Djordje Herceg

On Euler-like methods for the simultaneous approximation of polynomial zeros

In this paper we consider some iterative methods of higher order for the simultaneous determination of polynomial zeros. The proposed methods are based on Euler’s third order method for finding a zero of a given function and involve Weierstrass’ correction in the case of simple zeros. We prove that the presented methods have the order of convergence equal to four or more. Based on a fixed-point relation of Euler’s type, two inclusion methods are derived. Combining the proposed methods in floating-point arithmetic and complex interval arithmetic, an efficient hybrid method with automatic error bounds is constructed. Computational aspect and the implementation of the presented algorithms on parallel computers are given.

Means based modifications of Newton's method for solving nonlinear equations

In this paper we consider a family of six sets of means based modifications of Newtons method for solving nonlinear equations. Each set is a parametric class of methods. Some well-known methods belong to our family, for example, the arithmetic mean Newtons method [S. Weerakoon, T.G.I. Fernando, A variant of Newtons method with accelerated third order convergence, Appl. Math. Lett. 13 (2000) 87-93], the harmonic mean Newtons method, [A.Y. Ozban, Some new variants of Newtons method, Appl. Math. Lett. 17 (2004) 677-682], the geometric mean Newtons method, [T. Lukic, N.M. Ralevic, Geometric mean Newtons method for simple and multiple roots, Appl. Math. Lett. 21 (2008) 30-36], the power mean Newtons method [X. Zhou, A class of Newtons methods with third-order convergence, Appl. Math. Lett. 20 (2007) 1026-1030]. The convergence analysis shows third order of our family. Comparison of the family members shows that there are no big differences between them. Twelve numerical examples were tested, and two characteristic ones are presented.

A family of methods for solving nonlinear equations

We present a family of methods for solving nonlinear equations. Some well-known classical methods and their modifications belong to our family, for example Newton, Potra-Ptak, Chebyshev, Halley and Ostrowskis methods. Convergence analysis shows that our family contains methods of convergence order from 2 to 4. All our fourth order methods are optimal in terms of the Kung and Traub conjecture. Several examples are presented and compared.

Fourth-order finite-difference method for boundary value problems with two small parameters☆

Abstract We present a finite difference scheme for a class of linear singularly perturbed boundary value problems with two small parameters. The problem is discretized using a Bakhvalov-type mesh. It is proved under certain conditions that this scheme is fourth-order accurate and that its error does not increase when the perturbation parameter tends to zero. Numerical examples are presented which demonstrate computationally the fourth order of the method.

On a fourth-order finite-difference method for singularly perturbed boundary value problems

We present a fourth-order finite-difference method for singularly perturbed one-dimensional reaction-diffusion problem. The problem is discretized using a Bakhvalov-type mesh. We give a uniform convergence with respect to the perturbation parameter. Numerical examples are presented which demonstrate computationally the fourth order of the method.

Using Padé Approximation in Takács Hysteresis Model

The application of the modified Takács hysteresis model for numerical modeling of measured hysteresis loops of ferromagnetic material is investigated. In order to better approximate measured hysteresis loops, the Takács model is modified using parameterized Padé approximations of the hyperbolic tangent function. That way, several variants of the modified model are constructed. The experimental data are fitted with the original model, as well as with the modified model, and the quality of conformance between empirical data and the models is assessed. Three numerical criteria are used for comparison: 1) the coefficient of determination; 2) percent error of the residual vector; and 3) percent errors at the five characteristic points on the hysteresis loop. The proposed method shows improvement in cases when hysteresis loops are not reaching saturation, while maintaining accuracy in the saturation. The performance of the modified model is verified by theoretical data.

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 Halleys 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 Newtons 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.

A new family of methods for single and multiple roots

Abstract We present a new family of iterative methods for multiple and single roots of nonlinear equations. This family contains as a special case the authors’ family for finding simple roots from Herceg and Herceg (2015). Some well-known classical methods for simple roots, for example Newton, Potra–Ptak, Newton–Steffensen, King and Ostrowski’s methods, belong to this family, which implies that our new family contains modifications of these methods suitable for finding multiple roots. Convergence analysis shows that our family contains methods of convergence order from 2 to 4. The new methods require two function evaluations and one evaluation of the first derivative per iteration, so all our fourth order methods are optimal in terms of the Kung and Traub conjecture. Several examples are presented and compared. Through various test equations, relevant numerical experiments strongly support the claimed theory in this paper. Extraneous fixed points of the iterative maps associated with the proposed methods are also investigated. Their dynamics is explored along with illustrated basins of attraction for various polynomials.

Eighth order family of iterative methods for nonlinear equations and their basins of attraction

Abstract We present a new family of eighth order methods for solving nonlinear equations. The order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, demonstrate convergence and efficacy of our family of methods. For some combinations of parameter values, the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature. An exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials.