数值分析Numerical Analysis - Read.PPT

* 提问：数值分析是做什么用的？ 数值 分析 输入复杂问题或运算 ? ? ? ? 计算机 近似解 第一章 误差 /* Error */ §1 误差的背景介绍 /* Introduction */ 1. 来源与分类 /* Source Classification */ 从实际问题中抽象出数学模型 —— 模型误差 /* Modeling Error */ 通过测量得到模型中参数的值 —— 观测误差 /* Measurement Error */ 求近似解 —— 方法误差 (截断误差 /* Truncation Error */ ) 机器字长有限 —— 舍入误差 /* Roundoff Error */ §1 Introduction: Source Classification The following problem can be solved either the easy way or the hard way. Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown? The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. Thats all there is to it. When this problem was posed to John von Neumann, he immediately replied, 150 miles. It is very strange, said the poser, but nearly everyone tries to sum the infinite series. What do you mean, strange? asked Von Neumann. Thats how I did it! §1 Introduction: Source Classification 大家一起猜？ 1 1 / e 解法之一：将 作Taylor展开后再积分 S4 R4 /* Remainder */ | 舍入误差 /* Roundoff Error */ | = 0.747… … 由截去部分 /* excluded terms */ 引起 取 则 称为截断误差 /* Truncation Error */ 由留下部分 /* included terms */ 引起 例：近似计算 据说，美军 1910 年的一次部队的命令传递是这样的: 营长对值班军官: 明晚大约 8点钟左右，哈雷彗星将可能在这个地区看到，这种彗星每隔 76年才能看见一次。命令所有士兵着野战服在操场上集合，我将向他们解释这一罕见的现象。如果下雨的话，就在礼堂集合，我为他们放一部有关彗星的影片。 值班军官对连长: 根据营长的命令，明晚8点哈雷彗星将在操场上空出现。如果下雨的话，就让士兵穿着野战