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Article

Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems

1
Department of Mathematics, Cameron University, Lawton, OK 73505, USA
2
Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine
3
Department of Computational Mathematics, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(6), 978; https://doi.org/10.3390/sym12060978
Received: 25 April 2020 / Revised: 3 June 2020 / Accepted: 3 June 2020 / Published: 8 June 2020
(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)
A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the solution. In this article, such methods of higher convergence order are studied. In particular, we develop a two-step iterative method to solve large scale systems that does not require finding an inverse operator. Instead of the operator’s inverting, it uses a two-step Schultz approximation. The convergence is investigated using Lipschitz condition on the first-order derivatives. The cubic order of convergence is established and the results of the numerical experiment are given to determine the real benefits of the proposed method. View Full-Text
Keywords: nonlinear equation; iterative method; approximation of inverse operator; local convergence; order of convergence; Lipschitz condition nonlinear equation; iterative method; approximation of inverse operator; local convergence; order of convergence; Lipschitz condition
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MDPI and ACS Style

Argyros, I.K.; Shakhno, S.; Yarmola, H. Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems. Symmetry 2020, 12, 978. https://doi.org/10.3390/sym12060978

AMA Style

Argyros IK, Shakhno S, Yarmola H. Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems. Symmetry. 2020; 12(6):978. https://doi.org/10.3390/sym12060978

Chicago/Turabian Style

Argyros, Ioannis K., Stepan Shakhno, and Halyna Yarmola. 2020. "Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems" Symmetry 12, no. 6: 978. https://doi.org/10.3390/sym12060978

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