Evaluation of iterative methods for solving nonlinear scaler equations

This study is aimed at performing a comprehensive numerical evaluation of the iterative solution techniques without memory for solving nonlinear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes will be evaluated, in which the ability of the methods to solve different types of nonlinear equations will be studied. First, ۲۶ different iterative methods with the best performance are reviewed. These methods are selected based on performing more than ۴۶۰۰۰ analyses on ۱۶۶ different available nonlinear solvers. For the easier application of the techniques, consistent mathematical notation is employed to present reviewed approaches. After presenting the diverse methodologies suggested for solving nonlinear equations, the performances of the reviewed methods are evaluated by solving ۲۸ different nonlinear equations. In order to calculate novel computational efficiency indices and rank them accurately, the results of the obtained solutions are used. These data include the number of iterations, number of function evaluations, and convergence times. In addition, the successful runs for each process are used to rank the evaluated schemes. Although in general the choice of the method depends on the problem in practice, but t in practical applications, especially in engineering, changing the solution method for different problems is not feasible all the time, and accordingly, the findings of the present study can be used a s a guide to specify the fastest and most appropriate solution technique for solving nonlinear problems.

Keywords:

Nonlinear scalar equations , Iterative method , Efficiency index , Order of convergence , Initial guess