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：本文提出了一种求解右端项为低秩的西尔维斯特方程的算法。该方法基于投影到块有理Krylov子空间，相比现有技术有两项关键贡献。首先，我们展示了如何通过极点重排，使得在整个迭代过程中最后一个极点始终保持为无穷大，从而能够廉价地评估每一步的真实残差。其次，我们将[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到块情形。该推广将收敛性与在相关矩阵的谱或数值域上极小化一个小型有理矩阵范数的问题联系起来。这与非块情形形成对比，后者中的极小化问题是标量而非矩阵值的。将目标函数的范数替换为更易评估的函数，可得出若干自适应极点选择策略，为已知启发式方法提供了理论分析，也生成了有效的新技术。