On inexact alternating direction implicit iteration for continuous Sylvester equations,Numerical Linear Algebra with Applications

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On inexact alternating direction implicit iteration for continuous Sylvester equations
Numerical Linear Algebra with Applications ( IF 1.8 ) Pub Date : 2020-07-20 , DOI: 10.1002/nla.2320
Zhongyun Liu ₁ , Yang Zhou ₁ , Yulin Zhang ₂

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In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C, where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.

中文翻译：

关于连续Sylvester方程的不精确交替方向隐式迭代

在本文中，我们研究了交替方向隐式（ADI）迭代求解连续Sylvester方程AX + XB = C的情况，其中系数矩阵A和B假定是正半定矩阵（不一定是Hermitian），并且至少其中之一是正定矩阵。我们首先分析ADI迭代的收敛性以解决此类Sylvester方程，然后得出该ADI迭代的收缩因子的上限。为了降低其计算复杂度，我们进一步提出了ADI迭代的不精确变体，该方法在外部ADI迭代的每个步骤中都采用一些Krylov子空间方法作为其内部迭代过程。还对收敛进行了详细分析。数值实验说明了ADI和不精确ADI迭代的有效性。

更新日期：2020-07-20

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