一个求实对称矩阵的全部特征值及特征向量的新方法

A Method for Computing All Eigenvalues and Eigenvectors of Real Symmetric Matrix

本文是在正交投影方法、正幂法和带平移的反幂法的基础上引申出的一种求实对称矩阵的全部特征值和相应的特征向量的新方法。此方法可以按特征值的绝对值由大到小依次求出全部特征值和相应的特征向量。因每一步求解都是针对原始矩阵进行的,从而有效地抑制了误差的传递和积累。这一方法不但结构简单,收敛速度快,更有精度高等优点。经数值实验表明是十分成功的。

In this paper a new method for computing all eigenvalues and eigenvectors of real symmetric matrices is presented by combning the methods of orthogonal projection, power and antipower with origin's translation, By using this method all the eigenvalues and the corresponding eigenvectors can be obtained with higher accuracy progressively from the largest to the smallest according to the magnitude of the absolute values of the eigenvalues. Because each step of the computation is conducted on basis of the original matrix, the propagation and error accumulation are under effective control. This method is capable of a simpler structure, rapid convergence and more accurate results. The numberical example shows it is successful.