数值分析朱立永北京航空航天大学 数学与系统科学学院Email: numerical_analysishotmail.com Password:beihang 答疑时间：星期四下午2:305:30 答疑地点：主216Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析第十讲非线性方程的迭代解法第三章非线性方程与非线性方程组的迭代解法Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析这这这这一部分的主要任一部分的主要任务务务务是解是解其中其中f f（x x）是一个一元非线性函数。是一个一元非线性函数。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析非线性科学是当今科学发展的一个重要研究方 向，很多实际工程物理问题都归结为非线性方 程（组）的求解。例如：无穷组解无解一个解两个解四个解非线性方程的求根非常复杂。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析求根问题包括下面三个问题： 根的存在性：即f(x)=0有没有根？若有， 有几个根？ 哪儿有根？确定有根区间 根的近似求解Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析常用的求非线性方程根的方法 二分法（对分法、搜索法）二分法（对分法、搜索法） 不动点法不动点法 （简单迭代法、压缩映象法）及（简单迭代法、压缩映象法）及 其加速算法其加速算法 NewtonNewton方法及其变体方法及其变体Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析根，重根的定义：根，重根的定义： 1. 根：如果存在常数s，使得 f(s)=0则称s是f(x)=0的根（零点） ；2. 重根：如果 ， 称s 为m重根。特别地对 f(x) 是多项式，则有 其中 。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析二分法（对分法） 理论依据理论依据：设函数：设函数 f(x) f(x)Ca,b, Ca,b, f(a)f(b)0,运算停止，输出计算失败。如 果|f(a)| 而且kN, 输出计算失败，停止计算。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析简单迭代法及其收敛速度简单迭代法及其收敛速度 迭代法的构想迭代法的构想 从一个初值x0出发，计算如果 xk收敛，即存在x*, 使得 则由 得 即 x*是 (x)的不动点，也就是 f(x) 的根。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析问题问题： x xk k 收敛吗？怎样实现收敛吗？怎样实现 (x) (x) Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析简单迭代法的收敛速度简单迭代法的收敛速度 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析SteffensenSteffensen加速收敛方法加速收敛方法 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数值分析Newton 迭代法 非线性问题的最简单解法是线性近似.将非线性方程线性化，以线性方程的解逐步逼近非线性方程的解，这就是Newton法的基本思想.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011