本文選題：混合三角多項式 + 多項式方程組　； 參考：《大連理工大學(xué)》2015年碩士論文

【摘要】：混合三角多項式方程組(MTPS)是科學(xué)工程計算中常見的一類非線性方程組,它的每一個方程由一部分變元和其余變元為三角函數(shù)組成。就目前來講,對于求解這類方程組所有孤立解的數(shù)值方法主要分為兩大類：直接法和間接法。間接法是通過把三角函數(shù)部分轉(zhuǎn)化為多項式方程組進(jìn)行求解,而轉(zhuǎn)化的過程中又引進(jìn)了新的變量,從而會增大問題的規(guī)模；直接法的最大好處在于不需要引進(jìn)新的變量,直接對方組進(jìn)行求解,從而不會增大問題的規(guī)模,但已有的直接方法僅適用于求解稠密的或者具有特殊稀疏結(jié)構(gòu)的混合三角多項式方程組。在本論文中,我們構(gòu)造了直接多胞體同倫方法求解混合三角多項式方程組的全部解。首先構(gòu)造出一個初始混合三角多項式方程組,并給出初始方程組的求解方法。然后應(yīng)用這個初始混合三角多項式方程組,構(gòu)造出求解MTPS問題的同倫,并證明了算法的收斂性。數(shù)值實驗結(jié)果表明,我們的直接多胞體同倫方法優(yōu)于已有的求解MTPS全部解的數(shù)值方法。具體來說,本論文的內(nèi)容由如下幾部分構(gòu)成：第一章,首先介紹MTPS的概念及應(yīng)用,并給出幾個實際應(yīng)用中出現(xiàn)的簡單例子,介紹其基本的求解方法：直接和間接同倫方法,并且簡要的分析這兩種方法的各自優(yōu)點和缺點。第二章,具體的介紹如何求解混合多項式方程組全部解的同倫方法。介紹混合三角多項式方程組的基本形式及一些基本概念；論述如何構(gòu)造出一個好的同倫來對這類方程組進(jìn)行求解,重點介紹多胞體同倫方法,包括混合三角多項式方程組對應(yīng)的多胞體的混合體積及混合剖分的定義及數(shù)值計算、初始方程組的構(gòu)造及求解、多胞體同倫的構(gòu)造。第三章,具體給出了求MTPS問題全部解的直接多胞體同倫方法。通過構(gòu)造出初始方程組,從而進(jìn)一步構(gòu)造出多胞體同倫,并證明這個同倫是一個好的同倫。通過和已有算法的對比說明直接多胞體同倫方法的優(yōu)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.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O241.7

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相關(guān)博士學(xué)位論文 前5條

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5 于冉;幾種高精度求積公式的構(gòu)造與研究[D];大連理工大學(xué);2013年

相關(guān)碩士學(xué)位論文 前1條

1 李玉霞;混合三角多項式方程組孤立解個數(shù)上界估計[D];大連理工大學(xué);2010年

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本文編號：1913228