Extending the applicability of modified Newton-HSS method for solving systems of nonlinear equations
DOI:
https://doi.org/10.24193/subbmath.2018.2.09Keywords:
Modified Newton-HSS method, semilocal convergence, system of non- linear equations, generalized Lipschitz conditions, Hermitian method.Abstract
We present the semilocal convergence of a modified Newton-HSS method to approximate a solution of a nonlinear equation. Earlier studies show convergence under only Lipschitz conditions limiting the applicability of this method. The convergence in this study is shown under generalized Lipschitz- type conditions and restricted convergence domains. Hence, the applicability of the method is expanded. Moreover, numerical examples are also provided to show that our results can be applied to solve equations in cases where earlier study cannot be applied. Furthermore, in the cases where both old and new results are applicable, the latter provides a larger domain of convergence and tighter error bounds on the distances involved.
Mathematics Subject Classification (2010): 65F10, 65W05.
References
Amat, S., Busquier, S., Plaza, S., Dynamics of the King and Jarratt iterations, Aequa- tiones Math., 69(2005), 212–223.
Amat, S., Busquier, S., Plaza, S., Gutt´errez, J.M., Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. App. Math., 157(2003), 197–205.
Amat, S., Herna´ndez, M.A., Romero, N., A modified Chebyshev’s iterative method with at least sixth order of convergence, Appl. Math. Comput., 206(2008), 164–174.
Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer-Verlag, New York, 2008.
Argyros, I.K., Hilout, S., Computational Methods in Nonlinear Analysis, World Scientific Publ. Comp., New Jersey, 2013.
Argyros, I.K., George, S., Local convergence of some higher-order Newton-Like method with frozen derivative, SeMA, 70(2015), 47–59.
Argyros, I.K., Magren˜a´n, A´ .A., Ball convergence theorems and the convergence plans of an iterative methods for nonlinear equations, SeMA, 71(2015), 39–55.
Argyros, I.K., Sharma, J.R., Kumar, D., Local convergence of Newton-HSS methods with positive definite Jacobian matrices under generalized conditions, SeMA, (2017).
Axelsson, O., Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.
Bai, Z.Z., Golub, G.H., Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24(2003), 603–626.
Bai, Z.Z., Golub, G.H., Pan, J.Y., Preconditioned Hermitian and skew-Hermitian split- ting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98(2004), 1–32.
Bai, Z.Z., Guo, X.P., On Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices, J. Comput. Math., 28(2010), 235–260.
Cordero, A., Ezquerro, J.A., Herna´ndez-Veron, M.A., Torregrosa, J.R., On the local convergence of a fifth-order iterative method in Banach spaces, Appl. Math. Comput., 251(2015), 396–403.
Dembo, R.S., Eiscnstat, S.C., Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19(1982), 400–408.
Ezquerro, J.A., Herna´ndez, M.A., New iterations of R-order four with reduced compu- tational cost, BIT Numer. Math., 49(2009), 325–342.
Ezquerro, J.A., Herna´ndez, M.A., A uniparametric Halley type iteration with free second derivative, Int. J. Pure. Appl. Math., 6(2003), 99–110.
Guo, X.P., Duff, I.S., Semilocal and global convergence of Newton-HSS method for sys- tems of nonlinear equations, Numer. Linear Algebra Appl., 18(2011), 299–315.
Guti´errez, J.M., Herna´ndez, M.A., Recurrence relations for the super Halley Method, Comput. Math. Appl., 36(1998), 1–8.
Herna´ndez, M.A., Mart´ınez, E., On the semilocal convergence of a three steps Newton- type process under mild convergence conditions, Numer. Algor., 70(2015), 377–392.
Ortega, J.M., Rheinboldt, W.C., Iteraive Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
Shen, W.P., Li, C., Convergence criterion of inexact methods for operators with Ho¨lder continuous derivatives, Taiwanese J. Math., 12(2008), 1865–1882.
Wu, Q.B., Chen, M.H., Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations, Numer. Algor., 64(2013), 659–683.
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