Iterative Methods for Solving Linear Equations

We can use MATLAB programming to solve linear equation systems. In addition to direct solution methods, iterative approaches provide an alternative solution strategy. Iterative methods are based on stepwise approximation techniques that progressively approach the exact solution through multiple iterations of the equation system. In MATLAB, we can implement various iterative methods such as Jacobi iteration, Gauss-Seidel iteration, and Successive Over-Relaxation (SOR) method. These algorithms typically involve matrix decomposition and iterative updates using MATLAB's vectorized operations for efficient computation. Additionally, MATLAB's matrix operations can be utilized for direct solution of linear equations through matrix inversion. However, it's important to note that when dealing with high-dimensional matrices, the computational cost of matrix inversion becomes prohibitively large, making this approach unsuitable for large-scale systems. For better performance with high-dimensional problems, iterative methods with appropriate convergence criteria and preconditioning techniques are recommended.