# CONVERGENCE OF THE RELAXED NEWTON'S METHOD

• Argyros, Ioannis Konstantinos ;
• Gutierrez, Jose Manuel ;
• Magrenan, Angel Alberto ;
• Romero, Natalia
• Published : 2014.01.01
• 90 5

#### Abstract

In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < ${\lambda}$ < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter ${\lambda}$. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for ${\lambda}=1$.

#### Keywords

relaxed Newton's method;Banach space;Kantorovich hypothesis;majorizing sequence;local convergence;semilocal convergence

#### References

1. J. Appell, E. de Pascale, J. V. Lysenko, and P. P. Zabrejko, New results on Newton-Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 1-17.
2. I. K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Frechet derivative, Comput. Math. Appl. 37 (1999), no. 7, 109-115. https://doi.org/10.1016/S0898-1221(99)00091-7
3. I. K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comput. 80 (2011), no. 273, 327-343.
4. I. K. Argyros and S. Hilout, On the convergence of inexact Newton-type methods using recurrent functions, Panamer. Math. J. 19 (2009), no. 1, 79-96.
5. I. K. Argyros and S. Hilout, Inexact Newton methods and recurrent functions, App. Math. 37 (2010), no. 1, 113-126.
6. I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387. https://doi.org/10.1016/j.jco.2011.12.003
7. I. K. Argyros and S. Hilout, Estimating upper bounds on the limit pointss of majorizing sequences for Newton's method, Numer. Algor. 62 (2013), no. 1, 115-132. https://doi.org/10.1007/s11075-012-9570-1
8. I. K. Argyros and S. Hilout, On the semilocal convergence of damped Newton's method, Appl. Math. Comput. 219 (2012), no. 5, 2808-2824. https://doi.org/10.1016/j.amc.2012.09.011
9. I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Method for Equations and Its Applications, CRC Press/Taylor and Francis, New York, 2012.
10. Z.-Z. Bai and J.-L. Dong, A modified damped Newton method for linear complementarity problems, Numer. Algorithms 42 (2006), no. 3-4, 207-228. https://doi.org/10.1007/s11075-006-9028-4
11. X. J. Chen and L. Q. Li, A parameterized Newton method and a quasi-Newton method for nonsmooth equations, Comput. Optim. Appl. 3 (1994), no. 2, 157-179. https://doi.org/10.1007/BF01300972
12. R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), no. 2, 400-408. https://doi.org/10.1137/0719025
13. R. Fontecilla, T. Steihaug, and R. A. Tapia, A convergence theory for a class of quasi-Newton method for constrained optimization, SIAM J. Numer. Anal. 24 (1987), no. 5, 1133-1151. https://doi.org/10.1137/0724075
14. B. I. Epureanu and H. S. Greenside, Fractal basins of attraction associated with a damped Newton's method, SIAM Rev. 40 (1998), no. 1, 102-109. https://doi.org/10.1137/S0036144596310033
15. X. Guo, On semilocal convergence of inexact Newton method, J. Comput. Math. 25 (2007), no. 2, 231-242.
16. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
17. F. V. Haeseler and H. Kriete, Surgery for relaxed Newton's method, Complex Variables Theory Appl. 22 (1993), no. 1-2, 129-143. https://doi.org/10.1080/17476939308814653
18. B. T. Polyak, Newton-Kantorovich method and its global convergence, J. Math. Sci. (N. Y.) 133 (2006), no. 4, 1513-1523. https://doi.org/10.1007/s10958-006-0066-1
19. J. M. Ortega andW. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
20. A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, 1966.
21. A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, Nueva York, 1973.
22. L. B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger Publishing Company, Inc., California, 1979.
23. W. Shen and C. Li, Kantorovich-type convergence criterion for inexact Newton method, Appl. Numer. Math. 59 (2009), no. 7, 1599-1611. https://doi.org/10.1016/j.apnum.2008.11.002
24. T. Steihaug, Quasi-Newton methods for large scale nonlinear problems, Ph.D Thesis, Res. Rep. 49, School of Organization and Management, Yale University, New Hacen, CT, 1980.
25. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey, 1964.
26. S. Weerakon and T. G. I. Fernando, A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000), no. 8, 87-93.
27. T. Yamamoto, Historical developments in convergence analysis for Newton's and Newton-like methods, J. Comput. Appl. Math. 124 (2000), no. 1-2, 1-23. https://doi.org/10.1016/S0377-0427(00)00417-9
28. T. J. Ypma, Historical development of the Newton-Raphson method, SIAM Rev. 37 (1995), no. 4, 531-551. https://doi.org/10.1137/1037125
29. T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (1984), no. 3, 583-590. https://doi.org/10.1137/0721040

#### Cited by

1. A new tool to study real dynamics: The convergence plane vol.248, 2014, https://doi.org/10.1016/j.amc.2014.09.061
2. Modifications of Newton’s method to extend the convergence domain vol.66, pp.1, 2014, https://doi.org/10.1007/s40324-014-0020-y