Lyapunov解的快速奇异值衰减与非正常系数
Fast singular value decay for Lyapunov solutions with nonnormal coefficients
AI总结：
研究Lyapunov方程在低秩右端项下的解的奇异值衰减特性，揭示非正常系数对奇异值衰减的影响，发现非正常系数偏离度阈值外，奇异值衰减速度可能加快。
摘要：
具有低秩右端项的Lyapunov方程通常具有快速衰减的奇异值，这使得迭代方法能够生成低秩近似解。所有已知的奇异值衰减界都涉及与系数矩阵偏离正常性的二次函数，这些界暗示偏离度越大，奇异值衰减越慢。本文证明这种关系仅在阈值内成立，超过该阈值后，更大的偏离度反而可能导致奇异值更快衰减：如果奇异值衰减缓慢，则数值范围无法延伸到右半平面。
英文摘要：
Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities that depend quadratically on the departure of the coefficient matrix from normality: these bounds suggest that the larger the departure from normality, the slower the singular values will decay. We show this is only true up to a threshold, beyond which a larger departure from normality can actually correspond to faster decay of singular values: if the singular values decay slowly, the numerical range cannot extend far into the right-half plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
