Abstract
There are many current efforts toward developing mixed-precision numerical linear algebra algorithms, which can lead to speedups in real applications. Motivated by this, we aim to further the state-of-the-art in developing and analyzing mixed-precision variants of iterative methods.
In iterative methods based on Krylov subspaces, an orthogonal basis is generated by Arnoldi or Lanczos methods or their variants. In long recurrence methods such as GMRES, one needs to use an explicit orthogonalization scheme such as Gram-Schmidt to orthonormalize the vectors generated.
Block and s-step Krylov subspace methods improve the computational intensity versus their non-block counterparts. These algorithms require block orthogonalization schemes.
Recent efforts have focused on developing "low-synchronization" versions of block Gram Schmidt, which require only a single reduction per block. This synchronization avoidance, however, comes at the cost of a greater loss of orthogonality in finite precision.
In this talk, we explore the use of mixed precision within the reorthogonalized "low-sync" block Classical Gram-Schmidt algorithm. We show that the selective use of higher precision can lead to improved stability while retaining the desirable "low-sync" property.