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.
Highlights
Iterative methods for solving nonlinear equations and systems of nonlinear equations always represent an attractive research topic
It is worth noting that they used auxiliary parameters to change the path of converging sequences of successive approximations to obtain fast convergence. Inspired by their mathematical model for homotopy, we introduce a generalization of the two-parameter homotopy for solving a system of weakly nonlinear equations
We extend the model of iterative method Equation (22), by including more free-parameters, so that we can enhance the convergence order
Summary
Iterative methods for solving nonlinear equations and systems of nonlinear equations always represent an attractive research topic. In 2010, Yongyan et al [22] proposed a two-parameter homotopy method for the construction of iterative methods, for solving scalar nonlinear equations. The authors in [22] use the homotopy auxiliary parameters to change the dynamics of the proposed iterative methods for solving nonlinear equations and provide the optimal values of auxiliary parameters, for enhancing the convergence speed without altering the convergence order of the iterative method. It is worth noting that they used auxiliary parameters to change the path of converging sequences of successive approximations to obtain fast convergence Inspired by their mathematical model for homotopy, we introduce a generalization of the two-parameter homotopy for solving a system of weakly nonlinear equations. We use the homotopy techniques to develop mathematical models that we use for designing higher order iterative methods with the help of new parameters
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