# An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems
## 1 Introduction  
We consider the problem of finding a zero of a nonlinear function \(F:D\subset R^{m}\to R^{m}\), that is, a solution \(\alpha\) of the nonlinear system \(F(x)=0\) with \(m\) equations and \(m\) unknowns. [PERSON]'s method [1, 2] is the well-known method for solving nonlinear systems, which can be written as:  
\[x^{(k+1)}=x^{(k)}-F^{\prime}{(x^{(k)})}^{-1}F(x^{(k)}), \tag{1}\]  
where \(F^{\prime}(x^{(k)})\) is the Jacobian matrix of the function \(F\) evaluated in the \(k\)th iteration and \(F^{\prime}(x^{(k)})^{-1}\) is the inverse of \(F^{\prime}(x^{(k)})\). [PERSON]'s method is denoted by NM. [PERSON]'s method converges quadratically if \(F^{\prime}(\alpha)\) is nonsingular and \(F^{\prime}(x)\) is Lipschitz continuous on \(D\). The method (1) can be written as:  
\[\left\{ \begin{array}{c}F^{\prime}(x^{(k)})\gamma^{(k)}=F(x^{(k)}),\\ x^{(k+1)}=x^{(k)}-\gamma^{(k)},\end{array} \right. \tag{2}\]  
which requires \((m^{3}-m)/3\) multiplications and divisions in the LU decomposition (lower upper, also called LU factorization) and \(m^{2}\) multiplications and divisions for solving two triangular linear systems. So, the computational cost (multiplications and divisions) of the method (1) is \((m^{3}-m)/3+m^{2}\). In order to accelerate the convergence or to reduce the computational cost and function evaluation in each step of the iterative process, many efficient methods have been proposed for solving nonlinearsystems, see [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21] and the references therein. [PERSON] et al. [3; 4] developed some variants of [PERSON]'s method. One of the methods is the following fourth-order method [3]:  
\[\left\{ \begin{array}{l}y^{(k)}=x^{(k)}-F^{\prime}(x^{(k)})^{-1}F(x^{(k)}),\\ x^{(k+1)}=y^{(k)}-\big{[}2I-F^{\prime}(x^{(k)})^{-1}F^{\prime}(y^{(k)})\big{]} F^{\prime}(x^{(k)})^{-1}F(y^{(k)}),\end{array} \right. \tag{3}\]  
where \(I\) is the identity matrix. Method (3) is denoted by CM4 and requires LU decomposition of the Jacobian matrix only once per full iteration. Based on method (3), [PERSON] et al. [4] presented the following sixth-order method:  
\[\left\{ \begin{array}{l}y^{(k)}=x^{(k)}-F^{\prime}(x^{(k)})^{-1}F(x^{(k)}), \\ z^{(k)}=y^{(k)}-\big{[}2I-F^{\prime}(x^{(k)})^{-1}F^{\prime}(y^{(k)})\big{]} F^{\prime}(x^{(k)})^{-1}F(y^{(k)}),\\ x^{(k+1)}=z^{(k)}-F^{\prime}\big{(}y^{(k)})^{-1}F(z^{(k)}),\end{array} \right. \tag{4}\]  
where \(I\) is the identity matrix. Method (4) is denoted by CHM and requires two LU decompositions of the Jacobian matrix, one for \(F^{\prime}(x^{(k)})\) and one for \(F^{\prime}(y^{(k)})\). [PERSON] et al. [5] obtained the following sixth-order method:  
\[\left\{ \begin{array}{l}y^{(k)}=x^{(k)}-F[x^{(k)}+F(x^{(k)}),x-F(x^{(k)})]^{- 1}F(x^{(k)}),\\ z^{(k)}=y^{(k)}-\Big{\{}2F[x^{(k)},y^{(k)}]-F[x^{(k)}+F(x^{(k)}),x-F(x^{(k)})] \Big{\}}^{-1}F(y^{(k)}),\\ x^{(k+1)}=z^{(k)}-\Big{\{}2F[x^{(k)},y^{(k)}]-F[x^{(k)}+F(x^{(k)}),x-F(x^{(k)}) ]\Big{\}}^{-1}F(z^{(k)}),\end{array} \right. \tag{5}\]  
where \(F[\cdot,\cdot]\) denotes the first-order divided difference of \(D\). Method (5) is denoted by SNAM and requires two LU decompositions for solving the linear systems involved. It is well known that the computational cost of the iterative method greatly influences the efficiency of the iterative method. The number of LU decompositions that are used in the iterative method thus plays an important role when it comes to measuring the computational cost. So, we can reduce the computational cost of the iterative method by reducing the number of LU decompositions in each iteration. The purpose of this paper is to construct a new sixth-order iterative method for solving small- and medium-sized systems. The theoretical advantages of the new method are based on the assumption that the Jacobian matrix is dense and that LU factorization is used to solve systems with the Jacobian. This assumption is not correct for sparse Jacobian matrices.