We present an algorithm for the solution of Sylvester equations with right-hand side of low rank. The method is based on projection onto a block rational Krylov subspace, with two key contributions with respect to the state-of-the-art. First, we show how to maintain the last pole equal to infinity throughout the iteration, by means of pole reodering. This allows for a cheap evaluation of the true residual at every step. Second, we extend the convergence analysis in [Beckermann B., An error analysis for rational Galerkin projection applied to the Sylvester equation, SINUM, 2011] to the block case. This extension allows to link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field-of-values of the involved matrices. This is in contrast with the non-block case, where the minimum problem is scalar, instead of matrix-valued. Replacing the norm of the objective function with an easier to evaluate function yields several adaptive pole selection strategies, providing a theoretical analysis for known heuristics, as well as effective novel techniques.

翻译：本文提出一种求解低秩右端项Sylvester方程的算法。该方法基于块有理Krylov子空间投影，相对于现有技术具有两项关键贡献。首先，通过极点重排技术，我们展示了如何在迭代过程中始终保持最后一个极点趋于无穷大，从而实现对每次迭代真实残差的低成本评估。其次，我们将文献[Beckermann B., An error analysis for rational Galerkin projection applied to the Sylvester equation, SINUM, 2011]中的收敛性分析推广至块情形。该推广使得收敛性与在相关矩阵谱或数值域上极小化小型有理矩阵范数的问题相关联——这与非块情形（其中极小化问题是标量而非矩阵值）形成鲜明对比。将目标函数范数替换为更易评估的函数后，可衍生出多种自适应极点选取策略，既为已知启发式方法提供了理论分析，也提出了有效的新型技术。