Abstract
Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.
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