Hermit 插值：
function yi=hermite_wise(x,y,ydot,xi)
%分段hermit插值公式,其中
%这里x为向量，全部的插值节点;y为向量,插值节点处的函数值
%ydot为向量,插值节点处的导数值
%如果此处值缺省,则用均差代替导数
%端点用向前,向后均差,中间点用中心均差
%xi为标量,自变量x
%yi为xi处的函数估计值
%如果没有给出y的导数值,则用均差代替导数值
if isempty(ydot)==1
ydot=gradient(y,x);
end
n=length(x);m1=length(y);m2=length(ydot);
%输入x,y和y的导数的个数必须相同
if n~=m1|n~=m2|m1~=m2
error('The length of X,Y and Ydot must be equal!');
return;
end
for k=1:n-1
%插值节点必须互异
if abs(x(k)-x(k+1))<eps
error('The DATA is error!');
return;
end
if x(k)<=xi&xi<=x(k+1)
yi=y(k)*(1-2*(xi-x(k))/(x(k)-x(k+1)))...
*(xi-x(k+1))^2/(x(k)-x(k+1))^2+...
y(k+1)*(1-2*(xi-x(k+1))/(x(k+1)-x(k)))...
*(xi-x(k))^2/(x(k+1)-x(k))^2+...
ydot(k)*(xi-x(k))*(xi-x(k+1))^2/(x(k)-x(k+1))^2+...
ydot(k+1)*(xi-x(k+1))*(xi-x(k))^2/(x(k+1)-x(k))^2;
return
end
end
运行的
clear
a=-5;b=5;n=10;h=(b-a)/n;
x=a:h:b;y=1./(1+x.^2);
xx=a:0.01:b;yy=1./(1+xx.^2);
m=length(xx);z=zeros(1,m);
for i = 1:m
z(i)=hermite_wise(x,y,[],xx(i));
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