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In this paper, we proposed a new highly efficient two-point six-order family of
Halley type methods that do not require any second-order derivative evaluation for
obtaining simple roots of nonlinear equations, numerically. In terms of computational
cost, each member of the families requires two function and two first-order
derivative evaluations per iteration. On the account of the results obtained, it is
found that our proposed methods are efficient and show better performance than
existing six-order methods available in the literature. Further, It is also noted that
larger basins of attraction belong to our methods although the others methods are
slow and has darker basins while some of the method are too sensitive upon the
choice of the initial value.

Nonlinear equations. Simple roots. Halley’s method. Basins of attraction. Order of convergence.