MATLAB Source Code for Least Squares Fitting Based on Matrix QR Decomposition

This article provides an in-depth exploration of MATLAB source code for least squares fitting based on matrix QR decomposition. The program utilizes QR factorization to significantly enhance both the accuracy and computational efficiency of data fitting operations. We will examine the fundamental principles of QR decomposition and demonstrate its practical implementation in MATLAB for curve fitting applications. The implementation leverages MATLAB's built-in qr() function, which efficiently factorizes the design matrix A into an orthogonal matrix Q and an upper triangular matrix R, thereby solving the normal equations Ax = b more stably than direct methods. The discussion covers how to incorporate custom parameters into the fitting algorithm, including regularization terms and weight matrices for weighted least squares scenarios. We will also explore integration techniques with other MATLAB functions and toolboxes, such as Optimization Toolbox for constrained fitting and Statistical Toolbox for error analysis. The code structure includes proper handling of overdetermined systems through the backslash operator (\) which internally uses QR decomposition when solving linear systems. Furthermore, we analyze the program's advantages, including improved numerical stability for ill-conditioned matrices and reduced computational complexity compared to traditional methods like normal equations. The limitations discussion covers cases where Singular Value Decomposition (SVD) might be preferable for rank-deficient problems. Practical application examples demonstrate usage scenarios such as polynomial curve fitting, surface approximation, and time-series analysis, complete with code snippets showing parameter configuration and result visualization using MATLAB's plotting capabilities. The implementation follows MATLAB best practices with proper matrix preallocation, efficient memory usage, and clear documentation comments to ensure maintainability. Error handling mechanisms are included to address common issues like rank deficiency and matrix dimension mismatches.