MATLAB Implementation for Solving Ill-Conditioned Equation Systems

Solving ill-conditioned equation systems is a classic problem in numerical computation where these systems are extremely sensitive to minor perturbations, making conventional solution methods prone to significant errors. MATLAB offers various numerical approaches to address such problems, primarily categorized into direct methods and iterative methods. LU Decomposition Method As a representative direct method, LU decomposition factors the coefficient matrix into a lower triangular matrix L and an upper triangular matrix U. This method is suitable for small-to-medium-scale systems with relatively good condition numbers. MATLAB's built-in `lu` function efficiently performs this decomposition, though users should note that ill-conditioned matrices may lead to numerical instability. Jacobi Iteration A straightforward iterative technique that decomposes the coefficient matrix into diagonal and remainder components for iteration. While simple to implement, this method converges slowly for ill-conditioned problems and may even diverge. It works best for diagonally dominant matrices. Gauss-Seidel Iteration (GS Iteration) An improved version of Jacobi iteration that utilizes the most recently computed variable values in each iteration. Typically offering faster convergence than Jacobi, it still depends heavily on matrix properties. For ill-conditioned systems, careful adjustment of relaxation factors is recommended. SOR Iteration (Successive Over-Relaxation) Accelerates GS iteration convergence by introducing a relaxation factor ω. The choice of ω is critical: values greater than 1 enable over-relaxation (accelerating convergence), while values less than 1 provide under-relaxation (enhancing stability). Parameter tuning should consider the matrix's spectral radius. Implementation Recommendations Use an input parameter `M` to switch between algorithms (e.g., 1=LU, 2=Jacobi) through a `switch-case` control structure. For iterative methods, implement maximum iteration limits and tolerance error thresholds to prevent infinite loops. Preconditioning techniques (such as matrix scaling) are recommended for ill-conditioned problems to improve numerical stability. This program framework allows comparison of different algorithms' performance on ill-conditioned systems. Important considerations include: direct methods' susceptibility to rounding errors, and iterative methods' convergence dependence on matrix properties. Practical applications should incorporate condition number analysis and residual checking to verify result reliability.