On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

With an error corrector via principal branch of the mth root of a function-to-function ratio, we propose optimal quartic-order multiple-root finders for nonlinear equations. The relevant optimal order satisfies Kung-Traub conjecture made in 1974. Numerical experiments performed for various test equations demonstrate convergence behavior agreeing with theory and the basins of attractions for several examples are presented