Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography
The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical met...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2022/2596924 |
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| author | Amanullah Soomro Amir Naseem Sania Qureshi Nasr Al Din Ide |
| author_facet | Amanullah Soomro Amir Naseem Sania Qureshi Nasr Al Din Ide |
| author_sort | Amanullah Soomro |
| collection | DOAJ |
| description | The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absolute error, absolute functional value, and the machine time measured in seconds. The obtained results outperformed the chosen methods when applied models arising from physical and natural fields were solved. Finally, to observe the convergence graphically, some complex polynomials are plotted as polynomiographs, wherein the rapid convergence of the proposed method is guaranteed. |
| format | Article |
| id | doaj-art-ffa3f79dcfe94423b6f902bc9379eb43 |
| institution | Kabale University |
| issn | 1099-0526 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-ffa3f79dcfe94423b6f902bc9379eb432025-02-03T05:57:28ZengWileyComplexity1099-05262022-01-01202210.1155/2022/2596924Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex PolynomiographyAmanullah Soomro0Amir Naseem1Sania Qureshi2Nasr Al Din Ide3Department of Basic Sciences and Related StudiesDepartment of MathematicsDepartment of Basic Sciences and Related StudiesDepartment of MathematicsThe appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absolute error, absolute functional value, and the machine time measured in seconds. The obtained results outperformed the chosen methods when applied models arising from physical and natural fields were solved. Finally, to observe the convergence graphically, some complex polynomials are plotted as polynomiographs, wherein the rapid convergence of the proposed method is guaranteed.http://dx.doi.org/10.1155/2022/2596924 |
| spellingShingle | Amanullah Soomro Amir Naseem Sania Qureshi Nasr Al Din Ide Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography Complexity |
| title | Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography |
| title_full | Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography |
| title_fullStr | Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography |
| title_full_unstemmed | Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography |
| title_short | Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography |
| title_sort | development of a new multi step iteration scheme for solving non linear models with complex polynomiography |
| url | http://dx.doi.org/10.1155/2022/2596924 |
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