数值分析：误差完整版.pptVIP

数 值 分 析Numerical Analysis 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 非线性方程的数值解法 内容 线性方程组的数值解法 数值逼近 数值积分 微分方程数值解法 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 大家一起猜？ 1 1 / e 解法之一：将 作Taylor展开后再积分 S4 R4 /* Remainder */ 取 则 称为截断误差 /* Truncation Error */ | 舍入误差 /* Roundoff Error */ | = 0.747… … 由截去部分 /* excluded terms */ 引起 由留下部分 /* included terms */ 引起 §2 误差与有效数字 /* Error and Significant Digits */ ? 绝对误差 /* absolute error */ 其中x为精确值，x*为x的近似值。 Hey isn’t it simple? Oh yeah? Then tell me the absolute error of Oops! ，例如： 工程上常记为 ，称为绝对误差限 /* accuracy */， 的上限记为 注：e* 理论上讲是唯一确定的，可能取正，也可能取负。 e* 0 不唯一，当然 e* 越小越具有参考价值。 I can tell that this part’s diameter is 20cm?1cm. I can tell that distance between two planets is 1 million light year ±1 light year. Of course mine is more accurate ! The accuracy relates to not only the absolute error, but also to the size of the exact value. §2 Error and Significant Digits ? 相对误差 /* relative error */ Now I wouldn’t call it simple. Say … what is the relative error of 20cm±1cm? Don’t tell me it’s 5% because… But what kind of information does that 5% give us anyway? x 的相对误差上限 /* relative accuracy */ 定义为 A mathematician, a physicist, and an engineer were traveling through Scotland when they saw a black sheep through the window of the train. Aha, says the engineer, I see that Scottish sheep are black. Hmm, says the physicist, You mean that some Scottish sheep are black. No, says the mathematician, All we know is that there is at least one sheep in Scotland, and that at least one side of that one sheep is black! §2 Error and Significant Digits ?有效数字 /* significant digits */ 从一个数的左边第一个非0数字起，到末位数字止，所有的数字都是这个数的有效数