BIT Numerical Mathematics ( IF 1.5 ) Pub Date : 2021-06-19 , DOI: 10.1007/s10543-021-00847-2 Hyun-Min Kim , Jie Meng
This paper is concerned with the generalized Sylvester equation \(AXB+CXD=E\), where A, B, C, D, E are infinite size matrices with a quasi Toeplitz structure, that is, a semi-infinite Toeplitz matrix plus an infinite size compact correction matrix. Under certain conditions, an equation of this type has a unique solution possessing the same structure as the coefficient matrix does. By separating the analysis on the Toeplitz part with that on the correction part, we provide perturbation results that cater to the particular structure in the coefficient matrices. We show that the Toeplitz part is well-conditioned if the whole problem, without considering the structure, is well-conditioned. Perturbation results that are illustrated through numerical examples are applied to equations arising in the analysis of a Markov process and the 2D Poisson problem.
中文翻译:
无限尺寸拟托普利兹矩阵方程的结构化微扰分析及其应用
本文关注的是广义 Sylvester 方程\(AXB+CXD=E\),其中A , B , C , D , E是具有准 Toeplitz 结构的无限尺寸矩阵,即半无限 Toeplitz 矩阵加上无限尺寸紧致校正矩阵。在某些条件下,此类方程具有唯一解,其结构与系数矩阵相同。通过将 Toeplitz 部分的分析与校正部分的分析分开,我们提供了迎合系数矩阵中特定结构的扰动结果。我们表明,如果不考虑结构的整个问题是良好条件的,则 Toeplitz 部分是良好条件的。通过数值示例说明的扰动结果适用于分析马尔可夫过程和二维泊松问题时出现的方程。