The Newton method is one of the common methods for solving nonlinear equations
f : Rn →Rn,
f(x) = 0.
This method, even the appropriate initial guess, may fail or diverge. This problem becomes particularly
pronounced when the function f has irregular behavior or weak smoothness in the neighborhood of the
equation’s solution, or when the initial guess is chosen near the root of the derivative of f.
In this paper, to remedy these issues, we reduce the main equation to an equivalent equation by using
an auxiliary matrix function. Its auxiliary function feature is that it is differentiable and nonsingular in the
neighborhood of the solution of f(x) = 0, and it is chosen in such a way that the reduced function has
better smoothness compared to the original funtion. Then, we apply the Newton method for solving the
reduced equation to obtain the approximate solution of the original equation.
Now, there is this question. Do the the comutational cost and compexity of the problem increase by
applying the Newton method to the equivalent equation? In response to this question, the importance
of choosing the proper auxiliary function increases, and it should be selected in sucha way that reduces
computationalcosts. In this paper, by choosing a suitable auxiliary function and applying the proposed
method to solve the system of nonlinear equations, we investigate the efficiency of the new method.