###### Abstract  
A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations. Keywords:systems of nonlinear equations; frozen Jacobian; ordinary differential equations; partial differential equations; homotopy method +
Footnote †: journal:  
0000-0002-4800-3899-2000]M. O.  
method, but not for systems of nonlinear equations. The reason behind this fact is that usually, the constructed iterative methods, for solving nonlinear equations using different homotopy methods, contain higher order derivatives. From a computational point of view, the evaluation of higher order Frechet derivatives is expensive for a general system of nonlinear equations. However, when considering systems of weakly nonlinear equations, associated with wide classes of ordinary and partial differential equations, we find that all higher order Frechet derivatives possess a diagonal structure: as a consequence, the resulting computational cost is mild. The core idea of homotopy (or homotopy continuation) was developed for solving hard problems numerically. The nonlinear problem is approximated around some initial guess to construct a linear model or a model that can be solved easily. Once we have a linear model, we construct a homotopy function that maps linear model to a nonlinear model with the help of auxiliary parameters. The embedding of auxiliary parameters in the homotopy function helps us to solve stiff problems iteratively in a way that the solution of the less hard problem is used as an initial guess for the next harder problem. The gradual change in the parameters assists us in solving a hard problem softly in the mathematical sense. The above-described scenario is one of the pictures of the homotopy methods or, precisely speaking, homotopy continuation methods. The other picture is to use the homotopy method to approximate solution of nonlinear problems analytically [14; 15; 16; 17; 18; 19; 20]. Many authors have used [7; 21] homotopy methods for constructing iterative methods for solving nonlinear equations. For the construction of iterative methods, they build a homotopy between a linear model of the nonlinear equation and the nonlinear equation, by expanding it around some initial guess. The constructed homotopy contains auxiliary parameters. The auxiliary parameters can be used for different purposes. Some authors use the homotopy parameters to enhance the convergence by changing the dynamics of the iterative method and give estimates for the optimal parameters [22]. Now, we present in detail our proposals. [PERSON] [21] proposed the following homotopy iterative method for solving nonlinear equations. Let \(f(x)=0\) be a scalar nonlinear equation, and let \(x_{0}\) be an initial guess. We perform a linearization of \(f(x)\) around a point \(\gamma\):  
\[ \begin{split} f(x)&=f(\gamma)+f^{\prime}(\gamma)(x- \gamma)+g(x)=0\\ g(x)&=f(x)-f(\gamma)-f^{\prime}(\gamma)(x-\gamma) \end{split}  \tag{1}\]  
where \(f(\gamma)+f^{\prime}(\gamma)(x-\gamma)\) is the linear approximation of \(f(x)\) around \(\gamma\) and \(g(x)\) is the nonlinear part of \(f(x)\). Equation (1) can be written as:  
\[x=c+N(x) \tag{2}\]  
where \(c=\gamma-\frac{f(\gamma)}{f(\gamma)}\) and \(N(x)=-\frac{g(x)}{f(\gamma)}\).