【作者】 刘坤；

【导师】 巩增泰；

【作者基本信息】 西北师范大学 ， 应用数学， 2009， 硕士

【摘要】 在数学、物理学、工程计算和统计分析等领域的数学建模中,比较成熟也比较容易计算的是考虑能否将其转化为线性系统.然而,在具体的数学建模过程中经常涉及到参数的不确定性,这种不确定性又往往表现为一个模糊数.因此,涉及模糊数的线性系统,即模糊线性系统的求解问题是模糊数学的重要组成部分.本文对几类完全模糊线性系统的模糊近似解及其求解方法进行了研究和讨论.首先,扩展了LR-模糊数的概念,给出了广义LR-模糊数的定义,提出了一种四则运算的近似表示.其次,讨论了完全模糊线性系统的非负(正)模糊近似解,给出了其解的判定定理,并对两类特殊的完全模糊线性系统(模糊线性系统和齐次模糊线性系统)解的情况进行了研究,分别给出了解存在的判定定理和通解表示定理.而后,借助于系数矩阵与增广矩阵之间的秩关系,提出了广义相容完全模糊线性系统和广义不相容完全模糊线性系统的概念,分别讨论了其模糊近似解、模糊近似通解、最小二乘模糊近似解、最小二乘模糊近似通解和极小最小二乘模糊近似解.最后,针对对偶完全模糊线性系统,讨论了其模糊近似解,给出了对偶完全模糊线性系统模糊近似解的Cramer规则算法,并对直接算法和Cramer规则算法进行了比较.更多还原

【Abstract】 In the mathematical modeling of mathematics, physics, engineering computation and statistical analysis, the useful and easily estimated method is that it could be transferred into a linear system in some sense. However, the uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented as fuzzy numbers. So the investigation of solving the solutions for fuzzy linear systems whose elements of coefficient matrix or augmented matrix are fuzzy numbers, plays an important role in the fuzzy mathematics theory. In this thesis, the fuzzy approximate solution and its computation methods for some kinds of fully fuzzy linear systems are investigated.Firstly, the concept of the LR-fuzzy numbers is expanded, and the generalized LR-fuzzy numbers is defined. The approximate expressions of the fuzzy number arithmetic operations are given. Secondly, the non-negative and non-position fuzzy approximate solutions of the fully fuzzy linear system are discussed, and the judge theorem of its solutions is obtained. Furthermore, two special kinds of fully fuzzy linear systems including the fuzzy linear system and the homogenous fuzzy linear system are studied. A judge theorem of its solutions and a general solution representation for the fuzzy linear system and the homogenous fuzzy linear system are shown, respectively. In additions, a generalized consistent and inconsistent fully fuzzy linear system are further defined by using the relation between the ranks of coefficient matrix and augmented matrix. The fuzzy approximate solution, the fuzzy approximate general solution, the fuzzy least squares approximate and the fuzzy minimum least squares approximate solution about them are discussed. Finally, the dual fully fuzzy linear system is defined, its fuzzy approximate solution is discussed, and the Cramer ruler method of dual fully fuzzy linear system is pointed out. Both direct method and Cramer method are also obtained and compared by an example.更多还原