Relationship between the inexact Newton method and the continuous analogy of Newton's method

Authors

  • T. Zhanlav National University of Mongolia, Mongolia
  • O. Chuluunbaatar National University of Mongolia, Mongolia
  • G. Ankhbayar National University of Mongolia, Mongolia

DOI:

https://doi.org/10.33993/jnaat402-1047

Keywords:

inexact Newton method, continuous analogy of Newton's method
Abstract views: 262

Abstract

In this paper we propose two new strategies to determine the forcing terms that allow one to improve the efficiency and robustness of the inexact Newton method. The choices are based on the relationship between the inexact Newton method and the continuous analogy of Newton's method. With the new forcing terms, the inexact Newton method is locally \(Q\)-superlinearly and quadratically convergent. Numerical results are presented to support the effectiveness of the new forcing terms.

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References

J. Stoer and R. Bulirsch, Introduction to numerical analysis, Third edition, Springer, 2002. DOI: https://doi.org/10.1007/978-0-387-21738-3

E. Zeidler, Nonlinear functional analysis and its applications, part 1: Fixed-point theorems, Springer-Verlag, 1986. DOI: https://doi.org/10.1007/978-1-4612-4838-5_1

L.V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat. Nauk, 3:6(28), pp. 89-185, 1948, (in Russian).

M.K. Gavurin, Nonlinear functional equations and continuous analogues of iterative methods, Izv. Vyssh. Uchebn. Zaved. Mat., 5(6), pp. 18-31, 1958.

T. Zhanlav and I.V. Puzynin, The convergence of iterations based on the continuous analogy of Newton's method, Comput. Maths. Math. Phys., 32, pp. 846-856, 1992. (in Russian).

T. Zhanlav and O. Chuluunbaatar, A local and semilocal convergence of the continuous analogy of Newton's method for solving nonlinear equations, Bulletin of PFUR Series Mathematics. Information Sciences. Physics, 1, pp. 34-43, 2012.

R.S. Dembo, S.C. Eisenstat and T. Steihang, Inexact Newton methods, SIAM J. Numer. Anal., 19, pp. 400-408, 1982, https://doi.org/10.1137/0719025 DOI: https://doi.org/10.1137/0719025

E. Cătinaş, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models, Math. Comp., 74, pp. 291-301, 2004, https://doi.org/10.1090/s0025-5718-04-01646-1 DOI: https://doi.org/10.1090/S0025-5718-04-01646-1

X.-C. Cai, W.D. Gropp, D.E. Keyes and M.D. Tidriti, Newton-Krylov-Schwarz methods in CFD, Proceeding of the international workshop on numerical methods for the Navier-stokes equations, Vieweg, Braunschwieg, pp. 17-30, 1995, https://doi.org/10.1007/978-3-663-14007-8_3 DOI: https://doi.org/10.1007/978-3-663-14007-8_3

S.C. Eisenstat and M.F. Walker, Choosing the forcing term in an inexact Newton method, SIAM J. Sci. Comput., 17, pp. 16-32, 1996, https://doi.org/10.1137/0917003 DOI: https://doi.org/10.1137/0917003

H.-B. An, Z.-Y. Mo and X.-P. Liu, A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math., 200, pp. 47-60, 2007. https://doi.org/10.1016/j.cam.2005.12.030 DOI: https://doi.org/10.1016/j.cam.2005.12.030

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Published

2011-08-01

How to Cite

Zhanlav, T., Chuluunbaatar, O., & Ankhbayar, G. (2011). Relationship between the inexact Newton method and the continuous analogy of Newton’s method. Rev. Anal. Numér. Théor. Approx., 40(2), 182–189. https://doi.org/10.33993/jnaat402-1047

Issue

Vol. 40 No. 2 (2011)

Section

Articles

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