【作者】 张敏；

【导师】 王正盛；

【作者基本信息】 南京航空航天大学 ， 计算数学， 2017， 硕士

【摘要】 矩阵的伪谱作为分析矩阵(或算子)非正规性和特征值扰动的一种有效工具,在大气科学、控制理论、激光、水动力稳定性、微分方程数值解及矩阵迭代分析等诸多领域都有着重要的应用背景。本文研究矩阵伪谱的相关理论及其计算问题。首先,探讨了Companion矩阵伪谱问题,给出了Companion矩阵的一种伪谱定义和相应的一种快速计算方法,并利用Companion矩阵伪谱作为工具分析了Balance技术的作用;针对多项式零点扰动问题,进一步分析了Companion矩阵的结构伪谱问题。其次,探讨了矩阵Kronecker积的伪谱问题。针对两个矩阵的Kronecker积,分为四种不同的扰动情况,在每种情况下各提出了伪谱定义,并对其等价性进行了证明,同时推导出相应的伪谱计算方法。同理,给出了三个矩阵的Kronecker积在三种不同扰动情况下的伪谱定义。还给出了两个矩阵Kronecker积基于QR分解的伪谱定义、算法,以及实结构伪谱定义、算法。再次,探讨了矩阵伪谱计算区域的选取问题,给出了一种选择区域方法及区域选择的一般性建议,并改进了计算伪谱的区域排除法。最后,对文中提出的所有算法做了数值试验与比较,数值结果表明了所给出的理论和算法的正确性和有效性。更多还原

【Abstract】 The pseudospectra is a useful tool to explain the behavior of the systems associated with non-normal matrices and the eigenvalue perturbation theory,which have great backgrounds in many fields such as atmospheric science,control theory,lasers,hydrodynamic stability,numerical solution of differential equations and matrix iterations.In this paper,we study the theory of matrix pseudospectra and its computational problems.Firstly,the problem on the pseudospectra of companion matrix is discussed.A definition and a fast algorithm for the pseudospectra of companion matrix are analyzed,and the balance technique is presented using the pseudospectra as a tool.Moreover,the structured pseudospectra of companion matrix is discussed in order to analyse the perturbation of the zeros of polynomial.Secondly,the pseudospectra of Kronecker product is considered.For the Kronecker product of two matrices,the definitions of pseudospectra are given and the equivalence is proved,including four different disturbance situations.At the same time,the corresponding pseudospectral calculation method is deduced.Similarly,the pseudospectral definitions of Kronecker product of three matrices under three different disturbances are proposed.Furthermore,the definitions and algorithms of unstructured pseudospectra based on the QR factorization and real structured pseudospectra for Kronecker product of two matrices are derived.Besides,the selection of pseudospectral region is explored.A method of selecting region and general recommendations on regional options are given.Regional exclusion for calculating pseudospectra is improved.Finally,the numerical experiments and comparisons of the algorithms proposed in this paper are presented.The numerical results show the correctness and effectiveness of the proposed theory and algorithms.更多还原