*Result*: RANDOMIZED BLOCK GRAM-SCHMIDT PROCESS FOR THE SOLUTION OF LINEAR SYSTEMS AND EIGENVALUE PROBLEMS.

*This article introduces randomized block Gram-Schmidt process (RBGS) for QR decomposition. RBGS extends the single-vector randomized Gram-Schmidt algorithm and inherits its key characteristics, such as being more efficient and at least as stable as any deterministic (block) Gram-Schmidt algorithm. Block algorithms offer superior performance, as they are based on BLASS matrixwise 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 Arnold! procedure for the construction of a Krylov basis, which in turn is used in GMRES, the full orthogonalization method, and the Rayleigh-Ritz method 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. Finally, we extend the proposed methodology to Krylov s-step methods. [ABSTRACT FROM AUTHOR]*