On the Newton-Raphson method and its modifications

The Newton-Raphson method, also known as Newton's method, is a method for finding successively better approximations to the roots of a real-valued function, starting with an initial guess, being useful even for generating fractals when we consider complex functions. It is a fast method, but convergence is not guaranteed, which is the reason why several modifications of that method have been proposed. Here we present some modifications of the Newton-Raphson method, and we study the convergence of those methods through cases.