求半正定Hermitian矩阵特征向量的改进乘幂法  

An Improved Power Method for Calculating Eigenvectors of a Positive Semidefinite Hermitian Matrix

作　　者：张雷[1] 

机构地区：[1]中国西南电子技术研究所,成都610036

出　　处：《电讯技术》2015年第6期623-628,共6页Telecommunication Engineering

摘　　要：经典乘幂法结合压缩法是计算半正定Hermitian矩阵最大前n个特征值对应特征向量的重要方法,但其固定的迭代次数使得待分解矩阵的随机变化和初始向量的不同选择导致计算精度波动较大,同时,较大特征值对应特征向量的计算误差也会影响较小特征值对应特征向量的计算。为克服这些缺点,提出了一种将前后两次迭代所求向量的距离作为迭代终止条件的改进乘幂法,并证明了它在有误差传播时的收敛性。理论计算结果表明,对4阶半正定Hermitian随机矩阵,在相同计算精度前提下,所提方法比经典方法可至少降低一半计算复杂度。The combination of conventional power method and compression method is an important approach to calculate the eigenvectors corresponding to n maximal eigenvalues of a positive semidefinite Hermitian matrix. The approach has two main drawbacks： the random variation of matrix and the selection of initial vector fluctuate the computational accuracy much due to the fixed iteration number; besides, the computa- tional error of eigenvector corresponding to bigger eigenvalue deteriorates the solution of one corresponding to smaller eigenvalue. Thus, this paper propose an improved power method by making the distance of two successively obtained vectors as a condition of terminating iteration, and validates its convergence in the case of computational error propagation. The theoretical results show that for 4-dimension positive semidef- inite Hermitian random matrices, the proposed method can reduce the computational complexity by half at least compared with the conventional one in the condition of same computational accuracy.

关 键 词：半正定Hermitian矩阵 特征向量 乘幂法 向量距离 

分 类 号：TN92[电子电信—通信与信息系统]