无理数近似计算与分析

信息计算科学论文编号:XXLW007 论文字数:15816,页数:42

目  录
摘  要 I
Abstract II
目  录 III
前  言 V
第一章 现状与分析 1
第二章 计算历程与研究意义 2
2.1  的历史  2
2.2  现代科学家计算值的动力与意义 5
第三章 近似值的三种数值计算方法 7
3.1  蒙特卡洛（Monte Carlo）方法 7
3.1.1 蒙特卡罗(Monte Carlo)方法简介 7
3.1.2 蒙特卡罗(Monte Carlo)方法的基本原理 7
3.1.3 蒙特卡罗(Monte Carlo)解题主要步骤 8
3.1.3.1 构造或描述概率过程 8
3.1.3.2 实现从已知概率分布抽样 8
3.1.3.3 建立各种估计量 9
3.3.4 适用范围 9
3.1.5 蒙特卡洛方法的工作过程 9
3.1.6 蒙特卡洛方法在数学中的应用 9
3.2  数值积分法 12
3.2.1 常用求积公式 12
3.2.2 复化求积公式 15
3.2.3 变步长梯形方法 16
3.2.4求积公式的误差 17
3.2.5 应用 20
3.3  级数法 24
3.3.1 莱布尼茨级数求π 24
3.3.2 泰勒级数求π 26
3.3.3 应用 27
结 束 语 32
致  谢 33
参考文献 34
附录 35

摘  要
 数值计算方法是研究各种数学问题数值解的一种有效方法，基于其理论的重要性和应用的广泛性，它的基本思想、基本原理和基本方法已渗透到众多领域并得到深入的应用。随着现代科学技术的发展和计算机的广泛使用，计算机已为数学领域提供了一个科学的现代化环境，科学计算方法大大提高了数学学习的效率。本课题涉及到的数值计算方法是一种典型的科学计算方法，它是一种研究并解决数学问题的数值近似解的方法，是在计算机上解决数学问题的方法。本课题主要讨论无理数的近似计算，运用数值计算方法并通过Mathematica软件得到计算机输出的结果，观察比较、分析各种方法的收敛性、收敛速度、计算量、可取性及近似值的精确度。 在巩固已学知识的同时又有新的收获。

关键词： 数值计算 Mathematica软件    蒙特卡罗方法  收敛性

The Approximate Computation of the Irrational Number and Analysis
Student: Li Wen Advisor: Lei Jianguang

School of Science, Zhejiang University of Science and Technology
 
Abstract
 Numerical computation method is an effective method that is for studying the numerical solution of the various mathematical problems. Its basic ideology, basic principles and fundamental approach have been penetrated into many fields and have in-depth application and all these are based on the fundamentality of its theory and universality of its application. With the development of contemporary science and technology and the widespread use of computer, the computer has provided a scientific modern environment for the mathematical field, and scientific calculation method has greatly enhanced the efficiency of studying mathematics. The numerical computation method which the topic related to is a typical scientific calculation method, it is a method to investigate and solve the numerical approximation solution of mathematical problems, and it is also a ideal way to solve mathematical problems on a computer. This essay mainly discusses the irrational approximate calculation by the method of numerical computation, then some examples are given to show the idea. The results in this examples are get by the Mathematica software. Observs and compares analyzes the astringency, convergent rate, the amount of computation, and the desirability of the precise degree of approximation of the various methods. Acquires new harvest simultaneity in the consolidation of the knowledge.

Keyword: numerical computation; Mathematica software;; Monte Carlo method; astringency