On developing a higher-order family of double-Newton methods with a bivariate weighting function

Keywords: Sixth-order convergence Extraneous fixed point Asymptotic error constant Efficiency index Double-Newton method Basin of attraction a b s t r a c t A high-order family of two-point methods costing two derivatives and two functions are developed by introducing a two-variable weighting function in the second step of the classical double-Newton method. 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 variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development. In addition, this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials. A large number of high-order multipoint methods for a given nonlinear equation f ðxÞ ¼ 0 have been developed since Traub [29] initiated the qualitative as well as the quantitative analyses of iterative methods in the 1960s. Petkovicét al. [25] recently collected and updated the state of the art of multipoint methods. Other works on multipoint methods can The principal aim of this paper is to design a family of high-order methods costing only two derivatives and two functions. Described below in (1.1) is the well-known two-point fourth-order double-Newton method [15,29], which is a two-step Newton's method utilizing two derivatives and two functions: