Successive Over-Relaxation (SOR) Iterative Method for Solving Linear Equations

The Successive Over-Relaxation (SOR) method is an iterative technique for solving linear systems of equations. It is an improved version of the Gauss-Seidel method that introduces a relaxation factor ω during each iteration to enhance convergence stability and smoothness. This approach is particularly effective for linear systems with positive definite coefficient matrices, though it can sometimes be applied to non-positive definite matrices under certain conditions. Implementation steps for the SOR method: 1. Select the relaxation factor ω, typically within the range 0 < ω < 2. 2. Initialize the solution vector x^(0). 3. For iterations k = 0,1,2,..., repeat the following process: - For each component i = 1,2,...,n, compute the updated value using the formula: x_i^{(k+1)} = (1-ω)x_i^{(k)} + (ω/a_ii)[b_i - Σ_{j=1}^{i-1} a_ij x_j^{(k+1)} - Σ_{j=i+1}^{n} a_ij x_j^{(k)}] - Check convergence criteria (e.g., norm of residual below tolerance), terminating if satisfied. Key implementation considerations: The SOR method typically requires storing the coefficient matrix in a suitable format (dense or sparse) and implementing efficient summation operations. The convergence speed depends heavily on the optimal choice of ω, which can be determined through spectral radius analysis or experimental tuning. For large-scale systems, the algorithm benefits from sequential updates where new component values are immediately used in subsequent calculations within the same iteration. The primary advantage of SOR is its accelerated convergence rate compared to basic iterative methods, making it particularly suitable for large-scale linear systems. However, convergence behavior is highly sensitive to the relaxation factor selection, and multiple iterations may be required to achieve satisfactory accuracy in some cases.