Miodrag S Petkovic | University of Nis, Faculty of Electronic Engineering (original) (raw)

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Papers by Miodrag S Petkovic

Journal of Computational and Applied Mathematics, 2022

This foreward to the Virtual Special Issue dedicated to Luc Wuytack contains some historical info... more This foreward to the Virtual Special Issue dedicated to Luc Wuytack contains some historical information related to the founding of Journal CAM as well as to Luc's memory. It also serves as a preface to the collection of articles that have been selected to exemplify the field to which Luc devoted his career and around which the development of the journal CAM started.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1986

nung der bestapproximiereriden Losung benotigten Multiplikationen n3. Die Aufwandmngaben machen d... more nung der bestapproximiereriden Losung benotigten Multiplikationen n3. Die Aufwandmngaben machen deutlich, da13 die Verwendung der obigen Modifikationsformeln bei groflformatigen Aufgaben besonders zweckmiil3ig ist. Jedoch auch in der Variantenrechnung eroffnet ihr gezielter Einsatz neue Moglichkeiten.

In this paper we present an efficient iterative method of order six for the inclusion of the inve... more In this paper we present an efficient iterative method of order six for the inclusion of the inverse of a given regular matrix. To provide the upper error bound of the outer matrix for the inverse matrix, we combine point and interval iterations. The new method is relied on a suitable matrix identity and a modification of a hyper-power method. This method is also feasible in the case of a full-rank m × n matrix, producing the interval sequence which converges to the Moore-Penrose inverse. It is shown that computational efficiency of the proposed method is equal or higher than the methods of hyper-power's type.

Reliab. Comput., 2012

A new iterative method of Ostrowski’s type for the inclusion of one isolated simple or multiple c... more A new iterative method of Ostrowski’s type for the inclusion of one isolated simple or multiple complex zero of a polynomial is established in circular complex arithmetic. Cubic convergence is proved and computationally veriable initial condition that guarantees the convergence of this inclusion method is stated. In order to demonstrate convergence behavior of the proposed method, two numerical examples are given.

Applied Mathematics and Computation, 2019

In this paper, we propose a fifth-order family of iterative methods for approximation of all zero... more In this paper, we propose a fifth-order family of iterative methods for approximation of all zeros of a polynomial simultaneously. The new family is developed by combining Gander's third-order family of iterative methods with the second-order Weierstrass root-finding method. The aim of the paper is to state initial conditions that provide local and semilocal convergence of the proposed methods as well as a priori and a posteriori error estimates. In the case of semilocal convergence the initial conditions and error estimates are computationally verifiable which is of practical importance.

A general class of three-point iterative methods for solving nonlinear equations is constructed. ... more A general class of three-point iterative methods for solving nonlinear equations is constructed. Its order of convergence reaches eight with only four function evaluations per iteration, which means that the proposed methods possess as high as possible computational e-ciency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a spectacular convergence speed with only few function evaluations.

Numerical Algorithms, 2015

We present derivative free methods with memory with increasing order of convergence for solving s... more We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383-393, 2014). The order of convergence of new family is increased from 4 of the basic family to 2 + √ 5 ≈ 4.24 by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence 2 + √ 6 ≈ 4.45 is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the theoretical results. Keywords Systems of nonlinear equations • Iterative methods • Derivative free methods • Order of convergence • Computational efficiency Janak Raj Sharma

Two families of derivative free two-point iterative methods for solving nonlinear equations are c... more Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung–Traub hypothesis (1974) on the upper bound 2 of the order of multipoint methods based on n + 1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least 2þ ffiffiffi 5 p 4:236 and even 2þ ffiffiffi 6 p 4:449 in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory poss...

Applied Mathematics and Computation, 2011

A general way to construct multipoint methods for solving nonlinear equations by using inverse in... more A general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation is presented. The proposed methods belong to the class of multipoint methods with memory. In particular, a new two-point method with memory with the order (5 + √ 17)/2 ≈ 4.562 is derived. Computational efficiency of the presented methods is analyzed and their comparison with existing methods with and without memory is performed on numerical examples. It is shown that a special choice of initial approximations provides a considerably great accuracy of root approximations obtained by the proposed interpolatory iterative methods.

Abstract. The third order interval Gargantini method for the simultane-ous inclusion of polynomia... more Abstract. The third order interval Gargantini method for the simultane-ous inclusion of polynomial zeros was improved to the fourth order method by Carstensen and Petkovic ́ [An improvement of Gargantini’s simultaneous inclusion method for polynomial roots by Schroeder’s correction, Appl. Nu-mer. Math. 13 (1994), 453–468]. We investigate the numerical stability of this improved method in the presence of rounding errors. The dependence of the convergence rate of the considered method on the magnitude of rounding errors is studied. 1.

The numerical stability of the fourth order iterative method of La-guerre’s type for the simultan... more The numerical stability of the fourth order iterative method of La-guerre’s type for the simultaneous inclusion of polynomial zeros is analyzed in the presence of rounding errors. We state conditions under which the con-vergence order of the considered method is preserved. If these conditions are relaxed, the convergence rate reduces to three. 1.

arXiv: Numerical Analysis, 2018

Several root-ratio multipoint methods for finding multiple zeros of univariate functions were rec... more Several root-ratio multipoint methods for finding multiple zeros of univariate functions were recently presented. The characteristic of these methods is that they deal with mmm-th root of ratio of two functions (hence the name root-ratio methods), where mmm is the multiplicity of the sought zero, known in advance. Some of these methods were presented without any derivation and motivation, it could be said, out of the blue. In this paper we present an easy and entirely natural way for constructing root-ratio multipoint iterative methods starting from multipoint methods for finding simple zeros. In this way, a vast number of root-ratio multipoint methods for multiple zeros, existing as well new ones, can be constructed. For demonstration, we derive four root-ratio methods for multiple zeros. Besides, we study computational cost of the considered methods and give a comparative analysis that involves CPU time needed for the extraction of the mmm-th root. This analysis shows that root-ra...

arXiv: Numerical Analysis, 2017

A new one-parameter family of iterative method for solving nonlinear equations is constructed and... more A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known multiplicities. This family generates a variety of different third order methods, including Halley-like method as a special case. Four numerical examples are given to demonstrate convergence properties of the proposed methods for multiple zeros and various values of the parameter.

The construction of computationally verifiable initial conditions that provide both the guarantee... more The construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of a numerical method for solving nonlinear equations is one of the most important tasks in the field of iterative processes. A suitable convergence procedure, based partially on Smale’s “point estimation theory” from 1981, is applied in this paper to a new cubically convergent derivative free iterative method for the simultaneous approximation of simple zeros of polynomials. We have stated initial conditions which guarantee the convergence of this method. These conditions are of significant practical importance since they depend only on available data: the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.