Mudassir Shams¹, Nasreen Kausar^2,*, Shams Forruque Ahmed³, Irfan Anjum Badruddin⁴, Syed Javed⁴

CMC-Computers, Materials & Continua, Vol.74, No.3, pp. 5331-5347, 2023, DOI:10.32604/cmc.2023.033528

Abstract A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper. Convergence analysis demonstrates that the local order of convergence of the numerical method is five. The computer algebra system CAS-Maple, Mathematica, or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects. Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms. A dynamic evaluation of the presented methods is also presented utilizing basins… 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 >

Mudassir Shams^1,*, Naila Rafiq², Nazir Ahmad Mir^1,2, Babar Ahmad³, Saqib Abbasi¹, Mutee-Ur-Rehman Kayani¹

Computer Systems Science and Engineering, Vol.36, No.3, pp. 493-507, 2021, DOI:10.32604/csse.2021.014476

Abstract In this research article, we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation simultaneously. These modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to 3. Convergence analysis proves that the orders of convergence of the two newly constructed inverse methods are 3. Using computer algebra system Mathematica, we find the lower bound of the convergence order and verify it theoretically. Dynamical planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB (R2011b), to present the global convergence properties of inverse simultaneous iterative… 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 >

Yuming Chu¹, Naila Rafiq², Mudassir Shams^3,*, Saima Akram⁴, Nazir Ahmad Mir³, Humaira Kalsoom⁵

CMC-Computers, Materials & Continua, Vol.66, No.1, pp. 275-290, 2021, DOI:10.32604/cmc.2020.011907

Abstract In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine. Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes, numerical experiments and CPU time-methodology. Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous… More >