CONVERGENCE THEOREMS FOR NEWTON'S AND MODIFIED NEWTON'S METHODS
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In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modi¯ed Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modi¯ed Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.
Abstract
In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modi¯ed Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modi¯ed Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.
참고문헌 (7)
- I.K. Argyros, "On the Newton-Kantorovich hypothesis for solving equations", J. Comput.Appl. Math, 169, pp.315~332 (2004)
- I.K. Argyros, "Computational theory of iterative methods. Studies in Computational Mathe-matics, 15" (2007), Elsevier Publ. Co
- I.K. Argyros, "A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space", J. Math. Anal. Appl, 298(2), pp.374~397 (2004)
- I.K. Argyros, "A comparison between convergence theorems for Newton's and modi¯ed New-ton's methods", submitted for publication
- J.M. Gutierrez, "Accessibility of solutions by New-ton's method", Intern. J. Comput. Math, 57, pp.239~247 (1995)
- L.V. Kantorovich, "Functional Analysis" (1982), Pergamon Press
- M.A. Krasnosel0skii, "Approximate Solution of Operator Equations" (1972), Wolters-Noordhoff Publ