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Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

1
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
2
Department of Mathematical and Computational Sciences, National Institute of Technology, Karnataka 575 025, India
3
Departamento de Matemáticas y Computación, Universidad de la Rioja, 26006 Logroño, Spain
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 981; https://doi.org/10.3390/sym11080981
Received: 24 June 2019 / Revised: 17 July 2019 / Accepted: 25 July 2019 / Published: 2 August 2019
(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)
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Abstract

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton–Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299–315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection–diffusion equations further validate the theoretical results. View Full-Text
Keywords: Newton–HSS method; systems of nonlinear equations; semi-local convergence Newton–HSS method; systems of nonlinear equations; semi-local convergence
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Argyros, I.K.; George, S.; Godavarma, C.; Magreñán, A.A. Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method. Symmetry 2019, 11, 981.

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