  With the recent realization of exascale performace by Oak Ridge National
Laboratory's Frontier supercomputer, reducing communication in kernels like QR
factorization has become even more imperative. Low-synchronization Gram-Schmidt
methods, first introduced in [K. \'{S}wirydowicz, J. Langou, S. Ananthan, U.
Yang, and S. Thomas, Low Synchronization Gram-Schmidt and Generalized Minimum
Residual Algorithms, Numer. Lin. Alg. Appl., Vol. 28(2), e2343, 2020], have
been shown to improve the scalability of the Arnoldi method in high-performance
distributed computing. Block versions of low-synchronization Gram-Schmidt show
further potential for speeding up algorithms, as column-batching allows for
maximizing cache usage with matrix-matrix operations. In this work,
low-synchronization block Gram-Schmidt variants from [E. Carson, K. Lund, M.
Rozlo\v{z}n\'{i}k, and S. Thomas, Block Gram-Schmidt algorithms and their
stability properties, Lin. Alg. Appl., 638, pp. 150--195, 2022] are transformed
into block Arnoldi variants for use in block full orthogonalization methods
(BFOM) and block generalized minimal residual methods (BGMRES). An adaptive
restarting heuristic is developed to handle instabilities that arise with the
increasing condition number of the Krylov basis. The performance, accuracy, and
stability of these methods are assessed via a flexible benchmarking tool
written in MATLAB. The modularity of the tool additionally permits generalized
block inner products, like the global inner product.
