Numerical Stability Analysis of Matrix Decomposition in Solving Linear Systems

This paper investigates the numerical stability of QR decomposition, Singular Value Decomposition (SVD), and Cholesky decomposition in least squares problems. Through theoretical analysis and numerical experiments, the computational errors and efficiency of the three decomposition methods are compared for matrices with different condition numbers. The experimental results show that SVD decomposition exhibits the best robustness for ill-conditioned matrices, while Cholesky decomposition is the most efficient for well-conditioned matrices. Additionally, this paper compares the performance of direct solving (without decomposition) with decomposition methods, demonstrating that decomposition methods significantly outperform direct solving in terms of numerical stability and computational efficiency. To further validate the findings, we conduct experiments on both synthetic and real-world datasets, covering a range of matrix sizes and condition numbers. The results highlight the trade-offs between accuracy and computational cost, providing practical insights for selecting the appropriate decomposition method based on specific problem requirements. This study not only reinforces the theoretical understanding of matrix decompositions but also offers actionable guidelines for their application in scientific computing and data analysis