Enhancing Linear System Solving Through Third Refinement of Successive and Accelerated Over-Relaxation Methods
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Date
2024-03-27
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Çankaya University, Turkey
Abstract
One of the primary difficulties in linear algebra, considering its widespread application
in many different domains, is solving linear system of equations. It is nevertheless
apparent that there is a need for a quick, effective approach that can handle a variety of
linear systems. In the realm of large and sparse systems, iterative methods play a crucial
role in finding solutions. This research paper makes a significant contribution by
introducing an enhancement to the current methodology Successive and Accelerated
Over Relaxation methods, referred to as the "Third Refinement of Successive and
Accelerated Over Relaxation Methods." This new iterative approach demonstrates its
effectiveness when applied to coefficient matrices exhibiting properties such as 𝑀-
matrix, irreducible diagonal dominance, positive definiteness and symmetry
characteristics. Significantly, the proposed method substantially reduces the spectral
radius, resulting in fewer iterations and notably enhancing the convergence rate.
Numerical experiments were conducted to evaluate its performance compared to existing
second refinement of Successive and Accelerated Over Relaxation methods. The
outcomes underscore the "Third Refinement of Successive and Accelerated Over
Relaxation" methods potentially to boost the efficiency of solving linear systems, thus
rendering it a valuable asset within the arsenal of numerical methodologies utilized in
scientific and engineering realms
Description
This article presents two enhanced iterative methods—TRSOR and TRAOR—for efficiently solving linear systems, especially large and sparse ones. Derived as third refinements of existing SRSOR and SRAOR techniques, the proposed methods improve convergence speed and accuracy by significantly reducing spectral radius. Numerical experiments confirm their superiority, with TRAOR recommended for optimal performance.
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Citation
Audu, K. J., Oyetunji, M. O., & Essien, J. N. (2024). Enhancing Linear System Solving Through Third Refinement of Successive and Accelerated Over-Relaxation Methods. Cankaya Univerisity Journal of Science and Engineering, 21(1), 18-32.