【作者】 杨庆；

【导师】 朱元国；

【作者基本信息】 南京理工大学 ， 运筹学与控制论， 2017， 硕士

【摘要】 随着科学和工程技术的发展,线性代数中一般形式的线性系统已经无法达到研究应用的要求,变量为不确定的线性系统得到了众人的关注。李波和朱元国教授于2014年提出了不确定线性系统的概念,并给出了相关求解公式。本文在此基础上提出了全不确定线性系统(或全不确定线性方程)的概念,并定义了全不确定线性系统的解。随后,文章讨论了全不确定线性系统有解的充分条件。针对系统系数矩阵为方阵和非方阵的情况,本文分别给出了系统的求解公式以及系统有解的判断条件。考虑到求解时需要求矩阵逆,当系数矩阵维数较大时其计算量非常大,不便于应用,因此,本文进一步讨论了关于全不确定线性系统的数值迭代算法求解,并分析了几种迭代法在求解全不确定线性系统时其迭代格式的收敛性问题。最后,文章给出了关于全不确定线性系统的几个数值例子和一个应用实例。更多还原

【Abstract】 With the development of science and engineering technology,the general linear systems in linear algebra have been unable to meet the requirements of research and applications.Linear systems with uncertain variables have attracted some researchers’attention.Li and Zhu proposed the concept of uncertain linear systems and gave the corresponding solving formula in 2014.Base on it,the concept of fully uncertain linear systems(or fully uncertain linear equations)is defined in this paper.Furthermore,we discuss the sufficient conditions of the solution to the fully uncertain linear systems.For the case of the coefficients of a system are square and non-square,we give the formulas to find the solution of this system as well as in what conditions to guarantee solutions in this paper.Taking into account of the difficulty of computing inverse matrix and the large calculation,this paper discusses the numerical iterative algorithms for the fully uncertain linear systems and the convergence of those numerical iterative algorithms.Finally,several numerical examples and an application example are presented for the fully uncertain linear systems.更多还原