最小二乘法拟合多项式原理以及c++实现

转载博客：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 Vnum, int n);
double MutilSum(vector Vx, vector Vy, int n);
double RelatePow(vector Vx, int n, int ex);
double RelateMutiXY(vector Vx, vector Vy, int n, int ex);
void EMatrix(vector Vx, vector 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 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 Vnum, int n)
{
	double dsum=0;
	for (int i=0; i Vx, vector Vy, int n)
{
	double dMultiSum=0;
	for (int i=0; i Vx, int n, int ex)
{
	double ReSum=0;
	for (int i=0; i Vx, vector Vy, int n, int ex)
{
	double dReMultiSum=0;
	for (int i=0; i Vx, vector 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=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; 
}