  This article introduces randomized block Gram-Schmidt process (RBGS) for QR
decomposition. RBGS extends the single-vector randomized Gram-Schmidt (RGS)
algorithm and inherits its key characteristics such as being more efficient and
having at least as much stability as any deterministic (block) Gram-Schmidt
algorithm.
  Block algorithms offer superior performance as they are based on BLAS3
matrix-wise operations and reduce communication cost when executed in parallel.
Notably, our low-synchronization variant of RBGS can be implemented in a
parallel environment using only one global reduction operation between
processors per block. Moreover, the block Gram-Schmidt orthogonalization is the
key element in the block Arnoldi procedure for the construction of a Krylov
basis, which in turn is used in GMRES, FOM and Rayleigh-Ritz methods for the
solution of linear systems and clustered eigenvalue problems. In this article,
we develop randomized versions of these methods, based on RBGS, and validate
them on nontrivial numerical examples.
