One of the limitations of recycled GCRO methods is the large amount of computation required to orthogonalize the basis vectors of the newly generated Krylov subspace for the approximate solution when combined with those of the recycle subspace. Recent advancements in low synchronization Gram-Schmidt and generalized minimal residual algorithms, Swirydowicz et al.~\cite{2020-swirydowicz-nlawa}, Carson et al. \cite{Carson2022}, and Lund \cite{Lund2022}, can be incorporated, thereby mitigating the loss of orthogonality of the basis vectors. An augmented Arnoldi formulation of recycling leads to a matrix decomposition and the associated algorithm can also be viewed as a {\it block} Krylov method. Generalizations of both classical and modified block Gram-Schmidt algorithms have been proposed, Carson et al.~\cite{Carson2022}. Here, an inverse compact $WY$ modified Gram-Schmidt algorithm is applied for the inter-block orthogonalization scheme with a block lower triangular correction matrix $T_k$ at iteration $k$. When combined with a weighted (oblique inner product) projection step, the inverse compact $WY$ scheme leads to significant (over 10$\times$ in certain cases) reductions in the number of solver iterations per linear system. The weight is also interpreted in terms of the angle between restart residuals in LGMRES, as defined by Baker et al.\cite{Baker2005}. In many cases, the recycle subspace eigen-spectrum can substitute for a preconditioner.

翻译：回收的GCRO 方法的一个局限性是,如果将新生成的Krylov 子空间的基础矢量与循环子空间的基础矢量结合,则需要大量计算来对新生成的Krylov 子空间的基础矢量进行正向化,从而减轻基础矢量的损耗或振动的权重。在低同步Gram-Schmidt和通用最低残余算法方面最近的进展,Swirydowicz 等人的cite{202020-swirydowicz-nlawa}、Carson et al.\cite{Carson2022}和 Lund\cite{Lund2022},可以合并,从而减轻基础矢量矢量的基量矢量矢量的损耗或度的权重力。 强化的 Arnoldi 配制导致基质变形和相关的算法也可被视为 Qrylov 方法。 传统块-Schmidal 值算法中的经典和修改块-ral-lickral deal deal cal cal cal rodeal roup roup rouption rout rout rout rout rout 10 rout roduction roduction roduction roduction roducal roducal cal routs rout rout 10 roducs roduc) 中, 内, roduclex 内, 内, 内, 内,也可以以某种硬化法中, 10 。