Qingbo Cai¹, Reşat Aslan^2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.130, No.3, pp. 1479-1493, 2022, DOI:10.32604/cmes.2022.018338

Abstract The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind q-Bernstein operators related to Bézier basis functions with shape parameter . Firstly, we compute some basic results such as moments and central moments, and derive the Korovkin type approximation theorem for these operators. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz-type class and Peetre’s K-functional, respectively. Lastly, with the aid of Maple software, we present the comparison of the convergence of these newly defined operators to the certain function with some… More >

Obadah Said Solaiman¹, Samsul Ariffin Abdul Karim², Ishak Hashim^1,*

CMC-Computers, Materials & Continua, Vol.67, No.2, pp. 1951-1962, 2021, DOI:10.32604/cmc.2021.015344

Abstract There are several ways that can be used to classify or compare iterative methods for nonlinear equations, for instance; order of convergence, informational efficiency, and efficiency index. In this work, we use another way, namely the basins of attraction of the method. The purpose of this study is to compare several iterative schemes for nonlinear equations. All the selected schemes are of the third-order of convergence and most of them have the same efficiency index. The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees. As a comparison, we determine… More >

Obadah Said Solaiman, Ishak Hashim^*

Intelligent Automation & Soft Computing, Vol.27, No.2, pp. 379-390, 2021, DOI:10.32604/iasc.2021.015285

Abstract A new iterative technique for nonlinear equations is proposed in this work. The new scheme is of three steps, of which the first two steps are based on the sixth-order modified Halley’s method presented by the authors, and the last is a Newton step, with suitable approximations for the first derivatives appeared in the new scheme. The eighth-order of convergence of the new method is proved via Mathematica code. Every iteration of the presented scheme needs the evaluation of three functions and one first derivative. Therefore, the scheme is optimal in the sense of Kung-Traub conjecture. Several test nonlinear problems… More >

Obadah Said Solaiman, Ishak Hashim^*

CMC-Computers, Materials & Continua, Vol.66, No.2, pp. 1427-1444, 2021, DOI:10.32604/cmc.2020.012610

Abstract In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n − 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison… More >