【作者】 梁娟；

【导师】 陈桂芝；

【作者基本信息】 厦门大学 ， 计算数学， 2006， 硕士

【摘要】 本文研究求解大规模非对称矩阵特征问题的几个理论及算法。本学位论文共分四章。 第一章介绍大规模非对称矩阵特征问题的来源、解决这类问题的基本方法以及与论文有关的研究方向及发展动态，并概述了本文的主要工作。 第二章给出了调和Arnoldi方法的一种变形。经过m步Arnoldi过程，实际上产生的是m+1个基向量{ν_i}_i=1^m+1以及在这组基下的限制矩阵H_m。传统的调和Arnoldi方法用在空间span{ν₁，ν₂，…，ν_m}中求得的调和Ritz向量作为特征向量的近似，第m+1个基向量ν_m+1对于特征向量的计算没有任何贡献。改进后的调和Arnoldi方法保留调和Ritz值作为特征值的近似，而在近似特征向量的选取方面用产生的调和Ritz向量与第m+1个基向量ν_m+1的一种巧妙的线性组合作为特征向量的近似。理论分析表明了这种新的方法的有效性。最后，数值结果也验证了改进的调和Arnoldi方法的有效性。 第三章研究了改进的块调和Arnoldi方法。假设m步的Arnoldi过程产生的上Hessenberg矩阵H_m为不可约的，即H_m的次对角元都不为0，那么对相同的调和Ritz值，用Arnoldi方法只能得到一个线性无关的调和Ritz向量。不仅如此调和Arnoldi方法对于特征值稠密的情况，效果也比较差。为了能够更好地处理这一隋况，本文提出了改进的块调和Arnoldi方法。与标准的块Arnoldi方法相比，改进的块调和Arnoldi方法仍用调和Ritz值作为特征值的近似，而在特征向量的选取方面，充分利用Arnoldi过程所提供的基向量的信息，在m+1维块Krylov子空间中选取一个向量-称之为改进的调和向量-作为所求特征向量的近似。理论分析表明了这种新的方法更有效。 第四章我们对一种大规模特殊结构的矩阵-箭状矩阵的特征问题进行了讨论。更多还原

【Abstract】 This thesis presents several theories and algorithms for large unsymmetric matrix eigenproblems. It consists of four parts.Chapter one gives the background of large unsymmetric matrix eigenproblems and basic numerical algorithms for solving them. We review the state of the art of this subject. Finallly, we describe the work of this thesis.Chapter two investigates a variant on harmonic Arnoldi method. It is well known that the m-step Arnoldi process constructs an orthonomal basis of the Krylov subspace K_m+1（A,v₁） in practice as well as the restricted matrix H_m,, that is, the （m +1）th basis v_m+1 is already available. However, conventional harmonic Arnoldi method uses harmonic Ritz vectors as approximate eigenvectors. Thus the （m +1）th basis v_m+1 is wasted and it contributes nothing to the wanted eigenvectors. The modified harmonic Arnoldi methods retains harmonic Ritz value as the approximate eigenvalue, while the approximate eigenvectors are formed by an ingenious linear combination of the original harmonic Ritz vectors and the （m +1）th basis vector v_m+1. These new vectors are linear combinations of the harmonic Ritz vectors and v_m+1, such that the residual norms of the modified approximate eigenpairs are minimal in some degree. Theoretical analysis shows the efficiency of the new method. Finally, numerical results also confirm the efficiency of the modified harmonic method.In Chapter three we study the modified block harmonic Arnoldi method. Suppose that the upper Hessenberg matrix H_m which is obtained in the m-step Arnoldi process is irreducible, i. e. the offdiagonal elements are nonzero. Thenthe multiple harmonic Ritz values of A have only one harmonic Ritz vector associated with them, so harmonic Arnoldi method is inefficient for multiple eigenvalue problems. Moreover, harmonic Arnoldi method is also inefficient in computing clustered eigenvalues. In order to calculate the clustered eigenvalues that lie in the interior of the spectrum, block harmonic Arnoldi methods is presented. Comparing with the standard block Arnoldi method, modified block harmonic Arnoldi method utilizes harmonic Ritz values as the approximation of the required eigenvalues, while in the formation of approximate eigenvectors, it makes full use of the basis information and choose a new vector which is linear conbinations of harmonic Ritz vector and block basis Vm+i-called modified harmonic Ritz vector-to generate better approximation. Theoretical analysis shows the efficiency of the new modified method.In Chapter four we discuss the eigenproblems of large scale arrowhead matrices. An arrowhead matrix has such a property which is zeros except for its main diagonal and the zth row and the zth column. An algorithm for computing all the eigenvalues and eigenvectors of such matrices is presented. Taking advantage of the matrix structure, we convert the eigenproblems to a polynomial equation problems of the degree at most n. In addition, we point out that the computation for one simple eigenvalue is completely independent of others. Our algorithms can be implemented parallely. Rounding error analysis shows the stability of the new method.更多还原