Extended convergence analysis of Newton-Potra solver for equations

Ioannis Argyros; Stepan M. Shakhno; Yurii Shunkin; Halyna Yarmola

doi:10.33993/jnaat492-1186

Authors

Ioannis Argyros Cameron University, USA https://orcid.org/0000-0002-9189-9298
Stepan M. Shakhno Ivan Franko National University of Lviv, Ukraine https://orcid.org/0000-0002-3845-6260
Yurii Shunkin Ivan Franko National University of Lviv, Ukraine
Halyna Yarmola Ivan Franko National University of Lviv, Ukraine https://orcid.org/0000-0002-8986-2509

DOI:

https://doi.org/10.33993/jnaat492-1186

Keywords:

nonlinear equation, nondifferentiable operator, local and semi-local convergence, order of convergence, divided difference

Abstract views: 244

Abstract

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort.

Numerical experiments are carried out on the test examples with nondifferentiable operator.

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References

W.C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, SIAM, Philadelphia, 1998, DOI: https://doi.org/10.1137/1.9781611970012

L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, UK. (1982), https://doi.org/10.1016/b978-0-08-023036-8.50024-2 DOI: https://doi.org/10.1016/B978-0-08-023036-8.50024-2

P.P. Zabrejko, D.F. Nguen, The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optim., 9 (1987), 671–686, https://doi.org/10.1080/01630568708816254 DOI: https://doi.org/10.1080/01630568708816254

I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., (2004), 374-397, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008

I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, NY, USA, (2008), https://doi.org/10.1007/978-0-387-72743-1_5 DOI: https://doi.org/10.1007/978-0-387-72743-1_5

I.K. Argyros, A.A. Magrenan, A Contemporary Study of Iterative Methods, Elsevier (Academic Press), New York, NY, USA, (2018), https://doi.org/10.1016/B978-0-12-809214-9.00024-3 DOI: https://doi.org/10.1201/9781315153469-4

I.K. Argyros, S.M. Shakhno, Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions, Mathematics, 7(2) (2019) 207, https://doi.org/10.3390/math7020207 DOI: https://doi.org/10.3390/math7020207

E. Catinas, On some iterative methods for solving nonlinear equations, Revue d’Analyse Numerique et de Theorie de l’Approximation 23 (1994), 47–53 https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4

M.A. Hernandez, M.J. Rubio,The Secant method for nondifferentiable operators,J.Math. Anal. Appl., (2004), 374-397, https://doi.org/10.1016/s0893-9659(01)00150-1 DOI: https://doi.org/10.1016/S0893-9659(01)00150-1

R. Iakymchuk, S. Shakhno, H. Yarmola, Combined Newton-Kurchatov method forsolving nonlinear operator equations, Proc. Appl. Math. Mech., 16 (2016), 719-720, https://doi.org/10.1002/pamm.201610348 DOI: https://doi.org/10.1002/pamm.201610348

H. Ren, I.K. Argyros, A new semilocal convergence theorem for a fast iterative method with nondifferentiable operators, J. Appl. Math. Comp., 34 (2010) nos. 1-2., pp. 39-46, https://doi.org/10.1007/s12190-009-0303-0 DOI: https://doi.org/10.1007/s12190-009-0303-0

S.M. Shakhno, Convergence of combined Newton-Secant method and uniqueness of the solution of nonlinear equations, Visnyk Ternopil Nat. Tech., Univ. 69 (2013), 242–252. (In Ukrainian)

S.M. Shakhno, Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations, Journal of Computational and Applied Mathematics, 261, (2014), 378-386, https://doi.org/10.1016/j.cam.2013.11.018 DOI: https://doi.org/10.1016/j.cam.2013.11.018

S.M. Shakhno, O.P. Gnatyshyn, On an iterative algorithm of order 1.839. . . for solving the nonlinear least squares problems, Applied mathematics and computation, 161 (2005), (1), 253-264, https://doi.org/10.1016/j.amc.2003.12.025 DOI: https://doi.org/10.1016/j.amc.2003.12.025

S.M. Shakhno, I.V. Melnyk, H.P. Yarmola, Analysis of convergence of a combined method for the solution of nonlinear equations, Journal of Mathematical Sciences, 201 (2014) 1, 32-43, https://doi.org/10.1007/s10958-014-1971-3 DOI: https://doi.org/10.1007/s10958-014-1971-3

S.M. Shakhno, H.P. Yarmola, On the two-step method for solving nonlinear equations with nondifferentiable operator, Proc. Appl. Math. Mech., 12 (2012) 1, 617-618, https://doi.org/10.1002/pamm.201210297 DOI: https://doi.org/10.1002/pamm.201210297

S.M. Shakhno, H.P. Yarmola, On the convergence of Newton-Kurchatov method under the classical Lipschitz conditions, Journal of Computational and Applied Mathematics, 1 2016, 89-97.

S.M. Shakhno, H.P. Yarmola,Two-point method for solving nonlinear equation with nondifferentiable operator, Mat. Stud., 36 (2011) 2, 213–220. (in Ukrainian).

J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia. (1996), https://doi.org/10.1137/1.9781611971200 DOI: https://doi.org/10.1137/1.9781611971200

X. Wang, Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA Journal of Numerical Analysis., 20 (2000), 123-134, https://doi.org/10.1093/imanum/20.1.123 DOI: https://doi.org/10.1093/imanum/20.1.123

J.F. Traub, Iterative methods for the solution of equations, Prentice Hall, Englewood Cliffs, (1964).

F.A. Potra, On an iterative algorithm of order 1.839... for solving nonlinear operator equations, Numer. Funct. Anal. Optim., 7 (1984-1985) 1., 75-106, https://doi.org/10.1080/01630568508816182 DOI: https://doi.org/10.1080/01630568508816182

S.M. Shakhno, Iterative algorithm with convergence order 1,839... under the generalized Lipschitz conditions for the divided differences, Visnyk National University ”Lviv Politechnic” Ser. Phys.-Mat., 740 (2012), 62-65. (in Ukrainian).

S.M. Shakhno, O.M. Makukh, About iterative methods in conditions of Holder continuity of the divided differences of the second order, Matematychni Metody ta Fizyko-Mekhanichni Polya., 49 (2006) 2., 90-98. (in Ukrainian).

S.M. Shakhno, A.-V.I. Babjak, H.P. Yarmola, Combined Newton-Potra method for solving nonlinear operator equations, Journal of Computational and Applied Mathematics, Kyiv, 3 (2015), 120, 170-178. (in Ukrainian).

S.M. Shakhno, H.P. Yarmola, On convergence of Newton-Potra method under weak conditions, Visnyk Lviv Univ. Ser. Appl. Math. Inform., 25 (2017), 49–55. (In Ukrainian)

I.K. Argyros, A.A. Magrenan, Iterative Methods and Their Dynamics with Applications, CRC Press, New York, NY, USA, (2017), https://doi.org/10.1201/9781315153469 DOI: https://doi.org/10.1201/9781315153469