MATLAB Implementation of SOR Iterative Method for Numerical Computation

The SOR (Successive Over-Relaxation) iterative method is an effective numerical technique for solving linear equation systems, particularly suitable for large sparse matrices. As an accelerated version of the Gauss-Seidel method, it introduces a relaxation factor ω to adjust iteration step sizes, significantly improving convergence rates through strategic parameter tuning.

Fundamental Principles The core concept of SOR involves incorporating a relaxation factor ω (typically 1<ω<2) into the Gauss-Seidel iteration formula. By adjusting ω, the correction amount per iteration can be controlled: when ω=1, it reduces to standard Gauss-Seidel method; ω>1 enables over-relaxation for accelerated convergence; ω<1 provides under-relaxation, potentially enhancing stability for certain problem types.

Algorithm Key Points Relaxation factor selection: The optimal ω value usually requires theoretical estimation or trial-and-error determination, closely related to matrix properties Convergence conditions: Requires the coefficient matrix to be strictly diagonally dominant or symmetric positive definite, with ω in the range (0,2) Termination criteria: Typically uses residual norms or changes between successive iterative solutions as stopping conditions

MATLAB Implementation Considerations When implementing SOR in MATLAB, standard practices include: Matrix decomposition into lower triangular, diagonal, and upper triangular components Utilizing vectorized operations to avoid explicit loops for efficiency gains Dynamic residual calculation with real-time convergence monitoring Providing flexible input interfaces for the ω parameter adjustment

Application Scenarios SOR is particularly effective for numerical solutions of partial differential equations like heat conduction equations and Poisson equations, with extensive applications in image processing and computational fluid dynamics. Its convergence speed outperforms Jacobi and Gauss-Seidel methods, though actual performance heavily depends on proper ω selection.

Important Considerations Inappropriate ω values may lead to divergence Requires careful application for non-symmetric matrices Performance can be further optimized using modern iterative methods like conjugate gradient techniques