Search

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. [ABSTRACT FROM AUTHOR]

A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending King’s 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. [ABSTRACT FROM AUTHOR]

Construction of two-point sixth-order methods for simple root is an ambitious and challenging task in numerical analysis. Therefore, the main aim of this paper is to introduce a new highly efficient two-point sixth-order class of Chebyshev–Halley type methods free from second-order derivative for the first time. Each member of the proposed class requires only four functional evaluations (viz. two evaluations of function f and two of first-order derivative f  ′) per full iteration. A variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two-point/three-point sixth-order methods available in the literature. From their dynamical study, it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: A penta-parametric family of four-step multipoint iterative methods of order fifteen for nonlinear algebraic equations are developed and their convergence properties are established. The efficiency indices are all found to be 151/5 ≈1.71877, better than 141/5 ≈1.69522 of a family of fourteenth-order methods suggested by Neta . Numerical examples are demonstrated to verify the developed theory. [Copyright &y& Elsevier]

Abstract: A multi-parameter family of three-step eighth-order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants. [Copyright &y& Elsevier]

Abstract: In this paper, a new two-step biparametric family of sixth-order iterative methods free from second derivatives are proposed in this paper to find a simple root of a nonlinear algebraic equation. Their efficiency indices are all found to be . Convergence analysis together with numerical experiments confirms sixth-order convergence and asymptotic error constants. [Copyright &y& Elsevier]

Abstract: By extending the classical Newton’s irrational method defined by an iterationwe present a high-order k-fold pseudo-Newton’s irrational method for locating a simple zero of a nonlinear equation. Its order of convergence is proven to be at least k +3 and the convergence behavior of the asymptotic error constant is investigated near the corresponding simple zero. A root-finding algorithm is described as well as the introduction on the convergence of the fixed-point iterative method. Various numerical examples have successfully demonstrated a good agreement with the theory presented here. [Copyright &y& Elsevier]