第1章+插值法3

1、计算方法计算方法第一章第一章 插值法插值法3主讲人：罗莹莹主讲人：罗莹莹l函数的连续性函数的连续性l节点处的函数值相等节点处的函数值相等l插值的光滑度插值的光滑度l在节点处的导数值也相等在节点处的导数值也相等l导数的阶数越高则光滑度越高导数的阶数越高则光滑度越高l典型的例子典型的例子l现代的仿生学：在设计交通具的外形，就是参照海豚的现代的仿生学：在设计交通具的外形，就是参照海豚的标本上已知点及已知点的导数，做插值在计算机上模拟标本上已知点及已知点的导数，做插值在计算机上模拟海豚的外形制成飞机、汽车等外形。海豚的外形制成飞机、汽车等外形。3.5 Hermite插值插值构造二阶光滑度的插值函数构造

2、二阶光滑度的插值函数Hermite插值多项式插值多项式l构造H(x)njjjjnjjiiiiiiiifxyxxHnifxfxHxfxHxHnixffxfyxii00)()()(),.2, 1 ,0()()(),()()(),.,2, 1 ,0()(),(令满足希望，已知ijijxnixnixijijxLagrangeiijiijjj01)(4,.2, 1 ,00)(3,.2, 1 ,00)(201)(1）（）（）（）（插值函数我们设想由?)(?)(xxjj如何求0)(, 1)()(.)()(.)()(0:)()(.)()(.)()(0:)(11101110jjjnjjnjjjjjjjjxxxx

3、xxxxxxxxxxjjjjjjj而01112222201112222201112,.,.,( )() () .() () .()( )( )() () .() () .()( ) ( )( )21( )jjnjjjnjjjjjjjjnjxxxxxxxxxxxxxxxxxC xxxxxxxxxxxC x lxH xnC x则是的二重零点。所以令由于是次多项式，故为一次多项式。0)()(2)()(2)()()(0)()(10)(, 1)(22jjjjjjjjjjjjjjjjjjxlBAxAxlxlBAxxAlxBAxxlBAxxxjj即得：由)(21)(20)()(21jjjjjjjjjxlxB

4、xlAxlBAxABAx得由)()()(21 ()()(21)(2()(22xlxlxxxlxlxxxlxjjjjjjjjjjj故得：22( )( ) ( )( )() ( ) ( )jjjjxC x lxxAxB lxC xAxB令011110111101112()().()().()0 ()().()().()0 ()0,()1,.,.( ),( )() ( )jjjjjjjjjjjjjjjjjjjjjjnjjjxxxxxxxxxxxxx xxxxxxCxD lx由知道是的二重零点 所以设2()0()()2() () ()1jjjjjjjjjjjjxCxDxClxCxD lx lx2( )

5、() ()jjjxxx lx同理可得1jCDx 解得： 2( )() ( )jjjxxx lx所以Hermite插值余项插值余项特例（特例（n=1)2112111111313111331)()()21()()()21()()()()()(:)(,)()()(,)()()(,11313jjjjjjjjjjjjjjjjjjjjjjjjjjjjj-hxxfxxyhxxhxxfxxyhxxfxfxyxyxxHfxHfxHyxfxHyxfxHxHHermitexxjjjjj则满足条件：插值多项式上求二点三次对于区间)(),()()(! 41)()()(11221)4(33jjjjjjjxxxxhxxxx

6、fxHxfxR其中，例题例题算法实现算法实现).()(.).()().()(1)(),).()().()(1)()(11213132020102100nnnnnjjxxxxxxxxxxxxxxxxxxAxlxxxxxxAxxxxxxAxlxl则其中例如：）首先如何实现（nijnnnnxxxxxxxxxxxxxxxxxxxxxxxxxxAxl1002010102010030100302000)(1)(1.)(1)(1).()(.).()().()(1)(所以有njiijnjjjjjjjjjxxxxxxxxxxxxxl101110)(1)(1.)(1)(1.)(1)(1)(不是一般性算算 法法分段三次分段三次Hermite插值插值l分段线性插值曲线是折线，光滑性差，如果交通工具用这样的外形，则势必加大摩擦系数，增加阻力，因此用hermite分段插值更好。l例题2分段三次分段三次Hermite插值的程序设计插值的程序设计2221112122111111131)()()()()(21 ()()(21 ()()()()()()(,jjjjjjjjjjjjjjjjjjjjjjjjjjjjhxuxuuBhxuxuuBhxuhxuuAhxuhxuuAfxfxyxyxxHxxxHermite令时插值三次。输出则计算如果做对于输入插值点计算插值