A family of simultaneous methods for the determination of polynomial complex zeros

Using the development of a rational function by elementary fractions, a family of methods for the simultaneous determination of polynomial complex zeros is derived. All the methods of the family are cubically convergent for simple zeros. The known simultaneous procedures of the third order are included. The presented class of iteration functions is suitable for the parallel inclusion of polynomial complex zeros by circular regions. The family of methods, defined in complex circular arithmetic, gives a new interval method with cubic convergence. Numerical example is given.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

Fourth-order two-step iterative methods for determining multiple zeros of non-linear equations

In this paper, we describe and analyse two two-step iterative methods for finding multiple zeros of non-linear equations. We prove that the methods have fourth-order convergence. The methods calculate the multiple zeros with high accuracy. These are the first two-step multiple zero finding methods. The numerical tests show their better performance in the case of algebraic as well as non-algebraic equations     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37–55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.     

Sorting-based localization and stable computation of zeros of a polynomial. I

A method of stable address sorting is designed for localization and approximate computation of real and complex zeros of a polynomial. In an arbitrary domain, a software implementation of the method allows one to localize and to calculate with high accuracy all zeros of a polynomial, including the case when they are ill-separated. The vicinity of each zero dynamically decreases during localizing the boundaries of the domains of all zeros. Algorithms developed are formally described in the Object Pascal language and are implemented in the Delphi 7 environment. Some upper-bound estimates of a parallel version of the method are given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 165–182, January–February 2007.     

A family of simultaneous zero finding methods

A class of iterative methods with arbitrary high order of convergence for the simultaneous approximation of multiple complex zeros is considered in this paper. A special attention is paid to the fourth order method and its modifications because of their good computational efficiency. The order of convergence of the presented methods is determined. Numerical examples are given.     

An improvement on two iteration methods for simultaneous determination of the zeros of a polynomial

The Durand-Kerner and the Ehrlich? Methods of respective quadratic and cubic convergence are two of the current methods for determining simultaneously all zeros of a polynomial. By respectively including a Durand-Kerner and a Newton correction term in the above formulae, we establish two new methods-the Improved Durand-Kerner and the Improved Ehrlich.

We show that the improvement is reflected by an increase of unity in the order of convergence of each of the two methods.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

Convergence conditions of some methods for the simultaneous computation of polynomial zeros

The convergence of Newton's, Aberth's and Durand–Kerner's methods for polynomial root finding is analyzed; for each of them convergence theorems and error estimates are provided. Received: December 1995 / Revised version: May 1996     

On the local convergence of secant-type methods

In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed.     

On theR-order of some accelerated methods for the simultaneous finding of polynomial zeros

A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newton's and Halley's corrections. It is proved that the lower bounds of theR-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.     

Further discussion on the calculation of transmission zeros

The transmission zeros of a previous reported boiler system example [1,2] are computed, using double and extended precision arithmetic on an IBM 370/165 digital computer, by the algorithm of [2].     

The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros

In this paper, we give the convergence analysis of the Euler-like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial. The established initial conditions provide the safe convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. A procedure for the choice of initial inclusion disks is also given.     

A higher order method for multiple zeros of nonlinear functions

A method of order three for finding multiple zeros of nonlinear functions is developed. The method requires two evaluations of the function and one evaluation of the derivative per step.     

On a family of multipoint methods for non-linear equations

A new one-parameter family of methods for finding simple zeros of non-linear functions is developed. Each member of the family requires four evaluations of the given function and only one evaluation of the derivative per step. The order of the method is 16.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

On a third order family of methods for solving nonlinear equations

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468–483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223–239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |x _n+1?α|≤|x _n+1?x _n | for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared.     

Stabilization theory of iterative operator-splitting methods

In this paper we present a stabilization theory of iterative operator-splitting methods for linear and nonlinear differential equations. Continuous formulation is described and also the stability for linear and nonlinear cases. We apply linearization techniques to adduce proof of the linear theory. Iterative methods are applied to linearize and couple operator equations. A general theory is derived for linear and nonlinear iterative-splitting methods. Test examples verify the underlying theoretical results.