计算方法实验2插值法

1、一、实验目的与任务1. 掌握拉格朗日插值、牛顿插值、分段插值的基本原理；2. 理解各种插值法的优缺点和插值的误差；3. 熟悉插值法的一般过程。二、实验涉及的相关知识点线性插值函数的使用。三、实验内容与过程(1) 【实验1.1】利用C语言编程计算：已知sin0.32=0.,sin0.34=0.,sinO.36=0.,用线性插值法及抛物线插值法计算sin0.3367的值并估计截断误差。线性插值公式为：由(Xk,y k)、(Xk+1,y k+1)得I k(x)=(x-x k+1)/(X k-Xk+1), l k+1(X)=(X-X k)/(x k+1-X k)L«x)=y k*l k(x)

2、+y k+1*l k+1(x)抛物线插值公式为：由(Xk-1 ,y k-1)(x k,y k)、(x k+1,y k+1)三点可得插值公 式：l k-1 (x)=(x-x k)(x-x k+1 )/(x k-1 -X k)(x k-1 -X k+1)l k(x)=(x-X k-1)(x-X k+1)/(X k-x k-1 )(x-X k+1)l k+1 (x)=(x-x k-1 )(x-x k)/(X k+1 -X k-1 )(X k+1 -Xk)l_2(x)=y k-1 *l k-1 (x)+y k*l k(x)+y k+1*l k+1(x)(2) 【实验1.2】牛顿插值法：函数值与自变量的

3、差商就是均差，卅心®“如上迪一阶均差(或记作f X°,X1)；Jr 耳”孟 J二阶均差；'(或记作f X。, X1, X2)均差有两条常用性质：(1)均差用函数值的线性组合表示；(2)均差与插 值节点顺序无关。用均差为系数构造多项式，就是牛顿插值多项式M(X)=f(Xo)+f(Xo, Xi)( X-Xo)+f(Xo, Xi, X2)( X-Xo)( X- Xi)+ + +(Xo, Xi, X2,，Xn)( X -Xo)( X- Xi)( X-X2) ( X-Xn-i)牛顿插值多项式的余项为R(x)=f(x) M(x)=f(X, Xo, Xi, X2,，Xn)( X

4、Xo)( X Xi)( X X2) -(X Xn i)( X Xn)四、实验结果及分析【实验1.1】线性插值结果:抛物线插值结果:【实验1.2】线性插值结果:谕入插值右个数2谙輸天第!个插值怎的值二0 -0谙输A第丄个埋值点对应的函数值=0-4107S 谙输入第z个拯值怎的值：0-6-5请巒入誨空攝負恐寸应的函敎值=w *s?e±5 論-天需乘插荷点的聒0-55需求插值点的值«= W-E5需求擂值占奸应的函数值V- 0-51119 谙按任倉犍继獵-抛物线插值结果:'fay.：-o"：f：m.pO,gou"fm;o.""jm:：

5、；. a=f；.rke"".;5*p"ec.Ay.Fgl;r”ep:“.-iag-o”.b；a*:：s.- d.”:a4i8.:：. :：'：cp”:”.or值占:的值=i吉输A第工个辰值电刘.虫的函漿值=0-1075 诘宫俞入第“卜麻值点的值=0-55谙输人第2个桩值点:对应的越教值二e-S7®xs i.S.TA 第3沖-林值頁的悄=0-80五、实验相关说明有的容易实现，但在调式时就有很大问题，VC+不是英文输入法下作的都不行。VC+6.0提示错误时不够完善，我觉得用Microsoft Visual Studio 2008 中的VC+比较好。六

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