SVD Decomposition Method for Least Squares Estimation with Algorithm Implementation

This article presents the Singular Value Decomposition (SVD) method for solving least squares estimation problems. The implementation first converts the estimation task into an overdetermined system formulation Ax=b, where the matrix A may be rank-deficient. The core algorithm employs MATLAB's svd() function or numpy.linalg.svd() in Python to decompose matrix A into U, Σ, and V* matrices, enabling stable computation of pseudo-inverse solutions. Key implementation aspects include: 1) Automatic rank detection via thresholding singular values to handle ill-conditioned systems, 2) Memory-efficient batch processing for large datasets using economy-size SVD, and 3) Regularization techniques for noise resilience. The method demonstrates particular efficiency with massive datasets due to optimized LAPACK backend operations, achieving O(mn²) computational complexity for m×n matrices. Numerical stability is enhanced through condition number monitoring and gradual singular value decay handling, making it superior for noisy data scenarios compared to normal equations approaches. Consequently, this program provides fast, accurate least squares solutions with built-in robustness mechanisms ideal for large-scale and noise-contaminated data processing.