A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations

Abstract

We present two new families of iterative methods for obtaining simple roots of nonlinear equations. The first family is developed by fitting the model $m(x)=e^{px}(A{x}^2+Bx+C)$ to the function $f(x)$ and its derivative $f^{\prime }(x)$, $f^{″}(x)$ at a point $x_n$. In order to remove the second derivative of the first methods, we construct the second family of iterative methods by approximating the equation $f(x)=0$ around the point $(x_n,f(x_n))$ by the quadratic equation. Analysis of convergence shows that the new methods have third-order or higher convergence. Numerical experiments show that new iterative methods are effective and comparable to those of the well-known existing methods.