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Item Comparison of Refinement Accelerated Relaxation Iterative Techniques and Conjugate Gradient Technique for Linear Systems(Mathematical Association of Nigeria (MAN), 2022-10-05) Khaddejah James AuduIterative methods use consecutive approximations to get more accurate results. A comparison of three iterative approaches to solving linear systems of this type 𝑀𝑦=𝐵 is provided in this paper. We surveyed the Refinement Accelerated Relaxation technique, Refinement Extended Accelerated Relaxation technique, and Conjugate Gradient technique, and demonstrated algorithms for each of these approaches in order to get to the solutions more quickly. The algorithms are then transformed into the Python language and used as iterative methods to solve these linear systems. Some numerical investigations were carried out to examine and compare their convergence speeds. Based on performance metrics such as convergence time, number of iterations required to converge, storage, and accuracy, the research demonstrates that the conjugate gradient method is superior to other approaches, and it is important to highlight that the conjugate gradient technique is not stationary. These methods can help in situations that are similar to finite differences, finite element methods for solving partial differential equations, circuit and structural analysis. Based on the results of this study, iteration techniques will be used to help analysts understand systems of linear algebraic equations.Item An Accelerated Iterative Technique: Third Refinement of Gauss–Seidel Algorithm for Linear Systems(Multidisciplinary Digital Publishing Institute, Switszerland, 2023-04-28) Khaddejah James Audu; James Nkereuwem EssienObtaining an approximation for the majority of sparse linear systems found in engineering and applied sciences requires efficient iteration approaches. Solving such linear systems using iterative techniques is possible, but the number of iterations is high. To acquire approximate solutions with rapid convergence, the need arises to redesign or make changes to the current approaches. In this study, a modified approach, termed the “third refinement” of the Gauss-Seidel algorithm, for solving linear systems is proposed. The primary objective of this research is to optimize for convergence speed by reducing the number of iterations and the spectral radius. Decomposing the coefficient matrix using a standard splitting strategy and performing an interpolation operation on the resulting simpler matrices led to the development of the proposed method. We investigated and established the convergence of the proposed accelerated technique for some classes of matrices. The efficiency of the proposed technique was examined numerically, and the findings revealed a substantial enhancement over its previous modifications