Matrix Solution for Overdetermined Systems Using Least Squares Method

This article presents a comprehensive guide on solving overdetermined systems using the least squares method, with MATLAB-based matrix solution implementations. The least squares method is a widely-used fitting technique that finds optimal solutions for overfitting scenarios by minimizing the sum of squared residuals. When dealing with overdetermined equations (where there are more equations than unknowns), the matrix approach provides an efficient computational framework. We will detail the mathematical principles behind least squares optimization and demonstrate practical MATLAB implementation using key matrix operations.

The core algorithm involves solving the normal equations A^TAx = A^Tb, where A represents the coefficient matrix and b is the observation vector. In MATLAB, this can be efficiently implemented using the backslash operator (x = A\b) which automatically applies least squares solution for rectangular matrices. For enhanced numerical stability, we'll also explore QR decomposition and singular value decomposition (SVD) approaches using built-in functions like qr() and svd(). The implementation includes error handling for rank-deficient matrices and discusses computational efficiency considerations for large-scale problems.

Key MATLAB functions highlighted in our code examples include matrix transposition (A'), matrix multiplication, and specialized solvers for linear systems. We provide commented code segments showing step-by-step matrix operations, along with alternative implementations using pinv() (pseudo-inverse) for ill-conditioned systems. The article concludes with practical applications in curve fitting and data regression scenarios, enabling readers to both understand the theoretical foundations and apply these techniques effectively in scientific computing projects.