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Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

Computers, Materials & Continua 2021年第5期67卷 1951-1962页

作者： Obadah Said Solaiman samsul ariffin abdul karim Ishak Hashim Department of Mathematical Sciences Faculty of Science&TechnologyUniversiti Kebangsaan MalaysiaBangi Selangor43600Malaysia Department of Fundamental and Applied Sciences Center for Smart Grid Energy Research(CSMER)Institute of Autonomous SystemUniversiti Teknologi PETRONASBandar Seri IskandarSeri IskandarPerak DR32610Malaysia

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 *** this work,we use another way,namely the basins of attraction of the *** purpose of this study is to compare several iterative schemes for nonlinear *** the selected schemes are of the third-order of convergence and most of them have the same efficiency *** comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different *** a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all *** confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency ***,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

关键词： Nonlinear equations iterative methods basins of attraction order of convergence

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Newton-EGMSOR Methods for Solution of Second Order Two-Point Nonlinear Boundary Value Problems

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Journal of Mathematics and System Science 2012年第3期2卷 185-190页

作者： Jumat Sulaiman Mohd Khatim Hasan Mohamed Othman samsul ariffin abdul karim School of Science and Technology Universiti Malaysia Sabah Kota Kinabalu 88400 Sabah Malaysia School of Information Technology Universiti Kebangsaan Malaysia UKM Bangi 43600 Selangor Malaysia Department of Communication Technology and Network Universiti Putra Malaysia UPM Serdang 43400 Selangor Malaysia Department of Fundamental and Applied Sciences Universiti Teknologi Petronas Tronoh 31750 Perak Darul Ridzuan Malaysia

The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.

关键词： Explicit group MSOR iteration second order scheme two-point nonlinear boundary value problem.

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Scattered Data Interpolation Using Cubic Trigonometric Bézier

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Computers, Materials & Continua 2021年第10期 221-236页

作者： Ishak Hashim Nur Nabilah Che Draman samsul ariffin abdul karim Wee Ping Yeo Dumitru Baleanu Department of Mathematical Sciences Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaUKMBangi43600Selangor Darul EhsanMalaysia Department of Fundamental and Applied Sciences Universiti Teknologi PETRONASSeri Iskandar32610Perak Darul RidzuanMalaysia Department of Fundamental and Applied Sciences and Centre for Systems Engineering(CSE) Institute of Autonoumous System Universiti Teknologi PETRONASSeri Iskandar32610Perak Darul RidzuanMalaysia Faculty of Science Universiti Brunei DarussalamBandar Seri BegawanBE1410Brunei Darussalam Department of Mathematics Cankaya UniversityAnkaraTurkey Institute of Space Sciences Magurele-BucharestRomania Department of Medical Research China Medical University HospitalChina Medical UniversityTaichungTaiwan

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with C1 continuity *** derive the C1 condition on each adjacent *** each triangular patch,we employ convex combination method between three local *** final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data *** tested the proposed scheme with 36,65,and 100 data points for some well-known test *** scheme is also applied to interpolate the data for the electric *** compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell–Sabin(PS)and Clough–Tocher(CT)by measuring the maximum error,root mean square error(RMSE)and coefficient of determination(R^(2)).From the results obtained,our proposed method is competent with cubic Bézier,cubic Ball,PS and CT triangles splitting schemes to interpolate scattered data *** is very significant since PS and CT requires that each triangle be splitting into several micro triangles.

关键词： Cubic trigonometric Bézier triangular patches C1sufficient condition scattered data interpolation

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