Gaussian Elimination for Solving Linear Equations with MATLAB Implementation

In MATLAB, Gaussian elimination is a widely used method for solving linear equations. This algorithm systematically transforms the coefficient matrix into an upper triangular matrix or reduced row echelon form through sequential elimination steps. The process involves forward elimination to create zeros below the main diagonal, followed by back substitution to solve for the unknown variables. In practical implementations, MATLAB comments should include specifications for input matrix format (such as n×n coefficient matrix and n×1 constant vector) and output result format (solution vector). The MATLAB source code typically utilizes nested loops for elimination operations and implements partial pivoting to enhance numerical stability by preventing division by small pivot elements. Furthermore, through MATLAB source code analysis and optimization techniques like vectorization and pre-allocation, the algorithm's efficiency can be significantly improved while maintaining higher solution accuracy. Key functions often include matrix normalization, pivot selection, and backward substitution routines that ensure robust performance for various equation systems.