無理數近似計算與分析

信息計算科學論文編號:XXLW007 論文字數:15816,頁數:42

目  錄
摘  要 I
Abstract II
目  錄 III
前  言 V
第一章 現狀與分析 1
第二章 計算歷程與研究意義 2
2.1  的歷史  2
2.2  現代科學家計算值的動力與意義 5
第三章 近似值的三種數值計算方法 7
3.1  蒙特卡洛（Monte Carlo）方法 7
3.1.1 蒙特卡羅(Monte Carlo)方法簡介 7
3.1.2 蒙特卡羅(Monte Carlo)方法的基本原理 7
3.1.3 蒙特卡羅(Monte Carlo)解題主要步驟 8
3.1.3.1 構造或描述概率過程 8
3.1.3.2 實現從已知概率分布抽樣 8
3.1.3.3 建立各種估計量 9
3.3.4 適用范圍 9
3.1.5 蒙特卡洛方法的工作過程 9
3.1.6 蒙特卡洛方法在數學中的應用 9
3.2  數值積分法 12
3.2.1 常用求積公式 12
3.2.2 復化求積公式 15
3.2.3 變步長梯形方法 16
3.2.4求積公式的誤差 17
3.2.5 應用 20
3.3  級數法 24
3.3.1 萊布尼茨級數求π 24
3.3.2 泰勒級數求π 26
3.3.3 應用 27
結 束 語 32
致  謝 33
參考文獻 34
附錄 35

摘  要
 數值計算方法是研究各種數學問題數值解的一種有效方法，基于其理論的重要性和應用的廣泛性，它的基本思想、基本原理和基本方法已滲透到眾多領域并得到深入的應用。隨著現代科學技術的發展和計算機的廣泛使用，計算機已為數學領域提供了一個科學的現代化環境，科學計算方法大大提高了數學學習的效率。本課題涉及到的數值計算方法是一種典型的科學計算方法，它是一種研究并解決數學問題的數值近似解的方法，是在計算機上解決數學問題的方法。本課題主要討論無理數的近似計算，運用數值計算方法并通過Mathematica軟件得到計算機輸出的結果，觀察比較、分析各種方法的收斂性、收斂速度、計算量、可取性及近似值的精確度。 在鞏固已學知識的同時又有新的收獲。

關鍵詞： 數值計算 Mathematica軟件    蒙特卡羅方法  收斂性

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