On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case
In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the n...
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MDPI AG
2019-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/6/540 |
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| author | Zhang Yong Neha Gupta J. P. Jaiswal Kalyanasundaram Madhu |
| author_facet | Zhang Yong Neha Gupta J. P. Jaiswal Kalyanasundaram Madhu |
| author_sort | Zhang Yong |
| collection | DOAJ |
| description | In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>-continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section. |
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| format | Article |
| id | doaj.art-fd78a80841364b42b61f564ae76bc3f4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T14:38:19Z |
| publishDate | 2019-06-01 |
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| spelling | doaj.art-fd78a80841364b42b61f564ae76bc3f42022-12-21T23:41:41ZengMDPI AGMathematics2227-73902019-06-017654010.3390/math7060540math7060540On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative CaseZhang Yong0Neha Gupta1J. P. Jaiswal2Kalyanasundaram Madhu3School of Mathematics and Physics, Changzhou University, Changzhou 213164, ChinaDepartment of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, IndiaDepartment of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, IndiaDepartment of Mathematics, Saveetha Engineering College, Chennai 602105, IndiaIn this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>-continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section.https://www.mdpi.com/2227-7390/7/6/540Banach spacesemilocal convergenceω-continuity conditionJarratt methoderror bound |
| spellingShingle | Zhang Yong Neha Gupta J. P. Jaiswal Kalyanasundaram Madhu On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case Mathematics Banach space semilocal convergence ω-continuity condition Jarratt method error bound |
| title | On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case |
| title_full | On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case |
| title_fullStr | On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case |
| title_full_unstemmed | On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case |
| title_short | On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case |
| title_sort | on the semilocal convergence of the multi point variant of jarratt method unbounded third derivative case |
| topic | Banach space semilocal convergence ω-continuity condition Jarratt method error bound |
| url | https://www.mdpi.com/2227-7390/7/6/540 |
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