Direct Methods for Solving Linear Equation Systems

In this article, we will discuss the collection of commonly used algorithms in MATLAB language, specifically focusing on direct methods for solving linear equation systems. First, it is essential to understand what linear equation systems are and why solving them is important. Linear equation systems consist of sets of linear equations where each equation contains the same variables. Solving linear equation systems helps us determine the values of these variables, which is crucial for numerous engineering and scientific applications. We will then explore several widely-used direct solution algorithms including Gaussian Elimination, LU Decomposition, and Cramer's Rule. For each algorithm, we will provide detailed explanations of their fundamental principles and implementation approaches. The implementation discussion will include key MATLAB functions such as backslash operator (\), lu(), and det() for matrix operations, along with practical code examples demonstrating how to apply these methods effectively. The article will also cover MATLAB-specific optimizations and best practices for handling large-scale systems efficiently.