Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods
Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (function...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/318165 |
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| author | F. Soleymani S. Shateyi |
| author_facet | F. Soleymani S. Shateyi |
| author_sort | F. Soleymani |
| collection | DOAJ |
| description | Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes. |
| format | Article |
| id | doaj-art-a22bb76e690d45e19709aceebe58ff67 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-a22bb76e690d45e19709aceebe58ff672025-02-03T01:27:51ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/318165318165Two Optimal Eighth-Order Derivative-Free Classes of Iterative MethodsF. Soleymani0S. Shateyi1Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, IranDepartment of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South AfricaOptimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes.http://dx.doi.org/10.1155/2012/318165 |
| spellingShingle | F. Soleymani S. Shateyi Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods Abstract and Applied Analysis |
| title | Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods |
| title_full | Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods |
| title_fullStr | Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods |
| title_full_unstemmed | Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods |
| title_short | Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods |
| title_sort | two optimal eighth order derivative free classes of iterative methods |
| url | http://dx.doi.org/10.1155/2012/318165 |
| work_keys_str_mv | AT fsoleymani twooptimaleighthorderderivativefreeclassesofiterativemethods AT sshateyi twooptimaleighthorderderivativefreeclassesofiterativemethods |