Chein-Shan Liu¹, Essam R. El-Zahar^2,3, Yung-Wei Chen^4,*

CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 1111-1130, 2023, DOI:10.32604/cmes.2022.021655

Abstract How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations (NAEs). This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms. We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system. Through the maximal orthogonal projection concept, to minimize a merit function within a selected interval of splitting parameters, the optimal parameters can be quickly determined.… More > Graphic Abstract

Luyao Yang^1,#, Hao Chen^2,#, Haocheng Yu¹, Jin Qiu^1,*, Shuxian Zhu^1,*

CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.1, pp. 731-745, 2023, DOI:10.32604/cmes.2022.020656

Abstract Discrete Tomography (DT) is a technology that uses image projection to reconstruct images. Its reconstruction problem, especially the binary image (0–1 matrix) has attracted strong attention. In this study, a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructed model. The solution process can be divided into two procedures. First, the DT problem is reformulated into a polyhedron judgment problem based on lattice basis reduction. Second, the fixed-point iterative method of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program. All the… More >

Mudassir Shams¹, Naila Rafiq², Nasreen Kausar³, Nazir Ahmad Mir², Ahmad Alalyani^4,*

Computer Systems Science and Engineering, Vol.42, No.2, pp. 689-701, 2022, DOI:10.32604/csse.2022.022269

Abstract In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes… More >

Naila Rafiq¹, Saima Akram², Mudassir Shams^3,*, Nazir Ahmad Mir¹

CMC-Computers, Materials & Continua, Vol.69, No.2, pp. 2635-2651, 2021, DOI:10.32604/cmc.2021.018955

Abstract In this research article, we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation. We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab. Lower bound of convergence of simultaneous schemes is calculated using Mathematica. Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented. Some real life engineering applications along with some higher degree polynomials… 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^*

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 >

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 >

Syahmi Afandi Sariman¹, Ishak Hashim^{1, *}

CMC-Computers, Materials & Continua, Vol.65, No.1, pp. 69-85, 2020, DOI:10.32604/cmc.2020.010836

Abstract The classical iterative methods for finding roots of nonlinear equations, like the Newton method, Halley method, and Chebyshev method, have been modified previously to achieve optimal convergence order. However, the Householder method has so far not been modified to become optimal. In this study, we shall develop two new optimal Newton-Householder methods without memory. The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative. The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations. The efficiency… More >