A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

On a computer-aided approach to the computation of Abelian integrals

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four

On optimal parameter of Laguerre’s family of zero-finding methods

The article of record as published may be found at http://doi.org/10.1080/00207160.2018.1429598A one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre’s family, which is the main goal of this paper. Although methods from Laguerre’s family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski’s method possesses the best convergence behaviour for most polynomial equations.Serbian Ministry of Education and ScienceGrant 17402

Tchebychef-like method for the simultaneous finding zeros of analytic functions

AbstractUsing a suitable approximation in classical Tchebychef's iterative method of the third order, a new method for approximating, simultaneously, all zeros of a class of analytic functions in a given simple smooth closed contour is constructed. It is proved that its order of convergence is three. The analysis of numerical stability and some computational aspects, including a numerical example, are given. Also, the asynchronous implementation of the proposed method on a distributed memory multicomputer is considered from a theoretical point of view. Assuming that the maximum delay r is bounded, a convergence analysis shows that the order of convergence of this version is the unique positive root of the equation xr+1 − 2xr − 1 = 0, belonging to the interval (2, 3)

A note on some improvements of the simultaneous methods for determination of polynomial zeros

AbstractApplying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation

A class of simultaneous methods for the zeros of analytic functions

AbstractA family of iterative methods for simultaneously approximating simple zeros of analytic functions (inside a simple smooth closed contour in the complex plane) is presented. The order of convergence of the considered methods in m + 2 (m = 1, 2, …), where m is the order of the highest derivative of analytic function appearing in the iterative formula. A special attention is paid to the total-step and single-step methods with Newton's and Halley's corrections because of their high computational efficiency. Numerical examples are also included