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A Symplectic SL Algorithm of the Solution of the Symplectic Matrix Eigenvalues

Author: JiangXu
Tutor: ZhouGang
School: Shanghai Jiaotong University
Course: Computational Mathematics
Keywords: Hamilton systems symplectic matrix SL decomposition symplectic similarity transformation anti-Hamilton matrix SL algorithm
CLC: O241.6
Type: Master's thesis
Year: 2011
Downloads: 121
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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.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > Linear algebra method of calculating
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