一类二次特征值问题的向后误差分析

Backward Error Analysis for Some Quadratic Eigenvalue Problems

在高速列车的振动分析中,会遇到一类二次特征值问题(λ2 AT+λQ+A)z=0,其中A和Q为n×n复矩阵,且具有如下特殊结构:A和Q都是m×m的分块矩阵,每个块有k×k个元素,即n=m×k;此外,Q是块三对角阵,A只有位于(1,m)位置的一个块为非零块.本文主要讨论此类二次特征值问题的向后误差,并且证明了矩阵A的误差仅存在于它的非零块A13上.

In studying the vibration of fast trains,we encounter a palindromic quadratic eigenvalue problem （AZAT ＋AQ ＋A）Z=0, where A ,Q ∈ C^n×n and QT =Q. Moreover,thematrix Q is block tridiagonal and block Toeplitz, and the matrix A possesses only one nonzero block in the upper-right corner.We then discuss the backward error for this problem in this article and prove that the error of A is only on its non-zero block A13.