计算二次特征值问题单特征三元组二阶偏导数的单模态法  

Single-mode Method for Calculating the Second Partial Derivatives of a Single Eigentriple in a Quadratic Eigenvalue Problem

作　　者：王立群 卢欣 石丽伟 Li Qun WANG;Xin LU;Li Wei SHI(Department of Mathematics,College of Science,China University of Petroleum-Beijing,Beijing 102249,P.R.China;School of Information Management for Law,China University of Political Science and Law,Beijing 102249,P.R.China)

机构地区：[1]中国石油大学(北京)理学院数学系,北京102249 [2]中国政法大学法治信息管理学院,北京102249

出　　处：《数学学报（中文版）》2022年第3期559-570,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(61972415,12171482);中国石油大学(北京)科研基金(2462020YXZZ004);中国政法大学钱端升杰出学者支持计划资助项目(1000-10820721)。

摘　　要：本文提出了计算二次特征值问题单特征三元组的二阶偏导数的单模态法.该方法只需要用到待求二阶偏导数的特征三元组的信息;在计算特征向量二阶偏导数时只需求解一个线性方程组,该线性方程组的系数矩阵的阶数为n-1(n为二次特征值问题的规模),且系数矩阵的条件数恰好为其最大与最小非零奇异值的比值.本文给出三个例子进行了数值试验,并与已有的Nelson方法做了比较,数值结果表明,精度方面,单模态法计算特征值二阶导数的精度与Nelson方法相当,对某些例子计算特征向量二.阶导数的精度高于Nelson方法;CPU时间方面,第三个数值例子表明单模态法的CPU时间略低于Nelson方法,后者大约是前者的1.2倍.In this paper,a single mode method for calculating the second-order partial derivatives of a simple eigen-triplet of quadratic eigenvalue problems is presented.This method only needs the information of the eigen-triplet whose second-order partial derivatives are to be obtained;and to calculate the second-order partial derivative of the eigenvector,it only needs to solve a system of linear equations of order n-1(n is the size of quadratic eigenvalue problem).Most of all,the condition number of the coefficient matrix is exactly the ratio of its maximum to its minimum non-zero singular value.Three numerical examples are given and computed for the single mode method.Numerical results show that,compared with the Nelson method,the accuracy of the second derivative of eigenvalue calculated by the single mode method is almost the same,while the accuracy of second derivatives of eigenvector is higher;and the CPU time of single mode method is slightly less than that of the Nelson method,the latter is about 1.2 times as much as the former.

关 键 词：特征三元组 二阶偏导数 单模态法 条件数 Nelson方法 

分 类 号：O241.5[理学—计算数学]