Gauss-Seidel Iterative Method for Solving Linear Equations

The Gauss-Seidel iterative method is a numerical technique for solving systems of linear equations. It operates by progressively refining approximate solutions to approach the exact solution through successive iterations. This iterative algorithm works by sequentially updating each solution component using the most recently computed values of other components in each step, generating improved approximations. In MATLAB implementation, key programming considerations include: - Initial guess selection strategies (commonly using zero vectors or previous solutions) - Convergence criteria configuration (typically monitoring relative error changes between iterations) - Diagonal dominance verification to ensure method convergence - Implementation of iterative updates using matrix splitting techniques The core algorithm involves decomposing the coefficient matrix A into lower triangular (L), diagonal (D), and upper triangular (U) components, then applying the iterative formula: x^(k+1) = (L+D)^(-1) * (b - U*x^(k)) Programming implementation requires careful handling of: - Matrix inversion efficiency through forward substitution - Loop structures for component-wise updates - Tolerance settings for convergence detection - Maximum iteration limits to prevent infinite loops Proper debugging and validation against known solutions are essential to ensure computational accuracy and algorithm reliability. The method particularly benefits from MATLAB's vectorization capabilities for efficient large-scale system solutions.