最小二乘法推导过程

乘法推导过程"/>

最小二乘法推导过程

超定方程组最优解（最小二乘解）推导
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#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);//作用是将某一块内存中的内容全部设置为指定的值， 这个函数通常为新申请的内存做初始化工作。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; 
}