【作者】 禚卫东；

【导师】 汪晓虹；

【作者基本信息】 南京航空航天大学 ， 计算数学， 2008， 硕士

【摘要】 子空间方法是计算大型稀疏对称矩阵特征值问题的有效数值方法。初始迭代向量的选取和算法的收敛率直接影响到子空间方法的计算效率,很多加速的子空间方法都是从这两个方面考虑的。为了加速子空间方法的收敛,本文在总结已有的加速方法的基础上提出了以下几种加速策略:首先,为了选取最佳迭代向量数值使得总运算量最小,本文详细分析了影响算法总运算量的一些参变量,在此基础上,提出了根据这些参变量选取最佳迭代向量数的新方法。其次,为了选取好的初始迭代向量,本文根据圆盘定理提出了新的选取策略。接下来,针对Qian和Dhatt与Lam和Bertolini分别提出的重复逆迭代的加速思想,提出了改进方法。针对Lam和Bertolini根据收敛率确定逆迭代的迭代向量,本文为了改进这种方法带来的误差,提出根据相邻特征值选取迭代向量的策略,提出了一种新的确定最大迭代步数的方法。根据不同实际问题,利用上述思想给出了两种具体的算法。最后,根据超松弛加速策略所体现的思想提出了新的改进方法,克服了F.A. Akl和K.J.Bathe加速子空间方法中计算松弛因子的困难。此外,在各章的后面对各种改进的算法都做了具体的数值实验。更多还原

【Abstract】 Subspace iteration method is one of efficient numerical methods for solution of large sparse symmetric eigenproblems. The efficient of subspace iteration method is dependent on the selection of initial vectors and convergence rate. Many accelerated subspace iteration methods is based on these two aspects. For obtain a higher convergence rate, the paper presents following accelerated-strategies based on summary of the existing accelerated methods.Firstly, for selecting the best number of iterative vectors which cause the total operation to be smallest, the paper analysis the parameters influence the total operation of algorithm. According to these parameters, a new method is proposed. Secondly, in order to select best initial vectors, a new selection method is proposed based on the circle theorem.Then, using the accelerated-strategy of extra inverse iteration which is presented by Qian and Lam, the improvement method is proposed. Lam and Bertolini select vectors that need to be extra inverse iteration by convergence rate. For reduction the error of this method, the new method using consistent eigenvalues is proposed. And one new method for determination biggest iteration step is introduced. Two algorithms are presented for different problems.Finally, using the accelerated-strategy of over-relaxation method, a method which improves the difficulty of computing over-relaxation factors is proposed. And the numerical examples of different algorithms are presented at the end of every chapter.更多还原