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Gupta, Neha
- Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under HÖlder Continuity Condition
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Authors
Neha Gupta
1,
J. P. Jaiswal
1
Affiliations
1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, IN
1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, IN
Source
The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 56–69Abstract
The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x* and its uniqueness along with prior error bound are established. Also, the R-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.Keywords
Banach Space, Nonlinear Operator, Semilocal Convergence, Hölder Condition, Frèchet Derivative, Recurrence Relation, Error Bound.
References
- S. Amat and S. Busquier, Third-order iterative methods under Kantorovich conditions, J. Math. Anal. Appl. 336, 243-261 (2007).
- L. Chen, C. Gu and Y. Ma, Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces, J. Appl. Math., Volume 2011, Article ID 786306, 15 pages (2011).
- A. Cordero, M. A. Hern´andez, N. Romero and J. R. Torregrosa, Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces, J. Comput. Appl. Math. 273, 205-213 (2015).
- M. A. Hern´andez and M. A. Salanova, Sufficient conditions for semilocal convergence of a fourthorder multipoint iterative method for solving equations in Banach spaces, Southwest J. Pure Appl. Math. 1, 29-40 (1999).
- J. P. Jaiswal, Semilocal convergence and its computational efficiency of a seventh-order method in Banach spaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., DOI: 10.1007/s40010-018-0590-7 (2019).
- L. V. Kantorovich and G. P. Akilov, Functional Analysis. Pergamon Press, Oxford (1982).
- P. K. Parida and D. K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206, 873-887 (2007).
- M. Prashanth and D. K. Gupta, Recurrence relations for super-Halley’s method with H¨ older continuous second derivative in Banach spaces, Kodai Math. J. 36(1), 119-136 (2013).
- L. B. Rall, Computational solution of nonlinear operator equations. Robert E Krieger, New York (1979).
- X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algor. 57, 441-456 (2011).
- X.Wang and J. Kou, Analysis of convergence for improved Chebyshev-Halley methods under different conditions, J. Optimiz. Theory App., 162(3), 920-930 (2013).
- X. Xiao and H. Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations, Appl. Math. Comput. 26, 300-309 (2015).
- L. Zheng and C. Gu, Semilocal convergence of a sixth-order method in Banach spaces, Numer. Algor. 61, 413-427 (2012).
- L. Zheng and C. Gu, Fourth-order convergence theorem by using majorizing functions for superHalley method in Banach spaces, Int. J. Comp. Math. 90, 423-434 (2013).
- Ball Convergence of Modified Homeier-Like's Method in Banach Spaces under Weak Continuity Condition
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Authors
Source
The Journal of the Indian Mathematical Society, Vol 89, No 3-4 (2022), Pagination: 305-316Abstract
The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. $\omega$-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution.
Keywords
Banach Space, Lcal Cconvergence, Recurrence Relation, β-Continuity condition.References
- I. K. Argyros, Convergence and applications of Newton-type iterations, Springer, New York, 2008,
- I. K. Argyros and S. George, Local convergence of two competing third order methods in Banach spaces, Appl. Math., 41 (2016), 341–350.
- I. K. Argyros and S. George, Increasing the order of convergence for iterative methods in Banach space under weak conditions, Malaya J. Matematik, 6(2) (2018), 396–401.
- I. K. Argyros, D. Gonzalez and S. K. Khattri, Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces, Comment. Math. Univ. Carolin., 57 (2016), 289–300.
- I. K. Argyros and S. Hilout, Computational Methods in Nonlinear Analysis, World Scienti?c Publishing Company, New Jersey, 2013.
- I. K. Argyros and S. K. Khattri, Local convergence for a family of third order methods in Banach spaces, Punjab Univ. J. Math., 46 (2016), 52–63.
- A. Cordero, J. A. Ezquerro, M. A. Hern´andez, and J. Torregrosa, On the local convergence of a ?fth-order iterative method in Banach spaces, Appl. Math. Comput, 251 (2015), 396–403.
- H. H. H. Homeier, A modi?ed Newton method with cubic convergence: the multivariable case, J. Comput. Appl. Math., 169 (2004), 161–169
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
- E. Mart´?nez, S. Singh, J. L. Hueso and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.
- B. Panday and J. P. Jaiswal, On the local convergence of modi?ed Homeier-like method in Banach spaces, Num. Anal. Appl., 21(4) (2018), 419–433.
- L. B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E Krieger, New York, 1979.
- J. R. Sharma and I. K. Argyros, Local convergence of a Newton-Traub composition in Banach spaces, SeMA, 75(1) (2017), 57–68.
- J. R. Sharma and P. Gupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67(3) (2014), 591–601.
- J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, 1977.