Dynamics of four families of methods with the same weight function to solve nonlinear equations

We study the dynamics of four families of methods obtained with a weight function from a convex combination of Newton’s method and a Newton-Halley type method on polynomials with two roots. We find the analytical expressions for the fixed and critical points. We study the stable and unstable behavior of the strange fixed points. Also, parameters spaces for identify methods with good behavior are presented. Then, several dynamic planes are presented to confirm the results obtained. Finally, some periodic orbits with period two for a selected method are presented.

💡 Research Summary

The paper investigates the complex dynamics of four families of iterative root‑finding methods that are generated by a single weight function H(t) = A + 2(A − 1)² /