直接多胞体同伦求解混合三角多项式

发布时间：2018-05-20 04:34

本文选题：混合三角多项式 + 多项式方程组　；参考：《大连理工大学》2015年硕士论文

【摘要】：混合三角多项式方程组(MTPS)是科学工程计算中常见的一类非线性方程组,它的每一个方程由一部分变元和其余变元为三角函数组成。就目前来讲,对于求解这类方程组所有孤立解的数值方法主要分为两大类：直接法和间接法。间接法是通过把三角函数部分转化为多项式方程组进行求解,而转化的过程中又引进了新的变量,从而会增大问题的规模；直接法的最大好处在于不需要引进新的变量,直接对方组进行求解,从而不会增大问题的规模,但已有的直接方法仅适用于求解稠密的或者具有特殊稀疏结构的混合三角多项式方程组。在本论文中,我们构造了直接多胞体同伦方法求解混合三角多项式方程组的全部解。首先构造出一个初始混合三角多项式方程组,并给出初始方程组的求解方法。然后应用这个初始混合三角多项式方程组,构造出求解MTPS问题的同伦,并证明了算法的收敛性。数值实验结果表明,我们的直接多胞体同伦方法优于已有的求解MTPS全部解的数值方法。具体来说,本论文的内容由如下几部分构成：第一章,首先介绍MTPS的概念及应用,并给出几个实际应用中出现的简单例子,介绍其基本的求解方法：直接和间接同伦方法,并且简要的分析这两种方法的各自优点和缺点。第二章,具体的介绍如何求解混合多项式方程组全部解的同伦方法。介绍混合三角多项式方程组的基本形式及一些基本概念；论述如何构造出一个好的同伦来对这类方程组进行求解,重点介绍多胞体同伦方法,包括混合三角多项式方程组对应的多胞体的混合体积及混合剖分的定义及数值计算、初始方程组的构造及求解、多胞体同伦的构造。第三章,具体给出了求MTPS问题全部解的直接多胞体同伦方法。通过构造出初始方程组,从而进一步构造出多胞体同伦,并证明这个同伦是一个好的同伦。通过和已有算法的对比说明直接多胞体同伦方法的优越性。
[Abstract]:The mixed trigonometric polynomial equations (MTPS) is a class of nonlinear equations common in scientific engineering calculation. Each of its equations is composed of a part of variable elements and other variables as trigonometric functions. At present, the numerical methods for solving all the solitary solutions of these equations are divided into two main categories: direct method and indirect method. It can be solved by converting part of trigonometric function into a polynomial equation group, and a new variable is introduced in the process of transformation, which will increase the scale of the problem. The greatest advantage of the direct method is that it does not need to introduce new variables and solve the problem directly by the other group, but it will not increase the scale of the problem, but the existing direct method is only suitable. In this paper, we construct a direct multi cell homotopy method to solve all solutions of the mixed trigonometric polynomial equations in this paper. First, a set of initial mixed trigonometric polynomial equations is constructed, and the solution of the initial equations is given. With this initial mixed trigonometric polynomial equation, the homotopy of the MTPS problem is constructed and the convergence of the algorithm is proved. The results of the numerical experiment show that our direct multi cell homotopy method is superior to the existing numerical methods for solving all the MTPS solutions. This paper introduces the concept and application of MTPS, and gives some simple examples in practical applications, and introduces its basic solution methods: direct and indirect homotopy methods, and briefly analyses the respective advantages and disadvantages of the two methods. The second chapter introduces concretely the homotopy method of solving all solutions of the mixed polynomial equations. The basic forms and some basic concepts of the complex polynomial equations are discussed. It is discussed how to construct a good homotopy to solve this kind of equations. The emphasis is on the multi cell homotopy method, including the definition and numerical calculation of the mixed volume and mixed dissection of the multi cell body corresponding to the mixed trigonometric polynomial equation group, and the initial equation set. In the third chapter, the direct multi cell homotopy method for solving all solutions of MTPS problem is given. By constructing an initial equation set, we construct a multi cell homotopy and prove that the homotopy is a good homotopy. By comparing with the existing algorithms, the direct multi cell homotopy method is proved. Superiority.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.7

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