maltlab的埃尔米特插值问题请教

Question

求一个y=x^4的埃尔米特插值及误差的matlab编程
求一个y=x^4的埃尔米特插值及误差的matlab编程
求一个y=x^4的埃尔米特插值及误差的matlab编程

Answer 1

用MATLAB计算有关分段埃尔米特插值的误差
 
例6.6.8  设函数定义在区间上，取，按等距节点构造分段埃尔米特插值函数，用MATLAB程序在上计算和的误差公式和误差限.
解  在MATLAB工作窗口输入程序

syms h,x=-1:0.0001:1;
yxxxx=150000000./(1+25.*x.^2).^5.*x.^4-4500000./(1+25.*x.^2).^4.*x.^2+15000./(1+25.*x.^2).^3;
myxxxx=max(yxxxx), R=(h^4)* abs(myxxxx/384)
运行后输出的4阶导数在区间上绝对值的最大值myxxxx和在区间上的误差公式myxxxx为
myxxxx =                R =
15000              625/16*h^4
（4）在MATLAB工作窗口输入程序
h=0.2; R =625/16*h^4
运行后输出误差限为
R =
0.06250000000000
 
例6.6.9  设函数定义在区间上，取，按等距节点构造分段埃尔米特插值函数.
(1)用MATLAB程序计算各小区间中点处的值，作出节点，插值点，和的图形；
(2) 并用MATLAB程序计算各小区间中点处的值及其相对误差；
(3) 用MATLAB程序求和在区间上的误差公式和各插值的误差限.
解 （1）记节点的横坐标，插值点，.在MATLAB工作窗口输入程序
h=2*pi/9; x0=-pi:h:pi;
y0=tan(cos((sqrt(3)+sin(2*x0))./(3+4*x0.^2)));
xi=-pi+h/2:h:pi-h/2;
fi=tan(cos((3^(1/2)+sin(2*xi))./(3+4*xi.^2)));
b=max(x0); a=min(x0); x=a:0.001:b;
y=tan(cos((3^(1/2)+sin(2.*x))./(3+4*x.^2)));
Hi= hermitetx(x0,y0,xi,x,y);
Ri=abs((fi-Hi)./fi); xi,fi,Hi,Ri,i=[xi',fi',Hi',Ri']
title('函数y=tan(cos((sqrt(3)+sin(2x))/(3+4x^2)))及其分段埃尔米特插值函数，插值，节点(xi,yi) 的图形')
运行后屏幕显示各小区间中点xi处的函数值fi，插值Hi，相对误差值Ri，并且作出节点，插值点，和的图形(略).
（2）在MATLAB工作窗口输入程序
  syms  x
y=tan(cos((3^(1/2)+sin(2*x))/(3+4*x^2)));
yxxxx=diff(y,x,4),%simple(yxxxx)
运行后屏幕显示函数的4阶导数，然后将输出的编程求和及其在区间上的误差限的MATLAB程序如下
syms h,x=-pi:0.0001:pi;
yxxxx=-12.*(1.+tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2))).^2).^2.*sin((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2)).^3.*(2.*cos(2.*x)./(3.+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3.+4.*x.^2).^2.*x).^2.*(-4.*sin(2.*x)./(3.+4.*x.^2)-32.*cos(2.*x)./(3.+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3.+4.*x.^2).^3.*x.^2.-8.*(3.^(1./2)+sin(2.*x))./(3.+4.*x.^2).^2)+16.*(1.+tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2))).^2).^2.*sin((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2)).^4.*(2.*cos(2.*x)./(3.+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3.+4.*x.^2).^2.*x).^4.*tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2)))+8.*tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2))).^3.*(1.+tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^4.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3.+4.*x.^2).^2.*x).^4-8.*tan(cos((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2))).*(1.+tan(cos((3.^(1./2)+sin(2.*x))./(3.+4*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3.+4.*x.^2)).^2.*(2.*cos(2.*x)./(3.+4.*x.^2)-8*(3.^(1./2)+sin(2.*x))./(3.+4*x.^2).^2.*x).^4+6.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^2.*(-4.*sin(2.*x)./(3+4.*x.^2)-32.*cos(2.*x)./(3+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x.^2-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2)+(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^4-3.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(-4.*sin(2.*x)./(3+4.*x.^2)-32.*cos(2.*x)./(3+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x.^2-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2).^2-4.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).*(-8.*cos(2.*x)./(3+4.*x.^2)+96.*sin(2.*x)./(3+4.*x.^2).^2.*x+768.*cos(2.*x)./(3+4.*x.^2).^3.*x.^2-48.*cos(2.*x)./(3+4.*x.^2).^2-3072.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^4.*x.^3+384.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x)-(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(16.*sin(2.*x)./(3+4.*x.^2)+256.*cos(2.*x)./(3+4.*x.^2).^2.*x-3072.*sin(2.*x)./(3+4.*x.^2).^3.*x.^2+192.*sin(2.*x)./(3+4.*x.^2).^2-24576.*cos(2.*x)./(3+4.*x.^2).^4.*x.^3+3072.*cos(2.*x)./(3+4.*x.^2).^3.*x+98304.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^5.*x.^4-18432.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^4.*x.^2+384.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3)-12.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).^2*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^2.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^4.*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))-24.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^3.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^2.*(-4.*sin(2.*x)./(3+4.*x.^2)-32.*cos(2.*x)./(3+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x.^2-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2)-24.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2.*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^2.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^4.*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))+36.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^2.*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).*(-4.*sin(2.*x)./(3+4.*x.^2)-32.*cos(2.*x)./(3+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x.^2-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2)+6.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^2.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).^4+6.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^2.*(-4.*sin(2.*x)./(3+4.*x.^2)-32.*cos(2.*x)./(3+4.*x.^2).^2.*x+128.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x.^2-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2).^2+8.*tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).*(1+tan(cos((3.^(1./2)+sin(2.*x))./(3+4.*x.^2))).^2).*sin((3.^(1./2)+sin(2.*x))./(3+4.*x.^2)).^2.*(2.*cos(2.*x)./(3+4.*x.^2)-8.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^2.*x).*(-8.*cos(2.*x)./(3+4.*x.^2)+96.*sin(2.*x)./(3+4.*x.^2).^2.*x+768.*cos(2.*x)./(3+4.*x.^2).^3.*x.^2-48.*cos(2.*x)./(3+4.*x.^2).^2-3072.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^4.*x.^3+384.*(3.^(1./2)+sin(2.*x))./(3+4.*x.^2).^3.*x)
myxx=max(yxxxx), R=(h.^4).* abs(myxx./384)
将其保存为名为myxx.m的M文件，然后在MATLAB工作窗口输入该文件名
 myxx
运行后屏幕显示myxx =和在区间上的误差公式如下
myxx =                       R =
73.94706841647552   1734520780029061/9007199254740992*h^4
最后在MATLAB工作窗口输入
h=2*pi/9; R =1734520780029061/9007199254740992*h^4
运行后屏幕显示在区间上的误差限
R =
0.04574453029948