1. An improved Shifted CholeskyQR based on columns
- Author
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Fan, Yuwei, Guan, Haoran, and Qiao, Zhonghua
- Subjects
Mathematics - Numerical Analysis ,65F30 15A23 65F25 65G50 - Abstract
Shifted CholeskyQR3 is designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter $s$ to prevent failure during the initial Cholesky factorization step. The choice of this shift parameter $s$ is critical for the algorithm's effectiveness. Our goal is to identify a smaller $s$ compared to the traditional selection involving $\|X\|_2$. In this research, we propose a new norm $\|X\|_g$, which is based on the column properties of $X$, to obtain a reduced shift parameter $s$ for the Shifted CholeskyQR3 algorithm. We provide rigorous proofs of orthogonality and residuals for the improved algorithm with our proposed $s$. Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced $s$. Furthermore, we compare CPU times with other algorithms to assess performance improvements.
- Published
- 2024