关于经典Gram-Schmidt误差分析的一个注记
A note on the error analysis of classical Gram-Schmidt
AI总结：
本文针对满秩矩阵A的经典Gram-Schmidt分解，给出了一种误差分析结果，表明当R的对角元按类似Cholesky分解的方式计算时，计算得到的R满足向后误差界。
摘要：
给出了满秩矩阵$A$的经典Gram-Schmidt分解$A=QR$的误差分析结果，其中$Q$是左正交矩阵（具有标准正交列），$R$是上三角矩阵。本文表明，计算得到的$R$满足$ ormal{R}= ormal{A}+E$，其中$E$是适当小的向后误差，但仅当$R$的对角元以类似于法方程矩阵的Cholesky分解的方式计算时成立。类似的结果在[Giraud等人, Numer. Math. 101(1):87–100,2005]中已有陈述。然而，要使该结果成立，$R$的对角元必须按照本文推荐的方式计算。
英文摘要：
An error analysis result is given for classical Gram--Schmidt factorization of a full rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular. The work presented here shows that the computed $R$ satisfies $\normal{R}=\normal{A}+E$ where $E$ is an appropriately small backward error, but only if the diagonals of $R$ are computed in a manner similar to Cholesky factorization of the normal equations matrix. A similar result is stated in [Giraud at al, Numer. Math. 101(1):87--100,2005]. However, for that result to hold, the diagonals of $R$ must be computed in the manner recommended in this work.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
