SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1365-1380, January 2021.
The block version of the classical Gram--Schmidt (\tt BCGS) method is often employed to efficiently compute orthogonal bases for Krylov subspace methods and eigenvalue solvers, but a rigorous proof of its stability behavior has not yet been established. It is shown that the usual implementation of \tt BCGS can lose orthogonality at a rate worse than $O(\varepsilon) \kappa^{2}({$\mathcalX$})$, where $\mathcal{X}$ is the input matrix and $\varepsilon$ is the unit roundoff. A useful intermediate quantity denoted as the Cholesky residual is given special attention and, along with a block generalization of the Pythagorean theorem, this quantity is used to develop more stable variants of \tt BCGS. These variants are proven to have $O(\varepsilon) \kappa^2({$\mathcalX$})$ loss of orthogonality with relatively relaxed conditions on the intrablock orthogonalization routine satisfied by the most commonly used algorithms. A variety of numerical examples illustrate the theoretical bounds.

中文翻译：

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古典革兰氏块变体的稳定性--施密特

SIAM 矩阵分析与应用杂志，第 42 卷，第 3 期，第 1365-1380 页，2021 年 1 月。
经典 Gram--Schmidt (\tt BCGS) 方法的块版本通常用于有效计算 Krylov 子空间方法和特征值求解器的正交基，但尚未建立其稳定性行为的严格证明。结果表明，\tt BCGS 的通常实现会以比 $O(\varepsilon) \kappa^{2}({$\mathcalX$})$ 更差的速率失去正交性，其中 $\mathcal{X}$ 是输入矩阵和 $\varepsilon$ 是单位舍入。一个有用的中间量表示为 Cholesky 残差受到了特别关注，并且与勾股定理的块推广一起，这个量被用来开发更稳定的\tt BCGS 变体。这些变体被证明具有 $O(\varepsilon) \kappa^2({$\mathcalX$})$ 的正交性损失，并且在最常用的算法满足的块内正交化例程上相对宽松的条件。各种数值例子说明了理论界限。