Abstract
In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.
Highlights
Solving nonlinear equations is important in many applied mathematics and theoretical physics problems
Motivated by the recent results in [11], in this paper we introduce a generating function method for the construction of new two and three-point iterations with p-th order of convergence
In our previous paper [11] we have considered two and three-point iterative methods of solving nonlinear equation f (x) = 0 yn τn f, f τn f f, αn f . f
Summary
Solving nonlinear equations is important in many applied mathematics and theoretical physics problems. Motivated by the recent results in [11], in this paper we introduce a generating function method for the construction of new two and three-point iterations with p-th order of convergence. In our previous paper [11] we have considered two and three-point iterative methods of solving nonlinear equation f (x) = 0 yn. The iterative method (1) has fourth-order of convergence if and only if the parameter τn is given by τn = 1 + θn + 2θn2 + O(θn3), θn =. The three-point iterative methods (2) has an eighth-order of convergence if and only if the parameters τn and αn are given by τn = 1 + 2θn + βθn2 + γθn3 + · · · , τn τn − 1 θn. An extended version of the present paper will be published elsewhere
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