Unification of Well-known Numeric Methods for Solving Nonlinear Equations

Abstract

This article is a fairly comprehensive document on the numerical solution of nonlinear equations. The aim of this paper is to unify all numerical methods for solving nonlinear equations and complete the Najafi-Nikkhah method [1,2] and generalize the famous methods for solving systems of nonlinear equations. So, the available methods in this field, are being investigated and it will be indicated that how these techniques, despite the apparent dispersion, all are obtained from a unified idea, and this unified pattern would help find new techniques in a systematic way. All current methods require that the initial starting point or points to be close to the solution appropriately, but for the equations with complicated appearance finding the initial guess would not be easy. So, this article intends to provide an appropriate response for this fundamental issue for the first time. An algorithm is proposed to complete the Najafi-Nikkhah technique [1,2], and declares the procedure to make the initial guess become closer to the solution, even if they are far away from each other. Then, the procedure could be completed by one of the common methods available in this field. Dispersion of the current methods causes the confusion in the case of using them. Therefore, these methods should be compared with some criteria to determine the use of them in the practical applications. In the next section of this article, these criteria together with some comparisons including a series of tables and diagram would be provided. Finally, in the last section, generalization of the current methods for solving the nonlinear equation with a singular unknown to the methods for solving the systems of nonlinear equations would be investigated. Then, the generalization of the modified Najafi-Nikkhah method for the systems of nonlinear equations will be presented which can reduce the dependency of the initial guess to the solution, significantly.