Gauss-Seidel Iterative Method for Solving Linear Equation Systems

The Gauss-Seidel iterative method is a numerical computation technique for solving systems of linear equations, particularly effective for large sparse matrices. Similar to the Jacobi iteration method, it approaches the solution through iterative refinement, but distinguishes itself by immediately utilizing newly updated values during computation, resulting in generally faster convergence rates. ### Fundamental Concept The core principle involves reformulating each equation in the system to express the current unknown variable using the latest approximations of other variables. During each iteration, calculations proceed sequentially from the first equation to the last, with newly computed values immediately utilized in subsequent equations. This real-time utilization of updated information significantly accelerates convergence compared to methods that wait until completing a full iteration cycle. ### Algorithm Workflow Initialization: Start with an initial guess for the solution vector, typically设置为零向量 or based on heuristic estimates. Iterative Computation: For each unknown variable, compute using the most recently updated values from the current iteration, progressively updating the solution vector. Convergence Criteria: Determine termination through error comparison between successive iterations or by setting maximum iteration limits. ### MATLAB Implementation Key Points When implementing Gauss-Seidel in MATLAB, matrix operations can optimize computational efficiency through: Matrix Decomposition: Decompose the coefficient matrix into lower triangular, diagonal, and upper triangular components (L, D, U matrices). Iterative Solution Update: Use loop structures to update solution components sequentially, ensuring immediate use of newest values in subsequent calculations. Error Control: Implement precision checks through relative error (norm(x_new - x_old)/norm(x_new)) or absolute error thresholds. The method's convergence depends on the coefficient matrix properties – strict diagonal dominance or symmetric positive definiteness generally ensure convergence. Practical applications often incorporate relaxation techniques like Successive Over-Relaxation (SOR) to further accelerate convergence. MATLAB implementation typically involves while-loops for iterations and norm calculations for error monitoring, with precomputed matrix inverses of triangular components for computational efficiency.