On geometric constructions of two points methods for simple roots of nonlinear equations

The object of the present work is to give some geometric constructions of methods of two points for simple roots. Initially, two similar forms to the fourth order Chun's class are presented. This modification is made with the purpose of incorporating two-point methods with third-order convergence, which use the same weight functions of the Gander's class. Using these classes, diverse families of methods of two points with a convergence of third and fourth order for solving nonlinear equations with simple roots are presented. Then, a geometric construction of Chebyshev Halley type methods of two-point and other four obtained for the convex combination of the Chebyshev and Halley methods are developed. Finally, two geometric constructions for the two-point class based on the weight functions used in Gander's class are given.