Using Mixed Precision in Low‐Synchronization Reorthogonalized Block Classical Gram‐Schmidt

Mixed precision hardware has recently become commercially available, and more than 25% of the supercomputers in TOP500 list now have mixed capabilities. Using lower algorithms can be beneficial terms reducing both computation communication costs. There are many current efforts towards developing numerical linear algebra algorithms, which lead to speedups real applications. Motivated by this, we aim further state-of-the-art analyzing variants iterative methods. In methods based on Krylov subspaces, orthogonal basis is generated Arnoldi or Lanczos their variants. long recurrence such as GMRES, one needs use an explicit orthogonalization scheme Gram-Schmidt orthonormalize vectors generated. subspace methods, typically communication-bound modern machines; runtime usually dominated cost global synchronizations required for necessary orthogonalization. This motivated development various algorithmic attempt reduce while maintaining a stable algorithm. Recent work focused low-synchronization block orthogonalization, which, when used within communication-avoiding (s-step) number per block. this work, focus variant classical with reorthogonalization, call BCGSI+LS. We demonstrate that loss orthogonality produced exceed O(u)κ(