重正交化分块经典Gram-Schmidt算法
Reorthogonalized Block Classical Gram--Schmidt
AI总结：
提出一种新的重正交化分块经典Gram-Schmidt算法，用于QR分解，在浮点运算中保证Q的列正交性和分解的数值精度，并改进了现有CGS2算法的误差界。
摘要：
提出了一种新的重正交化分块经典Gram-Schmidt算法，该算法将满列秩矩阵$A$分解为$A=QR$，其中$Q$是左正交的（具有标准正交列），$R$是上三角且非奇异的。在对$R$的对角块进行适当假设的情况下，该算法在机器精度为$\macheps$的浮点运算中实现时，产生的$Q$和$R$满足$\| I- Q^{T} Q \|_2 =O(\macheps)$和$\| A-QR \|_2 =O(\macheps \| A \|_2)$。所得结果还改进了Giraud等人[Num. Math., 101(1):87-100, 2005]对Abdelmalek [BIT, 11(4):354--367, 1971]最初提出的CGS2算法的先前误差界。
英文摘要：
A new reorthogonalized block classical Gram--Schmidt algorithm is proposed that factorizes a full column rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular and nonsingular. With appropriate assumptions on the diagonal blocks of $R$, the algorithm, when implemented in floating point arithmetic with machine unit $\macheps$, produces $Q$ and $R$ such that $\| I- Q^{T} Q \|_2 =O(\macheps)$ and $\| A-QR \|_2 =O(\macheps \| A \|_2)$. The resulting bounds also improve a previous bound by Giraud et al. [Num. Math., 101(1):87-100,\ 2005] on the CGS2 algorithm originally developed by Abdelmalek [BIT, 11(4):354--367,\ 1971]. \medskip Keywords: Block matrices, Q--R factorization, Gram-Schmidt process, Condition numbers, Rounding error analysis.
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