Dragoslav Herceg

Uniform fourth order difference scheme for a singular perturbation problem

SummaryThe numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.

On rediscovered methods for solving equations

Abstract In the recent papers of Gerlach (SIAM Rev. 36 (1994) 272–276) and Ford and Pennline (SIAM Rev. 38 (1996) 658–659) a class of iterative methods for solving a single equation f ( x )=0, with arbitrary rate of convergence, has been presented. In this paper we show that this class is equivalent to five other classes of iterative methods, derived earlier in various ways and expressed in different forms. The proofs of equivalence of all considered iteration formulas 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.

Higher-order iterative methods for approximating zeros of analytic functions

Abstract Iterative methods with extremely rapid convergence in floating-point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions (inside a simple smooth closed contour in the complex plane) are presented. The R-order of convergence of the basic total-step and single-step methods, as well as their improvements which use Newtons and Halleys corrections, is given. Some hybrid algorithms that combine the efficiency of ordinary floating-point iterative methods with the accuracy control provided by interval arithmetic are also considered.

Point estimation of a family of simultaneous zero-finding methods

Abstract The construction of initial conditions which provide the safe convergence of iterative processes is one of the most important problems in solving equations. In this paper, we give initial conditions for a one parameter family of iterative methods for the simultaneous determination of all simple zeros of a polynomial zeros. This family has the order of convergence four and generates some new methods with good convergence properties. The presented initial conditions are of practical importance since depend only of available data: coefficients of a polynomial and initial approximations to the zeros.

On the extrapolation method and the USA algorithm

Abstract The paper is motivated by the paper of Khilnani and Tse (1985), ‘A fixed point algorithm with economic applications’, Journal of Economic Dynamics and Control 9, 125–137, where the method of updated successive approximations (USA) for the numerical solution of systems of nonlinear equations is presented. Some numerical examples show that the convergence statement of the USA method, which one can consider as a nonstationary extrapolation method, contains a flaw. In this paper we give some of these examples and discuss the variable extrapolation parameter used in the above cited paper, which cannot guarantee the convergence of the USA method. We also describe a nonlinear extrapolation method for solving nonlinear systems and give some sufficient conditions for the convergence of this method.

Extended covergence area for the (MSOR) method

Abstract In this paper we establish sufficient convergence conditions for the (MSOR) method when the matrix A of Ax = b belongs to the class C 1 , which contains the strictly diagonally dominant matrices. We also extend these results to the H -matrices. We get an improvement for the area of convergence for the (MSOR) method for the classes of matrices mentioned before.

The AOR method for solving linear interval equations

This paper is motivated by the paper [7], where the SOR method for solving linear interval equations was considered. It is known that sometimes the AOR method for systems of linear (“point”) equations converges faster than the SOR method. We give some sufficient conditions for the convergence of the interval AOR method for the same class of interval matrices which are considered in [7].ZusammenfassungDiese Arbeit ist durch die Arbeit [7] motiviert, wo das SOR-Verfahren zur Lösung von linearen Intervallgleichungen betrachtet wird. Es ist bekannt, daß das AOR-Verfahren für lineare (“Punkt”) Gleichungssysteme manchmal schneller als das SOR-Verfahren konvergiert. Wir geben einige hinreichende Konvergenzbedingungen für das Intervall-AOR-Verfahren für dieselbe Klasse von Matrizen, die in [7] betrachtet wurde.

On a numerical differentiation

We consider a numerical approximation of the mth derivative of a real-valued function of a real variable at a single point by the n-point rule. The coefficients of the n-point rule are obtained. In the general case this formula is of degree