一种求任意埃尔米特广义特征值的方法

An Algorithm for Hermitian Indefinite Generalized Eigenvalue Problems

本文利用五种复变换矩阵(其中四种为作者新提出),给出一种求解埃尔米特广义特征值问题Ax=λBx的方法,这里A,B为n阶任意埃尔米特阵.可说是[1]和[2]中方法的改进与推广,[1]中讨论了A、B实对称B非奇异的情形,[2]中的MDR法只能用于A,B实对称B半正定的情形.它们都不能解决B为奇异且不定的情形,也不能解决A,B为埃尔米特的情形.本文还对[1]中的中断情况作了改进,对MDR方法的改进在别处讨论,新方法称CHR法.

In this paper, five complex transformation matrices (four of them are proposed by the authers) and least square method are used, and an algorithm for Hermitian indefinite generalized eigenvalue problem Ax=λBx is established.When A and B are real symmetric and B is nonsingular, HR algorithm can be used to solve it.Here, A and B are Hermitian and B may be singular.The new algorithm is called 'CHR'.Four new complex transformation matrices are: Unitary diagonal exchange matrix T; Complex quasi-Givens matrix; Fast complex quasi-Givena transformation matrix and Complex quasi-House holder matrix.