【作者】 李长伟；

【导师】 卢琳璋；

【作者基本信息】 厦门大学 ， 计算数学， 2008， 硕士

【摘要】 本文研究了解大型稀疏非线性特征值问题的一类迭代投影法。非线性有理Krvlov法和非线性Arnoldi法适用于求解非线性问题在一个区域内的多个特征值。二者都属于迭代投影法,在本质上是等价的。在实现方式上却有所不同。非线性Arnoldi方法可以采用高效的算法来求解投影小问题,在实现上更为灵活。一般说来其实现性能要好于非线性有理Krylov方法。但非线性有理Krylov法在计算过程中不必明确给出投影小问题,可能比较适用于某些应用问题。非线性有理Krylov法将非线性问题线性化,只能近似求解投影特征值问题。本文考虑在算法中引入精化策略,用精化向量来取代Ritz向量,形成了精化非线性有理Krvlov方法。鉴于对大型问题,矩阵分解有可能无法实现或占用了过多的资源而影响算法的效率,本文采用一个内层迭代来求解非线性Arnoldi方法中的预条件系统,得到不精确的非线性Arnoldi方法。文中给出了这些方法的实用算法,最后列出的数值算例说明了精化非线性有理Krvlov方法和不精确的非线性Arnoldi方法比原方法高效。更多还原

【Abstract】 A class of numerical methods for nonlinear large sparse eigenvalue problems are studied in this thesis.Rational Krylov method and nonlinear Arnoldi method arc effective methods to find all eigenvalues of a nonlinear eigenvalue problem in some region.Both algorithms are belong to iterative projection inethods and do indeed resemble each other.The difference between them lies only in the way of implementation.Nonlinear Arnoldi method has the advantage over nonlinear Krylov method that it can take a flexible and more effieient way to solve the projected eigenvalue problem.But nonlinear rational Krylov method does not require the explicit form of the projected eigenvalue problem and may be more suitable for some particular applications.Nonliner Krylov method use a linearization to approximate the nonlinear problem and solve the projected eigenvalue problem approximately.By making use of the idea of refined projection and adopting refined vector instead of Ritz vector,we present the refined nonlinear rational Krylov method.In view of the considerable expense of LU factorization for large system,we nse a inner iteration to solve the precondition system and propose the inexact nonlinear Arnotdi method. Practical algorithms are given for these methods and numerical examples with them indicate the improved two methods presented in this thesis are superior in speed of convergence.更多还原