关于循环约简算法中非对角线奇异值衰减的研究
On the decay of the off-diagonal singular values in cyclic reduction
AI总结：
本文研究了循环约简算法生成的矩阵序列中非对角线块奇异值的指数衰减性质，给出了理论界限并提供估计工具，应用于求解大规模块三对角块Toeplitz系统和广义Sylvester方程。
摘要：
最近发现，循环约简算法生成的矩阵序列中非对角线块的奇异值呈指数衰减。这一性质被用于更高效地求解排队模型分析中出现的二次矩阵方程。本文提供了该指数衰减的理论界限，并基于有理插值问题提供了估计工具。展示了应用该方法求解n×n块三对角块Toeplitz系统以及具有n×n半分离块的广义Sylvester方程在O(n²logn)算术运算内的可行性。
英文摘要：
It was recently observed that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queueing models. In this paper, we provide a sharp theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Applications to solving $n\times n$ block tridiagonal block Toeplitz systems with $n\times n$ semiseparable blocks and certain generalized Sylvester equations in $O(n^2\log n)$ arithmetic operations are shown.
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