A new convergence theorem for the method of tangent hyperbolas in banach space
DOI:
https://doi.org/10.22199/S07160917.1999.0001.00001Keywords:
Banach space, Method of tangent hyperbolas, Fréchet - derivative, Newton - Kantorovich hypothesisAbstract
In this study we appmximate a locally unique solution of a non-linear operator equtation in Banach space using the method of tangent hyperbolas. A new semilocal convergence theorem is provided using Lipschitz conditions on the second Fréchet-derivative. Our conditions are different than earlier ones. Hence, they have theorctical and practical value. Numerical examples are also provoded.
References
[1] M. Altman, Concerning the method of tangent hyperbolas for operator equations, Bull. Acad. Polon. Sci. Ser. Math. Astr-. Phys. 9, pp. 633- 637, (1961).
[2] I.K. Argyros, On the convergence of an Euler-Chebysheff-type method under Newton-Kantorovich-type hypotheses, Pure Mathematics and Applications, 4 3, pp. 369- 373, (1993).
[3] I.K. Argyros, On the method of tangent hyperbolas, J. Appmx. Th. Appl. 12 1, pp. 78- 96, (1996).
[4] I.K. Argyros, Polynomial Operator- Equations in Abstract Spaces and Applications, CRC Press LLC, Boca Raton, Florida, (1998).
[5] I.K. Argyros and F. Szidarovszky, The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, Florida, (1993).
[6] J.A. Ezquerro, J.M. Gutierez, and M.A. Hernandez, A construction procedure of iterative methods with cubical convergence, Appl. Math. Comp. (to appear).
[7] L.M. Graves, Riemann integration and Taylor's theorem in general analysis, Trans. Amer. Math. Soc. 29, pp. 163- 177, (1927).
[8] J.M. Gutierez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145.
[9] S. Kanno, Convergence theorems for the method of tangent hyperbolas, Afath. Japanich, 37, 4, pp. 711- 722, (1992).
[10] L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, (1982).
[11] R.A. Safiev, The method of tangent hyperbolas, Sov. Math. Dokl. 4, pp. 482- 485, (1963).
[2] I.K. Argyros, On the convergence of an Euler-Chebysheff-type method under Newton-Kantorovich-type hypotheses, Pure Mathematics and Applications, 4 3, pp. 369- 373, (1993).
[3] I.K. Argyros, On the method of tangent hyperbolas, J. Appmx. Th. Appl. 12 1, pp. 78- 96, (1996).
[4] I.K. Argyros, Polynomial Operator- Equations in Abstract Spaces and Applications, CRC Press LLC, Boca Raton, Florida, (1998).
[5] I.K. Argyros and F. Szidarovszky, The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, Florida, (1993).
[6] J.A. Ezquerro, J.M. Gutierez, and M.A. Hernandez, A construction procedure of iterative methods with cubical convergence, Appl. Math. Comp. (to appear).
[7] L.M. Graves, Riemann integration and Taylor's theorem in general analysis, Trans. Amer. Math. Soc. 29, pp. 163- 177, (1927).
[8] J.M. Gutierez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145.
[9] S. Kanno, Convergence theorems for the method of tangent hyperbolas, Afath. Japanich, 37, 4, pp. 711- 722, (1992).
[10] L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, (1982).
[11] R.A. Safiev, The method of tangent hyperbolas, Sov. Math. Dokl. 4, pp. 482- 485, (1963).
Published
2018-04-04
How to Cite
[1]
I. K. Argyros, “A new convergence theorem for the method of tangent hyperbolas in banach space”, Proyecciones (Antofagasta, On line), vol. 18, no. 1, pp. 1-11, Apr. 2018.
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