Develop and Analyze Efficient Algorithms for Solving Large Linear SystemsUsing Analysis Techniques such as Numerical and Sparse Matrix Analysis

The study aims to develop effective numerical algorithms to solve large linear systems of the formula, as a separate matrix of huge dimensions is represented. This type of problem represents a real challenge in multiple areas such as numerical modeling, engineering analysis, and data analysis, due to its great mathematical ability. The study focuses on exploiting the structural characteristics of sporadic matrices with the aim of reducing the time cost and arithmetic resources. A group of numerical algorithms based on repetitive methods such as Jacobi, Gauss-Seidel,and Conjugate Gradient are designed, combining "pre-preparation" technologies to improve the rapprochement rate. Highly efficient software tools were used to implement the models, and a statistical analysis was made using SPSS to assess the performance ofalgorithms in terms of speed and accuracy. The sizes of systems that were tested ranged between 5,000 and 500,000 equations, and the results showed an improvement in the speed of treatment by up to 45, and the accuracy of solutions by 30 compared to traditional methods. The results of the statistical analysis indicate moral differences in support of the efficiency of the proposed models, which enhances their usefulness in applied mathematics applications.