Reducing Chaos and Bifurcations in Newton-Type Methods
We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/726701 |
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| author | S. Amat S. Busquier Á. A. Magreñán |
| author_facet | S. Amat S. Busquier Á. A. Magreñán |
| author_sort | S. Amat |
| collection | DOAJ |
| description | We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is
that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting
points for the original schemes. |
| format | Article |
| id | doaj-art-d08d690f33154876b7de3cf1d40d8f78 |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-d08d690f33154876b7de3cf1d40d8f782025-08-20T02:20:02ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/726701726701Reducing Chaos and Bifurcations in Newton-Type MethodsS. Amat0S. Busquier1Á. A. Magreñán2Departamento de Matemática Aplicada y Estadística, Universidad de Cartagena, 30202 Cartagena, SpainDepartamento de Matemática Aplicada y Estadística, Universidad de Cartagena, 30202 Cartagena, SpainDepartamento de Matemática Aplicada y Estadística, Universidad de Cartagena, 30202 Cartagena, SpainWe study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.http://dx.doi.org/10.1155/2013/726701 |
| spellingShingle | S. Amat S. Busquier Á. A. Magreñán Reducing Chaos and Bifurcations in Newton-Type Methods Abstract and Applied Analysis |
| title | Reducing Chaos and Bifurcations in Newton-Type Methods |
| title_full | Reducing Chaos and Bifurcations in Newton-Type Methods |
| title_fullStr | Reducing Chaos and Bifurcations in Newton-Type Methods |
| title_full_unstemmed | Reducing Chaos and Bifurcations in Newton-Type Methods |
| title_short | Reducing Chaos and Bifurcations in Newton-Type Methods |
| title_sort | reducing chaos and bifurcations in newton type methods |
| url | http://dx.doi.org/10.1155/2013/726701 |
| work_keys_str_mv | AT samat reducingchaosandbifurcationsinnewtontypemethods AT sbusquier reducingchaosandbifurcationsinnewtontypemethods AT aamagrenan reducingchaosandbifurcationsinnewtontypemethods |