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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.
Fan et al. (Mon,) studied this question.