中文摘要：

把原则，建议的胡和 Reichel 基于概括最小的剩余(GMRES ) 为 Sylvester 方程的一个最小的剩余算法。算法要求一个结构化的最少的广场问题的答案。他们形成最少的广场问题的正常方程然后由一个直接解答者解决它，因此它产生不稳定性。在这篇论文，由利用最少的广场问题的特殊结构并且直接研究这个问题，数字地稳定的 QR 分解基于算法为这个问题被介绍。新算法比胡和 Reichel 的正常方程算法更稳定。数字实验被报导证实新算法的优异稳定性。

英文摘要：

Based on the generalized minimal residual （GMRES） principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.