An improved Shifted CholeskyQR based on columns
<|reference_start|>An improved Shifted CholeskyQR based on columns: 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.<|reference_end|>
@article{fan2024an,
    title={An improved Shifted CholeskyQR based on columns},
    author={Yuwei Fan, Haoran Guan, Zhonghua Qiao},
    journal={arXiv preprint arXiv:2408.06311},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.06311},
    primaryClass={math.NA cs.NA}
}