Young Hee Geum

A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots

A uniparametric family of three-step eighth-order multipoint iterative methods requiring only a first derivative are proposed in this paper to find simple roots of nonlinear equations. Development and convergence analysis on the proposed methods is described along with numerical experiments including comparison with existing methods.

A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function

A biparametric family of four-step multipoint iterative methods of order sixteen to numerically solve nonlinear equations are developed and their convergence properties are investigated. The efficiency indices of these methods are all found to be 16^1^/^5~1.741101, being optimally consistent with the conjecture of Kung-Traub. Numerical examples as well as comparison with existing methods developed by Kung-Traub and Neta are demonstrated to confirm the developed theory in this paper.

A biparametric extension of King's fourth-order methods and their dynamics

A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending Kings fourth-order family of methods. With the introduction of an error corrector having a weight function dependent on a function-to-function ratio, higher-order convergence is obtained. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes.

A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

A class of three-point sixth-order multiple-root finders and the dynamics behind their extraneous fixed points are investigated by extending modified Newton-like methods with the introduction of the multivariate weight functions in the intermediate steps. The multivariate weight functions dependent on function-to-function ratios play a key role in constructing higher-order iterative methods. Extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction. Numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper. Relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials.

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

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 family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function

A new family of four-step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties. Numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory.

A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function

This paper proposes a biparametric family of three-step eighth-order multipoint iterative methods with optimal efficiency index in the sense of Kung-Traub for simple roots of nonlinear equations. We employ their third-step weighting function decomposed into Kings linear fractional function and a two-variable function to construct such a family of optimal methods. Development and convergence analysis on the proposed methods is fully described in addition to numerical experiments including comparison with existing methods.

A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. With the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction.

A cubic-order variant of Newton’s method for finding multiple roots of nonlinear equations

Abstract A second-derivative-free iteration method is proposed below for finding a root of a nonlinear equation f ( x ) = 0 with integer multiplicity m ≥ 1 : x n + 1 = x n − f ( x n − μ f ( x n ) / f ′ ( x n ) ) + γ f ( x n ) f ′ ( x n ) , n = 0 , 1 , 2 , … . We obtain the cubic order of convergence and the corresponding asymptotic error constant in terms of multiplicity m , and parameters μ and γ . Various numerical examples are presented to confirm the validity of the proposed scheme.