MATLAB Implementation of Gaussian Elimination Algorithm

In mathematics, Gaussian elimination is a fundamental algorithm for solving systems of linear equations. This method is typically applied to systems with lower dimensions since computational efficiency decreases significantly for higher-order matrices. The core algorithm consists of two main phases: forward elimination transforms the coefficient matrix into an upper triangular form through elementary row operations, while back substitution solves for variables starting from the last equation. The MATLAB implementation typically utilizes matrix operations and pivoting strategies to enhance numerical stability. However, this approach has limitations when dealing with ill-conditioned systems or equations requiring high precision, as rounding errors may accumulate during elimination steps. Key functions in MATLAB implementation include matrix indexing for row operations and nested loops for elimination procedures. Therefore, when selecting equation-solving methods, one must consider factors like matrix condition number, computational complexity (O(n³) for Gaussian elimination), and precision requirements to make appropriate trade-offs.