(untitled) (Al-Farooq Journal of Sciences, 2026)
The purpose of this work is to review numerical algorithms for the solution of linear systems and determine their convergence characteristics. Gaussian Elimination and LU Decomposition were found to be Direct methods while Jacobi Method, Gauss-Seidel Method and Conjugate Gradient Method were found to be Iterative methods. The study called for constructing and solving linear systems by using matrices of different dimensions and characteristics. Performance measures such as, number of iterations, execution time and the error norm were computed in order to assess the efficiency of the algorithms. The results proved that direct methods are ideal for small to medium size systems since the percentage accuracy is much higher than the iterative methods taking reasonable amount of time for execution. However, when compared to the direct methods, the iterative methods were more efficient for large and sparse systems of equations and the conjugate gradient method was the most efficient and fastest. This work also looked at certain effects like the condition numbers and distribution of eigenvalues and also the choice of the initial guess. This confirmed that preconditioning for instance boosts the convergence of iterative methods in a given structure. This work can be used as a guide for choosing algorithms suited to system size and characteristics and as a practical guide on how to enhance the numerical performance. Moreover, it calls for the additional study and extension of numerical algorithms and liability of employing contemporary technologies including artificial intelligence.