MATLAB Implementation of SOR Iterative Method for Numerical Computation

In numerical computation, iterative methods are widely used techniques for solving various problems, including linear system solutions, function root finding, and numerical integration. Among these, the Successive Over-Relaxation (SOR) iterative method is specifically designed for solving linear equation systems and can significantly accelerate convergence rates, particularly in large-scale problems. Implementing the SOR iterative method in MATLAB is straightforward, requiring only basic loop structures for computation. The algorithm typically involves initializing a solution vector, then iteratively updating each component using a weighted average of the previous iteration and the Gauss-Seidel method. The relaxation parameter ω (omega) plays a crucial role in convergence optimization - values between 1 and 2 typically enhance convergence speed. Key implementation aspects include: - Matrix decomposition into diagonal, lower, and upper triangular components - Implementation of the SOR update formula: x_i^{(k+1)} = (1-ω)x_i^{(k)} + (ω/a_ii)[b_i - Σ(a_ij*x_j^{(k+1)}) - Σ(a_ij*x_j^{(k)})] - Convergence checking using residual norms or maximum iteration counts - Parameter tuning for optimal performance across different problem types This MATLAB implementation allows for easy parameter adjustment to optimize algorithm performance, making SOR iteration an effective numerical approach for practical applications requiring efficient linear system solutions.