【作者】 蒋旭；

【导师】 周钢；

【作者基本信息】 上海交通大学 ， 计算数学， 2011， 硕士

【摘要】 大型辛矩阵特征值的计算在子结构链的动(静)力分析中十分重要,在离散时间最优控制大系统分析及金融数学中都有重要应用。在保证Hamilton结构始终不变的原则下来求解Hamilton矩阵的特征值是保证计算结果正确的最有效方式。常用的求解特征值的数值解法,只考虑数值精度,不考虑保证结构不变。保守体系有辛规律,Hamilton体系是保守体系,因此有必要探讨应用辛算法来计算各类Hamilton系统中的矩阵特征值。求解方式是多种多样的。VanLoan提出的平方约化法,保持了Hamilton结构,克服了常用的QR算法不能恰好保证每个半平面上都能求得n个特征值的缺陷。Benner提出了一个求解大型矩阵特征值问题的Lanczos方法,钟万勰建立了求解哈密尔顿矩阵特征值的共轭辛子空间逆迭代法和大型辛矩阵特征问题的逆迭代法。Bunse提出了求解实系数的Riccati方程的辛QR算法。本文的创新工作正是基于上述工作展开的。文中第1-2章是学术背景研究。第1章系统地介绍了Hamilton体系,并研究了Hamilton矩阵与辛矩阵的特性。第2章则介绍了矩阵特征值的常用数值计算方法。文中第3-4章主要是作者取得的一系列的创新成果,包括:从理论上建立了辛SL算法,分析了有效性和收敛性,以及如何用辛SL算法求解辛矩阵特征值。数值计算结果令人满意。文中的创新成果主要有:①就三种特殊形式的辛矩阵,建立了相应的求解特征值的算法。②针对上述三种矩阵,还建立了矩阵特征值的SL求解算法,并证明了算法的收敛性。③就反Hamilton矩阵,证明了也能应用SL算法有效求解特征值。本文的第5章,是作者按照“上海交通大学数学系硕士研究生毕业要求”的条例完成的,是在阅读、理解大量科技文献之后经思考提炼而撰写的综合报告。主要综述了辛算法在弹性力学、波动方程以及DNA弹性杆力学分析中的应用及隐式辛算法的稳定性分析。将不同背景下的实际问题转化为Hamilton系统,选取合适的辛差分格式求解。数值算例表明:Hamilton系统的辛算法数值解是十分可靠的,数值结果是收敛的。更多还原

【Abstract】 The calculation of the large symplectic matrix eigenvalue is very important to analyze the dynamic (static) force in sub-structural chain , it is also used in optimal control of large-scale systems in discrete-time as well as financial mathematics. The most effective way to make the answer correct is ensuring the Hamilton structure remains unchanged during the calculation process. The common numerical methods for solving eigenvalue, only consider the numerical accuracy, without ensuring the structure unchanged.A conservative system should be symplectic conservative. Hamilton system is a conservative system, so it is necessary to use symplectic algorithm to solve the Hamiltonian matrix eigenvalue or symplectic matrix eigenvalue. There are many varieties of solutions. VanLoan’s square reduced method maintains the structure of the Hamilton, and overcomes the defect of common QR algorithm and guarantees every half-plane can obtain n eigenvalues. Benner proposed a Lanczos method to solve the large matrix eigenvalue, Wanxie Zhong established a subspace-conjugate-inverse-iteration-method to solve the Hamiltonian matrix eigenvalue, and a inverse-iteration-method to solve the large symplectic matrix eigenvalue. Bunse proposed a symplectic QR algorithm to solve the Riccati equation with real coefficients.The chapters 1-2 in the text are academic background. The chapter 1 introduces the Hamilton system and the features of the Hamiltonian matrix and symplectic matrix. Chapter 2 describes the common numerical methods. The chapters 3-4 are a series of innovations achieved by author, including: the establishment of the symplectic SL algorithm , analyzing the effectiveness and convergence, and how to use symplectic SL algorithm to solve the symplectic matrix eigenvalue. The numerical results are satisfied.The innovations of the text are:①Propose three special symplectic matrices, then establish three corresponding eigenvalue algorithm.②For the three matrices, establish the symplectic SL algorithm, and prove the convergence of the algorithm.③Prove that the symplectic SL algorithm is effective to solve symplectic matrix eigenvalue.The chapter 5 are mainly defer to "Shanghaijiaotong University math department master the graduate sutdent to raise the rule "to complete. After reading and understanding massive science and technology literature the general report which after the ponder the refinement completes. It summary the applications of symplectic algorithm in elasticity, wave equation, the DNA elastic rod, and analyze the stability of the implicit symplectic algorithm. Transform different practical problems into Hamilton system, and select the appropriate symplectic difference scheme to solve. The numerical examples show: the symplectic numerical solutions of the Hamilton system are very reliable, numerical results are convergent.更多还原