Mathematics

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Mathematics

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Start date: 2020Subject: search.filters.subject.linear systems

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    Comparison of Refinement Accelerated Relaxation Iterative Techniques and Conjugate Gradient Technique for Linear Systems
    (Mathematical Association of Nigeria (MAN), 2022-10-05) Khaddejah James Audu
    Iterative 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.
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    Convergence of Triple Accelerated Over-Relaxation (TAOR) Method for M-Matrix Linear Systems
    (Islamic Azad University, Rasht, Iran, 2021-09-19) Khadeejah James Audu; Yusuph Amuda Yahaya; Rufus Kayode Adeboye; Usman Yusuf Abubakar
    In this paper, we propose some necessary conditions for convergence of Triple Accelerated Over-Relaxation (TAOR) method with respect to 𝑀 − coefficient matrices. The theoretical approach for the proofs is analyzed through standard procedures in the literature. Some numerical experiments are performed to show the efficiency of our approach, and the results obtained compared favourably with those obtained through the existing methods in terms of spectral radius of their iteration matrices