An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

We extend in this paper an optimal family of three-step eighth-order methods developed by Džunić et al. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials.