差分形式的牛顿插值法（c++）

本程序对cosx函数进行插值，取步长为0.1，因此x的值为0.00,0.10,0.20,0.30，对应的y值为cos(0.00),cos(0.10),cos(0.20),cos(0.30)，其实本程序Horner方法（又称秦九韶算法）效率更高，计算更加准确

#include
<iostream>

#include
<cmath>

using namespace std;

int factorial(int
n);      //声明阶乘函数

double average_deviation(double*
function_value,double& h,int
end);      //声明均差函数

int main()

{

double
h=0.10;      //计算的步长

double
x[4]={0.00,0.10,0.20,0.30};     
//用来存储已知x的值

double
y[4]={cos(0.00),cos(0.10),cos(0.20),cos(0.30)};     
//用来存储已知y的值

double
result=0;      //用来保存最后的插值结果

double
input_x;      //插值的x值

double
polynomial;      //保存多项式的值

cout<<"通过插值得到的函数为："<<endl;

for (int
i=0;i<4;i++)

{

cout<<average_deviation(y,h,i)/(factorial(i))/pow(h,i);

for (int
j=0;j<i;j++)

cout<<"*"<<"("<<"x-"<<x[j]<<")";

if
(i<3&&((average_deviation(y,h,i+1)/(factorial(i+1))/pow(h,i+1))>0))

cout<<"+";

}

cout<<endl;

cout<<"请输入插值的x值：";

cin>>input_x;

for (int
i=0;i<4;i++)

{

polynomial=1;

for (int
j=0;j<i;j++)

{

polynomial*=(input_x-x[j]);

}

result+=(average_deviation(y,h,i)/(factorial(i))/pow(h,i)*polynomial);

}

cout<<"插值得到的函数值为："<<result<<endl;

cout<<"函数的真实值为："<<cos(input_x)<<endl;

cout<<"误差为："<<100*((result-cos(input_x))/cos(input_x))<<"%"<<endl;

return
0;

}

int factorial(int n)

{

if
(n<0)

{

cout<<"请输入正整数值"<<endl;

exit(0);

}

else if
(n==0)

return 1;

else

return n*factorial(n-1);

}

double average_deviation(double*
function_value,double& h,int end)

{

double*
function_value_f;     
//function_value_f用来保存0到k-1的k阶差分

double*
function_value_b;     
//function_value_f用来保存1到k的k阶差分

double
value_f;

double
value_b;

if
(0==end)

{

return function_value[end];

}

else

{

function_value_f=new
double[end];

function_value_b=new
double[end];

for (int
i=1;i<=end;i++)

function_value_f[i-1]=function_value[i];

for (int
i=0;i<=end-1;i++)

function_value_b[i]=function_value[i];

value_f =
average_deviation(function_value_f,h,end - 1);

value_b =
average_deviation(function_value_b,h, end - 1);

delete[] function_value_f;

delete[] function_value_b;

return value_f-value_b;

}

}