数学与统计学院知识库

Toeplitz系统具有丰富的应用背景, 其广泛地产生于科学计算, 信息论以及工程等众多的应用领域. 由于其通常表现出很强的病态性, 因此设计高效
预处理子改善相关算法的求解效率是有效求解这类问题的关键. 本文借鉴若干已知的有效预处理子的构造思路, 设计了一类既适用于病态对称的Toeplitz系统, 也适用于病态非对称的Toeplitz系统的新的预处理子. 此外, 利用Courant-Fischer极大极小定理分析了预处理矩阵的谱分布, 并且分析了相应Krylov子空间的收敛性质. 最后, 通过数值实验验证了所构造的处理子用来求解病态对称和非对称Toeplitz系统问题的可行性和有效性.

Toeplitz systems have a rich background, which arise widely form many application areas, such as scientific computing, information theory and engineering. Due to its general ill-conditioning properties, it is critical to design efficient preconditioners to improve the efficiency of the related solution algorithms, so that we can solve these problems effectively. On the basis of the construction ideas of several efficient preconditioners, a class of new preconditioners are proposed, which are applicable for both the Hermitian and the non-Hermitian ill-conditioned Toeplitz systems. In addition, spectral distributions of the preconditioned matrices are analyzed based on
the Courant-Fischer minimax theorem, and the convergence properties of the corresponding Krylov space methods are studied. Finally, numerical experiments are presented to verify the feasibility and effectiveness of the constructed preconditioners for solving both the Hermitian and the non-Hermitian Toeplitz systems.