发表机构
School of Mathematics and Statistics, Anhui University of Finance and Economics（安徽财经大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出带保护的Anderson加速反射（SAAR）方法求解连续Sylvester方程，通过逐列反射内迭代与残差保护的外层Anderson加速，实现线性收敛并显著减少迭代次数与CPU时间。
AI 中文摘要
我们提出了带保护的Anderson加速反射（SAAR）方法，这是一种用于连续Sylvester矩阵方程$AX+XB=C$的两层迭代方法。该方法的核心是一个不动点重构：在每次外层迭代中，相关的线性矩阵方程被分解为独立的逐列线性系统，这些系统通过反射迭代近似求解。对于非奇异的$A$且$n\ge 2$，我们通过谱半径论证证明了反射内层迭代的收敛性，并将该结果推广到矩阵方程情形。由于实际中只使用有限次内层迭代，整体方案被分析为一种不精确不动点迭代，并在Frobenius范数下建立了未加速的不精确迭代线性收敛的充分条件。在此方案之上，SAAR将Anderson加速引入外层迭代，同时一种基于残差的保护机制拒绝那些Sylvester残差大于标准不精确反射迭代的Anderson候选解，从而增强了数值稳定性。数值实验表明，SAAR大幅减少了外层迭代次数，并且通常减少了CPU时间，同时对于所测试的问题，它提高了关于内层容差的经验鲁棒性。
英文摘要
We propose the safeguarded Anderson-accelerated reflection (SAAR) method, a two-level iterative method for the continuous Sylvester matrix equation $AX+XB=C$. The core of the method is a fixed-point reformulation: at each outer iteration, the associated linear matrix equation is decomposed into independent column-wise linear systems, which are solved approximately by a reflection iteration. For nonsingular \(A\) and \(n\ge 2\), we prove the convergence of the reflection inner iteration by a spectral-radius argument and extend the result to the matrix equation setting. Since only finitely many inner iterations are used in practice, the overall scheme is analyzed as an inexact fixed-point iteration, and a sufficient condition for linear convergence of the unaccelerated inexact iteration is established in the Frobenius norm. On top of this scheme, SAAR incorporates Anderson acceleration into the outer iteration, while a residual-based safeguard rejects Anderson candidates whose Sylvester residual is larger than that of the standard inexact reflection iterate, thereby enhancing numerical stability. Numerical experiments indicate that SAAR substantially reduces the number of outer iterations and often the CPU time, and it improves empirical robustness with respect to the inner tolerance for the tested problems.
