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# 第 2 章 插值法 PowerPoint PPT Presentation

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1、问题背景

§1 引 言

2、一般概念

P(xi)=yi (i=0,1,…,n)

3、插值多项式存在唯一性定理

P(xi)=yi, i=0,1,…,n

1、线性插值和抛物插值

L1(xk)=yk, L1(xk+1)=yk+1

§2 拉格朗日插值

2、拉格朗日插值多项式

Ln(x)——拉格朗日插值多项式

3、插值余项与误差估计

0.352274,用线性插值计算和抛物插值计算sin0.3367的值,

1、问题的引入

§3 差商与牛顿插值

2、差商定义

3、差商的基本性质

a0

a1

a2

a3

ak=f[x0,x1,x2,…,xk]

4、牛顿插值多项式

【例】f(x)的函数表如下，求4次牛顿差值多项式，并由此求f(0.596)的近似值

§4 埃尔米特插值

P(xi)=f(xi), (i=0,1,2), P′(x1)=f(x1)

H3(xk)=yk, H3(xk+1)=yk+1

H′3(xk)=mk, H′3(xk+1)=mk+1

P(xi)=f(xi), (i=0,1,2), P′(x1)=f(x1)

H3(xk)=yk, H3(xk+1)=yk+1

H′3(xk)=mk, H′3(xk+1)=mk+1

H3(xk)=yk, H3(xk+1)=yk+1

H′3(xk)=mk, H′3(xk+1)=mk+1

P36-例6

§5 分段低次插值

1、高次插值的病态性质

y=L10(x) 及y=1/(1+x2)在[-5,5]的对比图形

2、分段线性插值

3、分段三次埃尔米特插值

P48-ex2, ex4, ex7

P49-ex16, ex17

P48-ex8