	Avg-COC
MMϵ	7.676 × 10−15	0.2343	06	7.535
EMϵ	7.676 × 10−21	0.2343	04	6.535
BMϵ	8.675 × 10−33	0.1234	05	5.007
DSϵ1	9.654 × 10−42	1.1234	05	4.431
DSϵ2	3.004 × 10−36	2.7654	04	4.555
DSϵ3	0.095 × 10−46	2.1454	03	3.765

presents the overall behavior of the simultaneous schemes and demonstrates that ${\mathrm{DS}}^{{ϵ}_3}$ is more stable and consistent than ${\mathrm{DS}}^{{ϵ}_1}$ − ${\mathrm{DS}}^{{ϵ}_2}$, MM^ϵ, EM^ϵ, and BM^ϵ.

Example 5: Solving the diffusion equation [79]

Consider the heat diffusion equation with various boundary conditions as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \varphi }{\partial t}}=\alpha {\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}},0\le \gamma \le L;t\ge 0,\\ \varphi \left(\gamma ,0\right)={\varphi }_0\left(\gamma \right);\\ \varphi \left(0,t\right)={\sigma }_1,\varphi \left(L,t\right)={\sigma }_2,{\sigma }_1,{\sigma }_2\in \mathbb{R}.\end{array}\right.$$

(89)

To find the solution of (89), we set the diffusivity parameter as $\alpha ={\displaystyle \frac{1}{4}}$ and ensure the stability of the implicit finite difference scheme with $\vartheta =\alpha {\displaystyle \frac{\Delta t}{{\left(\Delta \gamma \right)}^2}}<1$:

$${\displaystyle \frac{\partial \varphi }{\partial t}}={\displaystyle \frac{{\varphi }_{i,j+1-}{\varphi }_{i,j}}{\Delta t}}+\Delta \left(t\right);{\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}}={\displaystyle \frac{{\varphi }_{i+1,j+1}-2{\varphi }_{i,j+1}+{\varphi }_{i-1,j+1}}{{\left(\Delta \gamma \right)}^2}}+\Delta \left(\gamma \right).$$

(90)

Applying approximations to (89), we derive the tridiagonal system of equations:

$${\varphi }_{i,j}=-\vartheta {\varphi }_{i-1,j+1}+\left(1+2\vartheta \right){\varphi }_{i,j+1}-\vartheta {\varphi }_{i=1,j+1}.$$

(91)

For different initial and boundary conditions, we obtain an additional set of partial differential equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \varphi }{\partial t}}=\alpha {\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}},0\le \gamma \le L;t\ge 0,\\ \varphi \left(\gamma ,0\right)=sin1\ast \left({\gamma }^3+0.3\right)tan\left(\gamma \right);\\ \varphi \left(0,t\right)=0,\varphi \left(L,t\right)=0.\end{array}\right.$$

(92)

Using (90) in (92), we derive a nonlinear system of equations similar to (91) after incorporating the initial and boundary conditions. The exact and approximate solutions obtained by BDS^ϵ₁ − DS^ϵ₃, MM^ϵ, EM^ϵ, and BM are presented in Figure 7.

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \varphi }{\partial t}}=\alpha {\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}},0\le \gamma \le L;t\ge 0,\\ \varphi \left(\gamma ,0\right)=sin1;\\ \varphi \left(0,t\right)=0,\varphi \left(L,t\right)=0.\end{array}\right.$$

(93)

Using (90) in (93), we obtain another nonlinear system of equations similar to (91) after incorporating the respective initial and boundary conditions. The exact and approximate solutions obtained by DS^ϵ₁ − DS^ϵ₃, MM^ϵ, EM^ϵ, and BM are presented in Figure 8.

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \varphi }{\partial t}}=\alpha {\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}},0\le \gamma \le L;t\ge 0,\\ \varphi \left(\gamma ,0\right)={\displaystyle \frac{e^{{\gamma }^5}}{{\gamma }^2+1}}+tan\left(\gamma \right);\\ \varphi \left(0,t\right)=0.5e^t,\varphi \left(L,t\right)=cos\left(t\right)\ast (t+1).\end{array}\right.$$

(94)

Using a random set of initial vectors ${\gamma }_1^{\left[0\right]}=\left[{\gamma }_{1,1}^{\left[0\right]},{\gamma }_{1,2}^{\left[0\right]}\right]$ from Appendix A Table A5, the numerical results from

Table 22. Numerical results of iterative techniques for solving (92).

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Iteration	103	115	123	87	66	65
Max-Err	0.13 × 10−20	4.3 × 10−17	4.3 × 10−10	0.13 × 10−20	4.31 × 10−33	4.2 × 10−49
Max-time	2.1234	4.1552	6.1872	5.2312	3.1313	3.1321
COC	7.2312	3.134	1.634	5.132	6.1325	4.4313

,

Table 23. Numerical results of iterative techniques for solving (93).

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Iteration	85	90	97	58	63	51
Max-Err	0.152 × 10−22	4.3 × 10−9	4.3 × 10−8	0.13 × 10−24	4.31 × 10−43	4.2 × 10−29
Max-time	2.5424	6.1638	7.4332	5.9872	3.0013	3.1321
COC	7.2312	3.984	1.934	5.132	6.1325	4.4313

and

Table 24. Numerical results of iterative techniques for solving (94).

Methods	MMϵ	EMϵ	BMϵ	DSϵ1	DSϵ2	DSϵ3
Iteration	97	115	53	82	63	45
Max-Err	1.43 × 10−20	4.3 × 10−13	0.66 × 10−23	0.54 × 10−20	4.31 × 10−33	0.2 × 10−19
Max-time	2.11234	4.1222	5.1003	5.1232	3.1645	3.5432
COC	7.2532	3.934	2.0004	5.1332	6.1875	4.4313

and Figure 7, Figure 8 and Figure 9 clearly show that the scheme DS^ϵ₃ is more stable and consistent than DS^ϵ₁ − DS^ϵ₂, MM^ϵ, EM^ϵ, and BM when solving (89) with different boundary and initial conditions.

The scheme DS^ϵ₁ − DS^ϵ₃ outperformed the MM^ϵ, EM^ϵ, and BM^ϵ. It also required less CPU time and fewer iterations, as evidenced by the numerical results of the iterative technique under various initial conditions shown in Table 22, Table 23 and Table 24. These tables depict the overall behavior of the simultaneous schemes and demonstrate that DS^ϵ₃ is more stable and consistent than DS^ϵ₁ − DS^ϵ₂, MM^ϵ, EM^ϵ, and BM^ϵ.

5. Conclusions

In this study, novel parallel numerical schemes are developed for solving nonlinear equations and their systems, including vectorial problems. A convergence analysis reveals that these parallel vertorial schemes achieve a high order of convergence, up to 10. Engineering applications using various random initial approximations were employed to assess the efficiency of MM^ϵ, PM^ϵ, BD^ϵ, BM^ϵ, DF^ϵ, and DS^ϵ. Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23 and Table 24 and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 present the numerical outcomes from these experiments. The results clearly demonstrate that the newly developed schemes exhibit greater stability and consistency with global convergence compared to previous methods documented in the literature. Table 2, Table 7, Table 12, Table 18 and Table 22 illustrate that utilizing initial approximations close to exact solutions enhances the convergence rate of DS^ϵ₁ − DS^ϵ₃, MM^ϵ, EM^ϵ, and BM. Future research will focus on developing similar higher-order simultaneous iterative methods to tackle more intricate engineering challenges involving nonlinear equations and associated systems [80,81].