Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-19T09:50:31.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Semilocal Convergence Theorem for a Newton-like Method

Part of: Nonlinear algebraic or transcendental equations Numerical linear algebra

Published online by Cambridge University Press:  07 September 2017

Rong-Fei Lin ,
Qing-Biao Wu ,
Min-Hong Chen ,
Lu Liu  and
Ping-Fei Dai
Rong-Fei Lin*
Affiliation:
Department of Mathematics, Taizhou University, Linhai 317000, Zhejiang, China
Qing-Biao Wu*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
Min-Hong Chen*
Affiliation:
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310012, Zhejiang, China
Lu Liu*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
Ping-Fei Dai*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
*
*Corresponding author. Email addresses:linrfei@tzc.edu.cn (R.-F. Lin), qbwu@zju.edu.cn (Q.-B. Wu), mhchen@zju.edu.cn (M.-H. Chen), liulu_flying@zju.edu.cn (L. Liu), pfdai@zju.edu.cn (P.-F. Dai)
*Corresponding author. Email addresses:linrfei@tzc.edu.cn (R.-F. Lin), qbwu@zju.edu.cn (Q.-B. Wu), mhchen@zju.edu.cn (M.-H. Chen), liulu_flying@zju.edu.cn (L. Liu), pfdai@zju.edu.cn (P.-F. Dai)
*Corresponding author. Email addresses:linrfei@tzc.edu.cn (R.-F. Lin), qbwu@zju.edu.cn (Q.-B. Wu), mhchen@zju.edu.cn (M.-H. Chen), liulu_flying@zju.edu.cn (L. Liu), pfdai@zju.edu.cn (P.-F. Dai)
*Corresponding author. Email addresses:linrfei@tzc.edu.cn (R.-F. Lin), qbwu@zju.edu.cn (Q.-B. Wu), mhchen@zju.edu.cn (M.-H. Chen), liulu_flying@zju.edu.cn (L. Liu), pfdai@zju.edu.cn (P.-F. Dai)
*Corresponding author. Email addresses:linrfei@tzc.edu.cn (R.-F. Lin), qbwu@zju.edu.cn (Q.-B. Wu), mhchen@zju.edu.cn (M.-H. Chen), liulu_flying@zju.edu.cn (L. Liu), pfdai@zju.edu.cn (P.-F. Dai)
Get access

Abstract

The semilocal convergence of a third-order Newton-like method for solving nonlinear equations is considered. Under a weak condition (the so-called γ-condition) on the derivative of the nonlinear operator, we establish a new semilocal convergence theorem for the Newton-like method and also provide an error estimate. Some numerical examples show the applicability and efficiency of our result, in comparison to other semilocal convergence theorems.

Keywords

Newton-like methodnonlinear equationNewton-Kantorovich theoremγ-conditionerror estimate

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices 65H10: Systems of equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 482 - 494
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Proinov, P.D., New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems, J. Complexity 26, 3–42 (2010).Google Scholar
[2] Ezquerro, J.A., González, D. and Hernández, M.A., A modification of the classic conditions of Newton-Kantorovich for Newton's method, Math. Comput. Modelling 57, 584–594 (2013).Google Scholar
[3] Shen, W.P. and Li, C., Kantorovich-type convergence criterion for inexact Newton methods, Appl. Numer. Math. 59, 1599–1611 (2009).Google Scholar
[4] Kantorovich, L., On Newton method (in Russian), Tr. Mat. Inst. Steklov. 28, 104–144 (1949).Google Scholar
[5] Smale, S., Newton's method estimates from data at one point, in The Merging of Disciplines: New Directions, Ewing, R., Gross, K., Martin, C. (Eds.), pp.185–196, Springer, New York (1986).Google Scholar
[6] Wang, X.H., Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42, 552–555 (1997).Google Scholar
[7] Kou, J.S., Li, Y.T. and Wang, X.H., Third-order modification of Newton's method, J. Comput. Appl. Math. 205, 1–5 (2007).Google Scholar
[8] Kou, J.S., Li, Y.T. and Wang, X.H., A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Comput. 186, 1258–1262 (2007).Google Scholar
[9] Darvishi, M.T. and Barati, A., A third-order Newton-type method to solve systems of nonlinear equations, Appl. Math. Comput. 187, 630–635 (2007).Google Scholar
[10] Chun, C., Construction of third-order modifications of Newton's method, Appl. Math. Comput. 189, 662–668 (2007).Google Scholar
[11] Ezquerro, J.A. and Hernández, M.A., On the R-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math. 197, 53–61 (2006).Google Scholar
[12] Frontini, M. and Sormani, E., Some variant of Newton's method with third-order convergence, Appl. Math. Comput. 140, 419–426 (2003).Google Scholar
[13] Frontini, M. and Sormani, E., Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149, 771–782 (2004).Google Scholar
[14] Frontini, M. and Sormani, E., Modified Newton's method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156, 345–354 (2003).Google Scholar
[15] Zhao, Y.Q. and Wu, Q.B., Convergence analysis for a deformed Newton's method with third-order in Banach space under γ-condition, Int. J. Comput. Math. 86, 441–450 (2009).Google Scholar
[16] Chen, M.H., Khan, Y., Wu, Q.B. and Yildirim, A., Newton-Kantorovich convergence theorem of a modified Newton's method under the Gamma-condition in a Banach Space, J. Optim. Theory Appl. 157, 651–662 (2013).Google Scholar
[17] Weerakoon, S. and Fernando, T.G.I., A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13, 87–93 (2000).Google Scholar
[18] Homeier, H.H.H., On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176, 425–432 (2005).CrossRefGoogle Scholar
[19] Gutiérrez, J.M. and Hernández, M.A., Newton's method under weak Kantorovich conditions, J. Numer. Anal. 20, 521–532 (2000).CrossRefGoogle Scholar
[20] Ezquerro, J.A. and Hernández, M.A., On Halley-type iterations with free second derivative, J. Comput. Appl. Math. 170, 455–459 (2004).Google Scholar
[21] Argyros, I.K., On Newton's method for solving equations containing Fréchet-differentiable operators of order at least two, Appl. Math. Comput. 215, 1553–1560 (2009).Google Scholar
[22] Wu, Q.B. and Zhao, Y.Q., Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput. 175, 1515–1524 (2006).Google Scholar
[23] Argyros, I.K. and Magreñán, A.A., On the convergence of inexact two-point Newton-like methods on Banach spaces, Appl. Math. Comput. 265, 893–902 (2015).Google Scholar
[24] Argyros, I.K., Cordero, A. and Magreñán, A.A., Torregrosa, J.R., On the convergence of a damped Newton-like method with modified right hand side vector, Appl. Math. Comput. 266, 927–936 (2015).Google Scholar
[25] Ezquerro, J.A., Gonzlez, D. and Hernndez-Vern, M.A., A semilocal convergence result for Newton's method under generalized conditions of Kantorovich, J. Complexity 30, 309–324 (2014).Google Scholar
[26] Argyros, I.K. and Hilout, S., On the local convergence of fast two-step Newton-like methods for solving nonlinear equations, J. Comput. Appli. Math. 245, 1–9 (2013).Google Scholar
[27] Ferreira, O.P., A robust semi-local convergence analysis of Newton's method for cone inclusion problems in Banach spaces under affine invariant majorant condition, J. Comput. Appl. Math. 279, 318–335 (2015).Google Scholar
[28] Argyros, I.K. and George, S., Unified convergence domains of Newton-like methods for solving operator equations, Appl. Math. Comput. 286, 106–114 (2016).Google Scholar