×

On a two-step Kurchatov-type method in Banach space. (English) Zbl 1411.65077

Summary: We present the semi-local convergence analysis of a two-step Kurchatov-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Argyros, I.K.: Computational Theory of Iterative Methods. Series: Studies in Computational Mathematics, 15, Editors: C.K.Chui and L. Wuytack. Elsevier, New York (2007)
[2] Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008) · Zbl 1153.65057
[3] Argyros, I.K., Magreñañ, A.A.: Iterative Methods and Their Dynamics with Applications: A Contemporary Study. CRC Press, Cambridge (2017) · Zbl 1360.65005 · doi:10.1201/9781315153469
[4] Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high order iterative methods for solving nonlinear models. Appl. Math. Comput. 303(15), 70-88 (2017) · Zbl 1411.65074
[5] Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218, 11496-11504 (2012) · Zbl 1278.65067
[6] Madru, K., Jayaraman, J.: Some higher order Newton-like methods for solving system of nonlinear equations and its applications. Int. J. Appl. Comput. Math. 3, 2213-2230 (2017) · Zbl 1397.65075 · doi:10.1007/s40819-016-0234-z
[7] Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 227-236 (2000) · Zbl 0952.47050 · doi:10.1007/s002459911012
[8] Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A.: Avoiding the computation of the second-Fréchet derivative in the convex acceleration of Newton’s method. J. Comput. Appl. Math. 96, 1-12 (1998) · Zbl 0945.65060 · doi:10.1016/S0377-0427(98)00083-1
[9] Ezquerro, J.A., Hernández, M.A.: Multipoint super-Halley type approximation algorithms in Banach spaces. Numer. Funct. Anal. Optim. 21(7&8), 845-858 (2000) · Zbl 0969.65047 · doi:10.1080/01630560008816989
[10] Ezquerro, J.A., Hernández, M.A.: A modification of the super-Halley method under mild differentiability condition. J. Comput. Appl. Math. 114, 405-409 (2000) · Zbl 0959.65074 · doi:10.1016/S0377-0427(99)00348-9
[11] Grau-Sanchez, M., Grau, A., Noguera, M.: Ostrowski type methods for solving systems of nonlinear equations. Appl. Math. Comput. 218, 2377-2385 (2011) · Zbl 1243.65056
[12] Gutiérrez, J.M., Magren̄án, A.A., Romero, N.: On the semi-local convergence of Newton-Kantorovich method under center-Lipschitz conditions. Appl. Math. Comput. 221, 79-88 (2013) · Zbl 1329.65101
[13] Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982) · Zbl 0484.46003
[14] Magreńãn, A.A.: Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 233, 29-38 (2014) · Zbl 1334.65083
[15] Magreńãn, A.A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 29-38 (2014) · Zbl 1338.65277
[16] Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Processes. In: Research Notes in Mathematics, Vol. 103, Pitman, Boston (1984) · Zbl 0549.41001
[17] Shakhno, S.M.: On a Kurchatov’s method of linear interpolation for solving nonlinear equations. PAMM Proc. Appl. Math. Mech. 4, 650-651 (2004) · Zbl 1354.65108 · doi:10.1002/pamm.200410306
[18] Shakhno, S.M.: On the difference method with quadratic convergence for solving nonlinear equations. Matem. Stud 26, 105-110 (2006). (In Ukrainian) · Zbl 1122.65357
[19] Sharma, J.R., Arora, H.: On efficient weighted-Newton methods for solving system of nonlinear equations. Appl. Math. Comput. 222, 497-506 (2013) · Zbl 1329.65106
[20] Sharma, J.R., Arora, H.: Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193-210 (2014) · Zbl 1311.65052 · doi:10.1007/s10092-013-0097-1
[21] Sharma, J.R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67, 591-601 (2014) · Zbl 1350.65048 · doi:10.1016/j.camwa.2013.12.004
[22] Sharma, J.R., Guha, R.K., Sharma, R.: An efficient fourth-order weighted Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307-323 (2013) · Zbl 1283.65051 · doi:10.1007/s11075-012-9585-7
[23] Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.