Solution of Nonlinear Equations Using the Modified Steffensen’s Type Method

The root-finding problem is one of the most important problems in Numerical Analyis. It arises in a wide variety of practical applications in physics, chemistry, biosciences, engineering, etc. As a matter of fact, determination of any unknown appearing implicitly led to the evolution of root-finding problem. Therefore, this paper focuses on the modification and analysis of a high order derivative-free iteration methods for finding roots of nonlinear algebraic equations of the form f (x) = 0. The methods require only one initial approximation. The proposed method is seen as an extension of the second-order Steffensen’s scheme, which is an Iterative method for approximating roots of non-linear equations which often breaks down when the derivative of the function value is zero or near zero at the point of iteration. This work therefore, seeks to introduce a method that overcomes such breakdown. The method herein is a combination of forward difference formula with Simpson’s quadrature in spirit of Steffensen. The idea is to modify the Steffensen’s method, which were recently developed to obtain derivative-free methods. The modified methods are shown to converge. We also describe how to obtain derivative-free methods to find solutions to multiple roots. Several numerical examples are provided to validate the theoretical order of convergence for nonlinear functions with simple roots and results obtained show the comparative advantage the proposed method has over well-known methods.