A third order convergent method for solving nonlinear equations

Abstract

Derivation of the Newton-Raphson method involves an indefinite integral of the derivative of the function, and the relevant area is approximated by a rectangle. In this study, the area under the curve which is appearing in the derivation of Newton-Raphson method is approximated by two points Gaussian quadrature formula. With the help of that an improvement to the Newton-Raphson method is presented for root finding of one variable nonlinear equation. This iterative method converges to the root faster than the Newton-Raphson method and the claim is proved by showing the new method is third order convergent. The Established theory is supported by computed results by applying the new method to a wide range of functions and comparing it with the Newton's method and some third order iterative methods.