A stable one-synchronization variant of reorthogonalized block classical Gram--Schmidt

The block classical Gram–Schmidt (BCGS) algorithm and its reorthogonalized variant are widely-used methods for computing the economic QR factorization of block columns $X$ due to their lower communication cost compared to other approaches such as modified Gram–Schmidt and Householder QR. To further reduce communication, i.e., synchronization, there has been a long ongoing search for a variant of reorthogonalized BCGS variant that achieves $O(u)$ loss of orthogonality while requiring only \emph{one} synchronization point per block column, where $u$ represents the unit roundoff. Utilizing Pythagorean inner products and delayed normalization techniques, we propose the first provably stable one-synchronization reorthogonalized BCGS variant, demonstrating that it has $O(u)$ loss of orthogonality under the condition $O(u){\kappa }^2(X)\le 1\mathrm{/}2$, where $\kappa (\cdot )$ represents the condition number. By incorporating one additional synchronization point, we develop a two-synchronization reorthogonalized BCGS variant which maintains $O(u)$ loss of orthogonality under the improved condition $O(u)\kappa (X)\le 1\mathrm{/}2$. An adaptive strategy is then proposed to combine these two variants, ensuring $O(u)$ loss of orthogonality while using as few synchronization points as possible under the less restrictive condition $O(u)\kappa (X)\le 1\mathrm{/}2$. As an example of where this adaptive approach is beneficial, we show that using the adaptive orthogonalization variant, $s$-step GMRES achieves a backward error comparable to $s$-step GMRES with BCGSI+, also known as BCGS2, both theoretically and numerically, but requires fewer synchronization points.