Block Gram-Schmidt Algorithms and Their Stability Properties

Erin Carson, Kathryn Lund, Miroslav Rozloznik, Stephen Thomas

Research output: Contribution to journalArticlepeer-review

14 Scopus Citations


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, 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.
Original languageAmerican English
Pages (from-to)150-195
Number of pages46
JournalLinear Algebra and Its Applications
StatePublished - 2022

NREL Publication Number

  • NREL/JA-2C00-80324


  • block Krylov subspace methods
  • Gram-Schmidt
  • Krylov
  • loss of orthogonality
  • stability


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