Convergence analysis of optimal iterative family for multiple roots and its applications

In this paper, we use weight function approach to construct a new King-like family of methods to solve nonlinear equations with multiple roots. Here the weight functions are chosen appropriately to reach the maximum convergence order eight and the family is optimal in the sense of Kung–Traub conjecture. Moreover, local convergence of a fourth order modified King’s family for multiple roots is also studied. Radii of convergence balls of fourth order schemes are computed and compared with an existing method. Numerical examples have been presented based on applications of some real life problems and the results obtained show the superiority of our eighth order schemes over the existing ones. To study the dynamical behaviour of the proposed schemes, basins of attraction have also been presented which verifies that proposed eighth order schemes have more convergent points and requires less number of iterations in comparison to the existing methods.