Toward a general theory of point to point iterative processes free of derivatives with quadratic convergence

In this work, we are concerned with the problem of developing a general theory about derivative-free iterative procedures with quadratic convergence. Newton’s method is the most used, well-known and studied in order to approximate the solution of a nonlinear problem. However, Newton’s method has the problem that the operator, whose root we intend to approximate, must be differentiable. Then, through the use of divided differences, we construct iterative processes that, while maintaining the efficiency of Newton’s method, allow us to approach solutions of non-linear problems raised from non-differentiable operators. Thus, in this work, we construct derivative-free iterative processes.