The Gearhart-Koshy acceleration for the Kaczmarz method for linear systems is a line-search with the unusual property that it does not minimize the residual, but the error. Recently one of the authors generalized the this acceleration from a line-search to a search in affine subspaces. In this paper, we demonstrate that the affine search is a Krylov space method that is neither a CG-type nor a MINRES-type method, and we prove that it is mathematically equivalent with a more canonical Gram-Schmidt-based method. We also investigate what abstract property of the Kaczmarz method enables this type of algorithm, and we conclude with a simple numerical example.

翻译：针对线性方程组Kaczmarz方法的Gearhart-Koshy加速是一种线搜索技术，其特殊之处在于不极小化残差而极小化误差。近期，其中一位作者将该加速方法从线搜索推广至仿射子空间搜索。本文论证该仿射搜索法既非CG类亦非MINRES类的Krylov子空间方法，并证明其在数学上等价于更经典的基于Gram-Schmidt的算法。我们进一步探究Kaczmarz方法中何种抽象性质支撑了此类算法，最后通过简单数值算例进行总结。