Improved Secant Method (Marouane’s Method) for Finding Roots of Equation

Improved Secant Method (Marouane’s Method) is a new method represents so much advantages in root finding algorithm for non-linear equations, compared to the secant method , this uses secant lines from 2 circles in each iteration, it then requires only one initial guess and its convergence is quadratic, this new method could replace the secant method in Brent’s and Zhang’s method to make the algorithm more quick and more efficient,some experimental tests presented in this study compares the performance of this new method to the secant method.

Given a function f , let x be such that f(x)=0 and let and xn be approximations to x. Assume r is a simple root (f(r)=0, f'(r) not equal zero) and f is twice continuously differentiable, Marouane’s method is:

This method taken from the following paper (http://vixra.org/pdf/1405.0013v1.pdf). The code solve the given equation and show the answer with the number of iterations and accuracy (residual), and chart the root-value-convergence with the # of iterations.