最小二乘法拟合多项式原理以及c++实现

转载请注明出处：http://blog.csdn.net/lsh_2013/article/details/46697625

最小二乘法（又称最小平方法）是一种数学优化技术。它通过最小化误差的平方和寻找数据的最佳函数匹配。

c++实现代码如下：

 #include
 #include
 #include
 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
     {
         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
     {
         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
     {
         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
     {
         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//消元过程
     {
         for(int i=k+1;i
         {
             double p1=0;

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

             for(int j=k;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;
 }