Computing Simple Roots by an Optimal Sixteenth-Order Class
The problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in wh...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/958020 |
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| author | F. Soleymani S. Shateyi H. Salmani |
| author_facet | F. Soleymani S. Shateyi H. Salmani |
| author_sort | F. Soleymani |
| collection | DOAJ |
| description | The problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index , by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newton's method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient. |
| format | Article |
| id | doaj-art-14b376f65ae048bb84beda8206c35bb8 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-14b376f65ae048bb84beda8206c35bb82025-02-03T06:13:41ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/958020958020Computing Simple Roots by an Optimal Sixteenth-Order ClassF. Soleymani0S. Shateyi1H. Salmani2Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, IranDepartment of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South AfricaDepartment of Civil Engineering, Islamic Azad University, Zahedan Branch, Zahedan, IranThe problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index , by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newton's method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient.http://dx.doi.org/10.1155/2012/958020 |
| spellingShingle | F. Soleymani S. Shateyi H. Salmani Computing Simple Roots by an Optimal Sixteenth-Order Class Journal of Applied Mathematics |
| title | Computing Simple Roots by an Optimal Sixteenth-Order Class |
| title_full | Computing Simple Roots by an Optimal Sixteenth-Order Class |
| title_fullStr | Computing Simple Roots by an Optimal Sixteenth-Order Class |
| title_full_unstemmed | Computing Simple Roots by an Optimal Sixteenth-Order Class |
| title_short | Computing Simple Roots by an Optimal Sixteenth-Order Class |
| title_sort | computing simple roots by an optimal sixteenth order class |
| url | http://dx.doi.org/10.1155/2012/958020 |
| work_keys_str_mv | AT fsoleymani computingsimplerootsbyanoptimalsixteenthorderclass AT sshateyi computingsimplerootsbyanoptimalsixteenthorderclass AT hsalmani computingsimplerootsbyanoptimalsixteenthorderclass |