MATLAB Implementation for Solving Ill-conditioned Equation Systems

This document presents a MATLAB implementation for solving ill-conditioned systems of equations. The program incorporates four distinct numerical algorithms: LU decomposition, Jacobi iteration, Gauss-Seidel iteration, and Successive Over-Relaxation (SOR) iteration. Users can select their preferred algorithm by specifying the input parameter M, which controls the method selection through a switch-case structure in the code implementation. LU decomposition is a matrix factorization technique that decomposes a coefficient matrix into a lower triangular matrix (L) and an upper triangular matrix (U). In MATLAB, this is implemented using the lu() function, which can handle partial pivoting to enhance numerical stability when dealing with ill-conditioned matrices. The solution is then obtained through forward and backward substitution. Jacobi iteration is an iterative method that approximates solutions through cyclic updates of all variables using previous iteration values. The MATLAB implementation typically involves initializing a solution vector and iterating until convergence criteria are met, with diagonal dominance verification for guaranteed convergence. Gauss-Seidel iteration is an enhanced version of Jacobi method that utilizes immediately updated values during iterations, leading to faster convergence. The code implementation optimizes this by overwriting solution components sequentially, reducing storage requirements and improving computational efficiency. SOR iteration introduces a relaxation parameter (ω) to accelerate convergence, where optimal ω values can significantly reduce iteration counts. The MATLAB code allows parameter tuning through experimental analysis or theoretical estimates, with conditional checks for convergence stability. This flexible MATLAB program enables users to select appropriate algorithms based on problem characteristics, providing easily modifiable code structure for customization and performance optimization in solving challenging ill-conditioned systems.