# Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
## 1 Introduction  
Finding the solutions of system of nonlinear equations \(F(x)=0\) is a hot problem with wide applications in sciences and engineering, wherein \(F:D\subset R^{m}\to R^{m}\) and \(D\) is an open convex domain in \(R^{m}\). Many efficient methods have been proposed for solving system of nonlinear equations, see for example [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] and the references therein. The best known method is the Steffensen method [1; 2], which is given by  
\[y^{(k)}=\psi_{1}(x^{(k)},w^{(k)})=x^{(k)}-[w^{(k)},x^{(k)};F]^{-1}F(x^{(k)}) \tag{1}\]  
where \(w^{(k)}=x^{(k)}+F(x^{(k)}),\quad[w^{(k)},x^{(k)};F]^{-1}\) is the inverse of \([w^{(k)},x^{(k)};F]\) and \([w^{(k)},x^{(k)};F]:D\subset R^{m}\to R^{m}\) is the first order divided difference on \(D\). Equation (1) does not require the derivative of the system \(F\) in per iteration. To reduce the computational time and improve the efficiency index of the Steffensen method, many modified high-order methods have been proposed in open literatures, see [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] and the references therein. [PERSON] _et al._[3] obtained a fourth-order derivative-free method for solving system of nonlinear equations, which can be written as  
\[\left\{ \begin{aligned} y^{(k)}=&\psi_{1}(x^{(k)},w^{( k)})\\ x^{(k+1)}=&\psi_{2}(x^{(k)},w^{(k)},y^{(k)})\\ =& y^{(k)}-[y^{(k)},x^{(k)};F]^{-1}([y^{(k)},x^{(k)}; F]-[y^{(k)},w^{(k)};F]\\ +&[w^{(k)},x^{(k)};F])[y^{(k)},x^{(k)};F]^{-1}F(y^{(k) })\end{aligned} \right. \tag{2}\]where \(w^{(k)}=x^{(k)}+F(x^{(k)})\). [PERSON] _et al._[4; 5] developed some efficient derivative-free methods. One of the methods is the following sixth-order method  
\[\left\{ \begin{aligned} y^{(k)}=& x^{(k)}-[w^{(k)},s^{(k)}; F]^{-1}F(x^{(k)})\\ z^{(k)}=& y^{(k)}-\left\{2[x^{(k)},y^{(k)};F]-[w^{(k)},s^{(k)};F]\right\}^{-1}F(y^{(k)})\\ x^{(k+1)}=& y_{3}(w^{(k)},s^{(k)},x^{(k)},y^{(k)},z^{(k)})\\ =& z^{(k)}-\left\{2[x^{(k)},y^{(k)};F]-[w^{(k)},s^{(k) };F]\right\}^{-1}F(z^{(k)})\end{aligned} \right. \tag{3}\]  
where \(w^{(k)}=x^{(k)}+F(x^{(k)})\) and \(s^{(k)}=x^{(k)}-F(x^{(k)})\). It should be noted that the Equations (2) and (3) need to compute two LU decompositions in per iteration, respectively. Some derivative-free methods are also discussed by [PERSON] _et al._ in [6] and by [PERSON] _et al._ in [7; 8]. The above multi-step derivative-free iterative methods can save the computing time in the High-precision computing. Therefore, it is meaningful to study the multi-step derivative-free iterative methods. It is well-known that we can improve the efficiency index of the iterative method and reduce the computational time of the iterative process by reducing the computational cost of the iterative method. There are many ways to reduce the computational cost of the iterative method. In this paper, we reduce the computational cost of the iterative method by reducing the number of LU (lower upper) decompositions in per iteration. Two new derivative-free iterative methods are proposed for solving system of  
**Theorem 1**. : _Let \(\alpha\in R^{m}\) be a solution of the system \(F(x)=0\) and \(F:D\subset R^{m}\to R^{m}\) be sufficiently differentiable in an open neighborhood \(D\) of \(\alpha\). Then, for an initial approximation sufficiently close to \(\alpha\), the convergence order of iterative Equation (4) is four with the following error equation_  
\[\varepsilon=(4A_{2}^{2}-A_{3}-A_{3}F^{\prime}(\alpha)^{2})E\varepsilon^{2}+A_{2 }E^{2}+O(\varepsilon^{5}) \tag{7}\]  
_where \(e=x^{(k)}-\alpha\) and \(E=y^{(k)}-\alpha\). Iterative Equation (5) is of sixth order convergence and satisfies the following error equation_  
\[e_{n+1}=2A_{2}E\varepsilon-(A_{3}+A_{3}F^{\prime}(\alpha)^{2}-4A_{2}^{2})e^{2} \varepsilon+O(\varepsilon^{7}) \tag{8}\]  
_where \(e_{n+1}=x^{(k+1)}-\alpha\)_  
Proof.