  Communication-avoiding and pipelined variants of Krylov solvers are critical
for the scalability of linear system solvers on future exascale architectures.
We present low synchronization variants of iterated classical (CGS) and
modified Gram-Schmidt (MGS) algorithms that require one and two global
reduction communication steps. Derivations of low synchronization iterated CGS
algorithms are based on previous work by Ruhe. Our main contribution is to
introduce a backward normalization lag into the compact $WY$ form of MGS
resulting in a ${\cal O}(\eps)\kappa(A)$ stable GMRES algorithm that requires
only one global synchronization per iteration. The reduction operations are
overlapped with computations and pipelined to optimize performance. Further
improvements in performance are achieved by accelerating GMRES BLAS-2
operations on GPUs.

Numerical Analysis