非线性特征值问题的数值方法及其应用

发布时间：2019-01-30 17:07

【摘要】：非线性特征值问题出现在量子点数值模拟、计算电磁场、粘弹性结构的振动分析、时滞系统的稳定性分析等许多领域,其研究具有重要的理论意义和应用价值。本文研究求解非线性特征值问题的数值方法,并将围道积分方法应用于判定n维时滞系统的稳定性,主要创新工作如下:提出了求解非线性特征值问题的逐次m次近似方法,给出了该方法的收敛性结果。为解决逐次m次近似方法产生的大规模多项式特征值问题,结合求解多项式特征值问题的部分正交投影方法,发展了求解非线性特征值问题的一个有效的逐次m次近似方法。为降低基于奇异值分解的Newton方法的计算量,采用逆迭代技术,每步近似计算最小奇异值以及相应的左右奇异向量,发展了改进的Newton方法,并证明了改进的Newton方法具有局部二次收敛性。研究了确定某一区域内非线性特征值问题的特征值代数重数的方法。将幅角原理推广到有理数情形,给出了求非线性特征值问题在某一区域内所有特征值及相应代数重数的数值方法;为了计算多项式特征值问题在一个半开平面的所有特征值,基于分式线性映射,建立了一个多项式特征值问题在半开平面内的所有特征值与求另一个多项式特征值问题在开圆盘中所有特征值的关系,提出了计算多项式特征值问题在半开平面内所有特征值的围道积分方法。对n维线性多时滞系统,利用围道积分方法,提出了计算该系统最右端特征值的数值方法,发展了判定n维线性多时滞系统在正平衡点处局部稳定性的新方法;对n维非线性多时滞Lotka-Volterra系统,给出了判定Lotka-Volterra系统在平衡点处局部稳定性的数值方法。数值结果说明了本文所给求解非线性特征值问题的数值方法都是有效的。
[Abstract]:Nonlinear eigenvalue problems appear in many fields, such as numerical simulation of quantum dots, calculation of electromagnetic fields, vibration analysis of viscoelastic structures, stability analysis of time-delay systems, and so on. In this paper, the numerical method for solving nonlinear eigenvalue problems is studied, and the circumferential integral method is applied to determine the stability of n-dimensional time-delay systems. The main innovations are as follows: a successive m-order approximation method for solving nonlinear eigenvalue problems is proposed. The convergence results of the method are given. In order to solve the large scale polynomial eigenvalue problem generated by the successive m approximation method, an effective successive m order approximation method for solving nonlinear eigenvalue problems is developed by combining the partial orthogonal projection method for solving the polynomial eigenvalue problems. In order to reduce the computational complexity of the Newton method based on singular value decomposition, the inverse iteration technique is used to calculate the minimum singular value and the corresponding left and right singular vectors in each step, and the improved Newton method is developed. The local quadratic convergence of the improved Newton method is proved. The method of determining the algebraic multiplicity of nonlinear eigenvalue problems in a certain region is studied. The amplitude-angle principle is extended to the case of rational numbers, and a numerical method for finding all eigenvalues and corresponding algebraic multiplicity of nonlinear eigenvalue problems in a certain region is given. In order to calculate all eigenvalues of the polynomial eigenvalue problem in a semi-open plane, based on fractional linear mapping, The relation between all eigenvalues of a polynomial eigenvalue problem in a semi-open plane and all eigenvalues of another polynomial eigenvalue problem in an open disk is established. A method for calculating all eigenvalues of polynomial eigenvalue problems in semi-open plane is presented. For n-dimensional linear multi-delay systems, a numerical method for calculating the eigenvalues at the right end of the system is presented by means of the circumscribed integral method, and a new method for determining the local stability of n-dimensional linear multi-delay systems at the positive equilibrium point is developed. For n-dimensional nonlinear multi-delay Lotka-Volterra systems, a numerical method for determining the local stability of Lotka-Volterra systems at the equilibrium point is presented. The numerical results show that the numerical methods for solving nonlinear eigenvalue problems are effective.
【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.6

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