Local convergence of a Hansen-Patrick-like family of optimal fourth order methods. (English) Zbl 1433.65096
Summary: We present a local convergence analysis of an optimal fourth order Hansen-Patricklike family of methods in order to approximate a solution of a nonlinear equation. Earlier studies use hypotheses involving derivatives up to the third order to show convergence although only the first derivative appears in these methods. In the present study we use only hypotheses on the first derivative. We also provide computable error bounds on the distances involved based on Lipschitz constants. This way we expand the applicability of these methods. Numerical examples are also given in this study.
MSC:
| 65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
| 65H05 | Numerical computation of solutions to single equations |
References:
| [1] | Amat, S., Hern´andez, M. A., Romero,N., (2012), Semilocal convergence of a sixth order iterative method for quadratic equations, Applied Numerical Mathematics, 62, pp.833-841. · Zbl 1387.65049 |
| [2] | Argyros, I. K., (2007), Computational Theory of Iterative Methods. Series: Studies in Computational Mathematics, 15, Editors: C.K.Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A. · Zbl 1147.65313 |
| [3] | Argyros, I. K., (2011), A semilocal convergence analysis for directional Newton methods. Math.Comput., 80, pp.327-343. · Zbl 1211.65057 |
| [4] | Argyros, I. K., Hilout. S., (2012), Weaker conditions for the convergence of Newton’s method. J. Complexity, 28, pp.364-387. · Zbl 1245.65058 |
| [5] | Argyros, I. K., Hilout, S., (2013), Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications, World Scientific. · Zbl 1279.65062 |
| [6] | Argyros, I.K., Ren, H., (2012), Improved local analysis for certain class of iterative methods with cubic convergence, Numerical Algorithms, 59, pp.505-521. · Zbl 1317.65127 |
| [7] | Chun, C., Lee, M. Y., Neta, B., Dzunic, J., (2012), On optimal fourth order iterative methods free from second derivative and their dynamics, Appl. Math. Comput., 218, pp.6427-6438. · Zbl 1277.65031 |
| [8] | Cordero, A., Torregrosa, J. R., Vasileva, M. P., (2013), Increasing the order of convergence of iterative schemes for solving nonlinear systems, J. Comput. Appl. MAth., 252, pp.86-94. · Zbl 1288.65070 |
| [9] | Ezquerro, J. A., Gonzalez,D., Hern´andez, M. A., (2013), A modification of the classic conditions of NewtonKantorovich for Newtons method, Math. Comput. Model., 57, pp.584-594. · Zbl 1305.65144 |
| [10] | Ezquerro, J. A., Gonzalez, D., Hern´andez, M. A., (2012), A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type, Appl. Math. Comput., 218, pp.9536-9546. · Zbl 1245.65179 |
| [11] | Guti´errez, J. M., Magre¯n´an, A. A., Romero, N., (2013), On the semi-local convergence of Newton-Kantorovich method under center-Lipschitz conditions, Applied Mathematics and Computation, 221, pp.79-88. · Zbl 1329.65101 |
| [12] | Hansen, E., Patrick, M., (1977), A family of root finding methods, Numer. Math. 27, pp.257-269. · Zbl 0361.65041 |
| [13] | Hern´andez, M. A., Salanova, M. A., (2000), Modification of the Kantorovich assumptions for semi-local convergence of the Chebyshev method, Journal of Computational and Applied Mathematics, 126, pp.131-143. · Zbl 0985.65057 |
| [14] | Kansal, M., Kanwar. V., Bhatia, S., (2015), New modifications of Hansen-Patrick’s family with optimal fourth and eighth orders of convergence, Appl. Math. Comput., 269(15), pp.507-519. · Zbl 1410.65163 |
| [15] | Kantorovich, L. V., Akilov, G. P., (1982), Functional Analysis, Pergamon Press, Oxford. · Zbl 0484.46003 |
| [16] | Kou, J. S., Li. Y. T., Wang, X. H., (2006), A modification of Newton method with third-order convergence, Appl. Math. Comput., 181, pp.1106-1111. · Zbl 1172.65021 |
| [17] | Magrenan, A. A., (2014), Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput., 233, pp.29-38. · Zbl 1334.65083 |
| [18] | Petkovic, M. S., Neta, B., Petkovic, L., Dˇzuniˇc, J., (2013), Multipoint Methods for Solving Nonlinear Equations, Elsevier. · Zbl 1286.65060 |
| [19] | Rheinboldt, W. C., (1977), An adaptive continuation process for solving systems of nonlinear equations, In: Mathematical models and numerical methods (A.N.Tikhonov et al. eds.) pub.3, 129-142 Banach Center, Warsaw Poland. · Zbl 0378.65029 |
| [20] | Traub, J. F., (1982), Iterative Methods for the Solution of Equations, AMS Chelsea Publishing. · Zbl 0472.65040 |
| [21] | Tu˘g, O., Ba¸sar, F., (2016), On the spaces of N¨orlund null and N¨orlund convergent sequences, TWMS J. Pure Appl. Math., 7(1), pp.76-87. · Zbl 1508.40006 |
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