最小二乘法拟合多项式原理以及c++实现

#include <iostream>
#include <vector>
#include <cmath>
using namespace std;

//最小二乘拟合相关函数定义
double sum(vector<double> Vnum, int n);
double MutilSum(vector<double> Vx, vector<double> Vy, int n);
double RelatePow(vector<double> Vx, int n, int ex);
double RelateMutiXY(vector<double> Vx, vector<double> Vy, int n, int ex);
void EMatrix(vector<double> Vx, vector<double> Vy, int n, int ex, double coefficient[]);
void CalEquation(int exp, double coefficient[]);
double F(double c[],int l,int m);
double Em[6][4];

//主函数，这里将数据拟合成二次曲线
int main(int argc, char* argv[])
{
double arry1[5]={0,0.25,0,5,0.75};
double arry2[5]={1,1.283,1.649,2.212,2.178};
double coefficient[5];
memset(coefficient,0,sizeof(double)*5);
vector<double> vx,vy;
for (int i=0; i<5; i++)
{
vx.push_back(arry1[i]);
vy.push_back(arry2[i]);
}
EMatrix(vx,vy,5,3,coefficient);
printf("拟合方程为：y = %lf + %lfx + %lfx^2 \n",coefficient[1],coefficient[2],coefficient[3]);
return 0;
}
//累加
double sum(vector<double> Vnum, int n)
{
double dsum=0;
for (int i=0; i<n; i++)
{
dsum+=Vnum[i];
}
return dsum;
}
//乘积和
double MutilSum(vector<double> Vx, vector<double> Vy, int n)
{
double dMultiSum=0;
for (int i=0; i<n; i++)
{
dMultiSum+=Vx[i]*Vy[i];
}
return dMultiSum;
}
//ex次方和
double RelatePow(vector<double> Vx, int n, int ex)
{
double ReSum=0;
for (int i=0; i<n; i++)
{
ReSum+=pow(Vx[i],ex);
}
return ReSum;
}
//x的ex次方与y的乘积的累加
double RelateMutiXY(vector<double> Vx, vector<double> Vy, int n, int ex)
{
double dReMultiSum=0;
for (int i=0; i<n; i++)
{
dReMultiSum+=pow(Vx[i],ex)*Vy[i];
}
return dReMultiSum;
}
//计算方程组的增广矩阵
void EMatrix(vector<double> Vx, vector<double> Vy, int n, int ex, double coefficient[])
{
for (int i=1; i<=ex; i++)
{
for (int j=1; j<=ex; j++)
{
Em[i][j]=RelatePow(Vx,n,i+j-2);
}
Em[i][ex+1]=RelateMutiXY(Vx,Vy,n,i-1);
}
Em[1][1]=n;
CalEquation(ex,coefficient);
}
//求解方程
void CalEquation(int exp, double coefficient[])
{
for(int k=1;k<exp;k++) //消元过程
{
for(int i=k+1;i<exp+1;i++)
{
double p1=0;

if(Em[k][k]!=0)
p1=Em[i][k]/Em[k][k];

for(int j=k;j<exp+2;j++)
Em[i][j]=Em[i][j]-Em[k][j]*p1;
}
}
coefficient[exp]=Em[exp][exp+1]/Em[exp][exp];
for(int l=exp-1;l>=1;l--)   //回代求解
coefficient[l]=(Em[l][exp+1]-F(coefficient,l+1,exp))/Em[l][l];
}
//供CalEquation函数调用
double F(double c[],int l,int m)
{
double sum=0;
for(int i=l;i<=m;i++)
sum+=Em[l-1][i]*c[i];
return sum;
}