用简单反迭代法计算三对角矩阵的特征向量

与特征值计算的算法丰富多彩相比,在已知比较精确的特征值的情况下,求其相应的特征向量的算法却不多见,已有的算法有基本反迭代法1]2]4]5]、交替法3]等.到目前为止,计算特征向量的算法都是基于反迭代法的,衡量算法是否收敛都是以残量的大小为标准,本文的算法也不例外.本文的目的就是计算不可约实对称三对角矩阵T=bj-1,aj,bj]的相应于某个特征值λi（已得到其近似λ）的特征向量.首先我们来看下面的例子:例1 我们取T为201阶的Wilkinson负矩阵,λ取计算的最大特征值,分别令迭代的初始向量是e1,e100,e201,e=（1,1,…,1）T.图1反映了反迭代的收敛速度.

COMPUTING EIGENVECTOR OF SYMMETRIC TRIDIAGONAL MATRIX BY USING SIMPLE INVERSE ITERATION

The goal of this paper is to compute the eigenvector of symmetrictridiagonal matrix T corresponding to the given approximate eigenvalue λ, by us-ing one step of inverse iteration for certain chosen right vector ek. Careful analysis shows that for suitable chosen k, say kmax, the simple inverse iteration, onestep of inverse iteration, can give a high accurate eigenvector, and its accuracy isachieved for the approximate eigenvalue λ too. Some sufficient conditions of thatkd＝kmax are given. We also give an estimate to show that kd still works very welleven in the case when the sufficient conditions are not exactly satisfied.