一类二次矩阵方程的条件数和后向误差  

Condition number and backward error of quadratic matrix equation

作　　者：刘兰冬[1] 

机构地区：[1]中国矿业大学(北京)理学院,北京100083

出　　处：《应用数学与计算数学学报》2014年第4期424-431,共8页Communication on Applied Mathematics and Computation

基　　金：国家自然科学基金资助项目(11371364);中央高校基本科研业务费资助项目(2009QS09)

摘　　要：主要讨论一类二次矩阵方程X^2-EX-F=0的条件数和后向误差,其中E是一个对角矩阵,F是一个M矩阵.这类二次矩阵方程来源于Markov链的噪声Wiener-Hopf问题.实际问题中人们感兴趣的是它的M矩阵的解.应用Rice创立的基于Frobenius范数下的条件数理论,导出此类二次矩阵方程的M矩阵解的条件数的显式表达式.同时,也给出近似解的后向误差的定义以及一个可计算的表达式.最后,通过数值例子验证理论结果是有效的.This paper is devoted to the condition number and backward error for the quadratic matrix equation X^2- EX- F = 0,where E is a diagonal matrix and F is an M-matrix.The quadratic matrix equation of this type arises in noisy Wiener-Hopf problems for Markov chains.The solution of the practical interest is a particular M-matrix solution.We apply the condition number theory developed by Rice to define the condition number in the Frobenius norm.The explicit expression of the condition number is derived in a uniform manner.Meanwhile,the backward error is defined and evaluated by an explicit formula.The theoretical results are illustrated by using the simple numerical examples.

关 键 词：二次矩阵方程 M-矩阵 扰动分析 条件数 后向误差 

分 类 号：O151.21[理学—数学]