论文标题
块革兰氏 - schmidt方法及其稳定性的概述
An overview of block Gram-Schmidt methods and their stability properties
论文作者
论文摘要
在许多科学计算应用中，Block Gram-Schmidt算法是必不可少的内核，但是对于许多常用的变体，对其稳定性的严格处理仍然开放。这项工作提供了对块革兰氏 - schmidt算法的全面分类，尤其是在Krylov子空间方法中使用的算法，一次用于构建正统基础一个块向量。已知的稳定性结果组装，并为重要的交流变体总结或猜想了新的结果。此外，得出了新的低同步变体的新块版本，并且在广泛的挑战范围内证明了它们的功效和稳定性。数值示例是使用https://github.com/katlund/blockstab上托管的多功能MATLAB软件包计算的，并提供了用于复制本文中所有结果的脚本。讨论了流行软件包中的块革兰氏 - 施密特实现，以及许多开放问题。提供了包含所有算法类型集的附录，以统一的方式。
Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This work provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. Known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Numerical examples are computed with a versatile MATLAB package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.
