Example of Least Squares Method in Numerical Computation with MATLAB Implementation

In this example, we will demonstrate how to implement the least squares method for numerical computation in MATLAB. The least squares method is a widely used mathematical technique for data fitting and parameter estimation, with applications spanning economics, physics, and engineering disciplines. We will begin by generating synthetic data points, then employ least squares to fit a simple linear model. The implementation will highlight MATLAB's capabilities for linear algebra and matrix operations, utilizing built-in functions like the backslash operator (\) for efficient parameter calculation. Key steps include: 1. Constructing the design matrix using Vandermonde structure for linear regression 2. Applying the normal equations solution: β = (XᵀX)⁻¹Xᵀy 3. Evaluating model quality through residuals calculation and R-squared metrics The example will demonstrate practical code segments for: - Data generation with controlled noise using randn() function - Matrix formulation for linear least squares problems - Parameter estimation via MATLAB's mldivide algorithm - Fit assessment using residual analysis and statistical measures Through this exercise, you will gain practical experience in implementing least squares methodology in MATLAB, building foundational skills for advanced numerical computations. The approach emphasizes computational efficiency through MATLAB's optimized matrix operations and provides insights into algorithm selection for different data characteristics.