## Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces.(English)Zbl 1238.65047

Summary: We study a modified Newton’s method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators
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### References:

 [1] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. · Zbl 0241.65046 [2] L. B. Rall, Computational Solution of Nonlinear Operator Equations, John Wiley & Sons Inc., New York, 1969. · Zbl 0186.21101 [3] A. Y. Özban, “Some new variants of Newton’s method,” Applied Mathematics Letters, vol. 17, no. 6, pp. 677-682, 2004. · Zbl 1065.65067 [4] M. Frontini and E. Sormani, “Some variant of Newton’s method with third-order convergence,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 419-426, 2003. · Zbl 1037.65051 [5] M. A. Noor and K. I. Noor, “Modified iterative methods with cubic convergence for solving nonlinear equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 322-325, 2007. · Zbl 1114.65048 [6] J. Kou and Y. Li, “The improvements of Chebyshev-Halley methods with fifth-order convergence,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 143-147, 2007. · Zbl 1118.65036 [7] J. Kou, “The improvements of modified Newton’s method,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 602-609, 2007. · Zbl 1122.65332 [8] F. Cianciaruso and E. de Pascale, “Estimates of majorizing sequences in the Newton-Kantorovich method: a further improvement,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 329-335, 2006. · Zbl 1123.65049 [9] J. Chen, I. K. Argyros, and R. P. Agarwal, “Majorizing functions and two-point Newton-type methods,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1473-1484, 2010. · Zbl 1191.65063 [10] I. K. Argyros, Y. J. Cho, and S. Hilout, “On the midpoint method for solving equations,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2321-2332, 2010. · Zbl 1198.65095 [11] C. Chun, P. St\uanic\ua, and B. Neta, “Third-order family of methods in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 6, pp. 1665-1675, 2011. · Zbl 1217.65101 [12] J. A. Ezquerro, M. A. Hernández, and M. A. Salanova, “Recurrence relations for the midpoint method,” Tamkang Journal of Mathematics, vol. 31, no. 1, pp. 33-41, 2000. · Zbl 1002.47049 [13] M. A. Hernández, “Chebyshev’s approximation algorithms and applications,” Computers & Mathematics with Applications, vol. 41, no. 3-4, pp. 433-445, 2001. · Zbl 0985.65058 [14] X. Ye and C. Li, “Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 294-308, 2006. · Zbl 1100.47057 [15] P. K. Parida and D. K. Gupta, “Recurrence relations for a Newton-like method in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 873-887, 2007. · Zbl 1119.47063 [16] X. Wang, C. Gu, and J. Kou, “Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces,” Numerical Algorithms, vol. 56, no. 4, pp. 497-516, 2011. · Zbl 1226.65052 [17] I. K. Argyros, J. A. Ezquerro, J. M. Gutiérrez, M. A. Hernández, and S. Hilout, “On the semilocal convergence of efficient Chebyshev-Secant-type methods,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3195-3206, 2011. · Zbl 1215.65102
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