## 1 Introduction  
The design of iterative algorithms for solving a nonlinear system of equations is a most needed and challenging task in the domain of numerical analysis. Nonlinearity is a phenomenon that occurs in physical and natural events. Phenomena which have an inherent quality of nonlinearity occur frequently in fluid and plasma mechanics, gas dynamics, combustion, ecology, biomechanics, elasticity, relativity, chemical reactions, economics modeling problems, transportation theory, etc. Due to this frequency, many works related to the current mathematical research provide importance for the development and analysis of methods for nonlinear systems. Hence, finding a solution \(\alpha\) of the nonlinear system \(\Phi(x)=0\) is a classical and difficult problem that unlocks the behavior pattern of many application problems in science and engineering, wherein \(\Phi:D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a sufficiently Frechet differentiable function in an open convex set \(D\). In the last decade, many iterative techniques have been proposed for solving a system of nonlinear equations. See, for instance, [1; 2; 3; 4; 5; 6; 7] and the references therein. The popular method for getting a solution \(\alpha\in D\) is the [PERSON]'s method (\(2^{nd}NM\))  
\[x^{(r+1)}=G_{2^{nd}NM}(x^{(r)})=x^{(r)}-[\Phi^{\prime}(x^{(r)})]^{-1}\Phi(x^{( r)}),\ r=0,1,2,... \tag{1}\]  
where \(G_{method}\) denotes any iterative scheme and \(\Phi^{\prime}(x^{(r)})^{-1}\) denotes the inverse of first Frechet derivative \(\[PERSON]^{\prime}(x^{(r)})\). This \(2^{nd}NM\) method needs evaluation of one function, same number of derivative and matrix inversion per iteration which produces second-order convergence.  
Another classical scheme for solving system of nonlinear equations is a two-step third-order variant of [PERSON]'s method (\(3^{rd}TM\)) proposed by [PERSON] [8] is given by  
\[x^{(r+1)}=G_{3^{rd}TM}(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-[\[PERSON]^{\prime}(x^{(r)})]^{- 1}\[PERSON](G_{2^{nd}NM}(x^{(r)})).\]  
The fourth-order [PERSON]'s method (\(4^{th}NR\)) formed using two steps is given by  
\[x^{(r+1)}=G_{4^{th}NR}(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-\[PERSON]^{\prime}[G_{2^{nd}NM }(x^{(r)})]^{-1}\[PERSON](G_{2^{nd}NM}(x^{(r)})), \tag{2}\]  
was proposed by [PERSON] et al. [5] based on the variational iteration technique. [PERSON] et al. [1] combined the [PERSON] and [PERSON] method to obtain a fifth-order method (\(5^{th}ACT\)) with three steps, given as below:  
\[x^{(r+1)}=G_{5^{th}ACT}(x^{(r)})=G_{3^{rd}TM}(x^{(r)})-[\Phi^{\prime}(G_{2^{nd} NM}(x^{(r)}))]^{-1}\Phi(G_{3^{rd}TM}(x^{(r)})). \tag{3}\]  
[PERSON] and [PERSON] [6] proposed an eighth-order three-step method (\(8^{th}SA\)) given by  
\[x^{(r+1)}=G_{8^{th}SA}(x^{(r)})=z(x^{(r)})-\Big{(}\frac{7}{2}I-G(x^{(r)})\Big{(} 4I-\frac{3}{2}G(x^{(r)})\Big{)}\Big{)}[\Phi^{\prime}(x^{(r)})]^{-1}\[PERSON](z^{(r) }),\]  
\[ \begin{split}&\text{where }\ z(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-\Big{(} \frac{13}{4}I-G(x^{(r)})\Big{(}\frac{7}{2}I-\frac{5}{4}G(x^{(r)})\Big{)}\Big{)} [\[PERSON]^{\prime}(x^{(r)})]^{-1}\[PERSON](G_{2^{nd}NM}(x^{(r)}),\\ &\text{ }G(x^{(r)})=[\[PERSON]^{\prime}(x^{(r)})]^{-1}\[PERSON]^{\prime}(G_{2^{ nd}NM}(x^{(r)})).\end{split}  \tag{4}\]  
Recently, [PERSON] et al. [9] (\(5^{th}MBJ\)) proposed a fifth-order two-step method given by  
\[ \begin{split}&\text{ }x^{(r+1)}=G_{5^{th}MBJ}(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-H_{1}(x^{(r)})[\[PERSON]^{ \prime}(x^{(r)})]^{-1}\[PERSON](G_{2^{nd}NM}(x^{(r)})),\\ & H_{1}(x^{(r)})=2I-\tau(x^{(r)})+\frac{5}{4}(\tau(x^{(r)})-I)^{2},\quad\tau(x^{(r)})=[\[PERSON]^{\prime}(x^{(r)})]^{-1}\[PERSON]^{\prime}\left(G_{2^{nd} NM}(x^{(r)})\right).\end{split}  \tag{5}\]  
Furthermore, this method was extended to produce \(3k+5\) (\(k\geq 1\)) order of convergence known as \((3k+5)^{th}MBJ\) method using an additional function evaluation and it is given below  
\[