（1）、（2）
#include <iostream>
#include <cmath>
using namespace std;
double f(double x)
{
return pow(x,2);
}
int main()
{
double a,b,e;
int n;
cin>>a>>b>>e;
cin>>n;
double h=(b-a)/n;
double x[1000];
// x[0]=a;另外一种形式
double result_ti,result_simpson;
for(int i=0;i<=n;i++)
{
x[i]=a+i*h;
}
for(int k=0;k<n;k++)
{
result_ti+=(h/2)*(f(x[k])+f(x[k+1]));
result_simpson+=(h/6)*(f(x[k])+4*f((x[k]+x[k+1])/2)+f(x[k+1]));
}
/*
for(int k=1;k<n;k++)
{
x[k]=a+h*k;
mid_ti=(h/2)*(f(a)+2*f(x[k])+f(b));
// result_simpson=(b-a)/6*(f(a)+4f((a+b)/2)+f(b));
}
result_ti=(h/2)*(f(a)+mid_ti+f(b));
*/
cout<<"Ti result: "<<result_ti<<endl;
cout<<"Simpson result: "<<result_simpson<<endl;
cout<<"The exact result: "<<(pow(b,3)-pow(a,3))/3<<endl;
return 0;
}
（3）
//Sin(x)/x in [0,1]
#include <iostream>
#include <cmath>
#define eps 1e-12
using namespace std;
double a, b, h;
double fx(double x)
{
if (x==0)
return 1;
return sin(x)/x;
}
int main()
{
cin>>a>>b;
h = fabs(b-a);
double S1, S2, T1, T2, C1, C2, R1, R2;
double x, f;
T2 = T1 = (fx(a)+fx(b)) * h/2;
int k = 0;
R1=0; R2=1; h *= 2;
while (fabs(R2-R1)>eps)
{
cout<<T2<<" ";
if (k>1)
cout<<S2<<" ";
if (k>2)
cout<<C2<<" ";
if (k>3)
cout<<R2;
cout<<endl;
T1 = T2;
S1 = S2;
if (k>1)
C1 = C2;
if (k>2)
R1 = R2;
k++; h/=2;
f = 0; x = a + h/2;
while (x<b)
{
f += fx(x);
x += h;
}
T2 = (T1 + h * f) / 2;
S2 = T2 + (T2 - T1)/3;
if (k==1)
continue;
C2 = S2 + (S2 - S1)/15;
if (k==2)
continue;
R2 = C2 + (C2 - C1)/63;
}
return 0;
}
1. 拉格朗日插值在高次插值时同原函数偏差大、存在龙格现象，高次插值多项式不收敛。
2. 低次插值时，拉格朗日插值精度较高，与原值误差较小
分段插值时，一般分段越多，精度相应越高
找规律 #include <iostream> using namespace std;