A rigorous inexact Newton method with applications to boundary value problems

Autores

  • Eduardo Ramos
  • Victor Hugo Nolasco
  • Marcio Gameiro

DOI:

https://doi.org/10.5540/03.2021.008.01.0345

Palavras-chave:

Differential equations, Inexact Newton Method, Newton-Kantorovich Theorem, Rigorous Numerics

Resumo

We introduce a semi-local theorem for the feasibility and convergence of the inexact Newton method  Uk+i = Uk — DF(uk)~rF(uk) + rfc, where rk represents the error in each step. Unlike the previous  results of this type in the literature, we prove the feasibility of the inexact Newton method under the  minor hypothesis that the error rk is bounded by a small constant to be computed, and moreover we  prove results concerning the convergence of the sequence Uk to the solution under this hypothesis.  We present an application of the method to rigorously compute zeros for two-point boundary value  problems.

Downloads

Não há dados estatísticos.

Biografia do Autor

Eduardo Ramos

Instituto de Ciências Matemáticas e de Computação, USP, São Carlos, SP

Victor Hugo Nolasco

Instituto de Ciências Matemáticas e de Computação, USP, São Carlos, SP

Marcio Gameiro

Instituto de Ciências Matemáticas e de Computação, USP, São Carlos, SP

Referências

Argyros, I.K., Saíd Hilout, Ángel A. Magrehán. Robust semí-local convergence analysis for inexact Newton method, Applied Mathematics and Computation, 227:741-754, 2014. DOI: 10.1016/j.amc.2013.11.076.

Argyros, I.K., George, S., Senapati, K. Extending the applicability of the inexact Newton-HSS method for solving large systems of nonlinear equations, Numerical Algorithms, 83:333-353, 2020. DOI: 10.1007/sll075-019-00684-z.

Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer-Verlag New York, New York, 2011. DOI: 10.1007/978-0-387-70914-7.

Chow, S. N., Hale, J. K. Methods of bifurcation theory. Springer Science and Business Media, Grundlehren der mathematischen Wissenschaften, volume 251, number 1, 1982. DOI: 10.1007/978-1-4613-8159-4.

Ferreira, O. P., Svaiter, B. F. A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance. Journal of Complexity, 28(3):346-363, 2012. DOI: 10.1007/10.1016/j.jco.2012.02.002.

Kantorovich, L. V., Akilov, G. P. Functional analysis second edition. Pergamon Press, 1982. DOI: 10.1016/C2013-0-03044-7.

Ramos, E., Gameiro, M., Nolasco, V. Rigorous Enclosures of Solutions of Neumann Boundary Value Problems, arXiv preprint arXiv:2005.02755v2, 2020. DOI: to appear.

Shen, W., Li, C. Kantorovich-type convergence criterion for inexact Newton methods, Applied Numerical Mathematics, 59(7):1599-1611, 2009. DOI: 10.1016/j.apnum.2008.11.002.

Wu, M. A new semi-local convergence theorem for the inexact Newton methods. Applied mathematics and computation, 200(l):80-86, 2008. DOI: 10.1016/j.ame.2007.10.057.

Downloads

Publicado

2021-12-20

Edição

Seção

Trabalhos Completos