An overview of Block Gram-Schmidt methods and their stability properties

  • Date:
  • Time: 14:00 - 15:30
  • Address:
    Sokolovská 83, Praha
  • Room: Online via Zoom
  • Speaker: Kathryn Lund

Block Gram-Schmidt algorithms comprise essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This talk walks through a survey providing a comprehensive categorization of block Gram-Schmidt algorithms, especially those used in Krylov subspace methods to build orthonormal bases one block vector at a time. All 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, including s-step-like matrices. For every method, we numerically confirm known theoretical bounds in plots with varying condition numbers. A MATLAB package for reproducing the results and experimenting with new block Gram-Schmidt methods is hosted at a publicly available repository