1、1设已知某个函数关系的列表函数值我们所知道仅仅上述的表列值我们所知道仅仅上述的表列值 ,它们常常是间接求得它们常常是间接求得的的 .例如是由实验例如是由实验 (观测观测 )得来的得来的 ,或者是从级数或微分方或者是从级数或微分方程求得的。问应该如何估算任意一点程求得的。问应该如何估算任意一点 x的值的值 ?如何估算如何估算函数关系函数关系 y=f(x)第三章 : 插值 (Interpolation)2设设 y=f(x) 是实变量是实变量 x的单值函数，且已知在给定的的单值函数，且已知在给定的n+1个互异点处的值个互异点处的值 ,即即插值的基本问题是插值的基本问题是 , 寻求多项式寻求多项式 p
2、(x), 使得使得3Linear interpolationParabolic Interpolation The Lagrange interpolationThe Aitken methodThe Newton InterpolationHermite Interpolation插值方法4Consider a given discrete data set (xi , yi) with i=1,2,3, , n+1. The simplest way to obtain the approximation of y for xxi , xi+1 is to construct a stra
3、ight line between xi and xi+1 . Then y is given by Linear interpolation线性插值5With being a parameter determined by the specific form y(x) of the given discrete data set (xi, yi). If one draws a quadratic curve pass through f(x), f(xi), f(xi+1), then 6axi , xi+1 7Examplef(x)=sin(x) ; xi =/4, xi+1 =/2 L
4、et us use linear interpolation scheme to find the approximate value of f(3/8). f(/4)=0.7071 f(/2)=1.0000 8Parabolic InterpolationLinear interpolation9yi =f(/4)=0.7071 yi+1 =f(/2)=1.0000 y(x)=0.7071+(x- /4) (1.0000-0.7071) /(/2- /4)Sin (3/8)=0.9239y(3/8) = 0.853610Parabolic InterpolationLinear interpolation
