一，原理
二，根据原理得到的 python 源码。
import matplotlib.pyplot as plt
import math
import numpy
import random
fig = plt.figure()
ax = fig.add_subplot(111)
#阶数为9阶
order=9
#生成曲线上的各个点
x = numpy.arange(-1,1,0.02)
y = [((a*a-1)*(a*a-1)*(a*a-1)+0.5)*numpy.sin(a*2) for a in x]
#ax.plot(x,y,color='r',linestyle='-',marker='')
#,label="(a*a-1)*(a*a-1)*(a*a-1)+0.5"
#生成的曲线上的各个点偏移一下，并放入到xa,ya中去
i=0
xa=[]
ya=[]
for xx in x:
yy=y[i]
d=float(random.randint(70,130))/100
#ax.plot([xx*d],[yy*d],color='m',linestyle='',marker='.')
i+=1
xa.append(xx*d)
ya.append(yy*d)
'''''for i in range(0,5):
xx=float(random.randint(-100,100))/100
yy=float(random.randint(-60,60))/100
xa.append(xx)
ya.append(yy)'''
ax.plot(xa,ya,color='m',linestyle='',marker='.')
#plt.show()
#print(ax)
#进行曲线拟合
matA=[]
for i in range(0,order+1):
matA1=[]
for j in range(0,order+1):
tx=0.0
for k in range(0,len(xa)):
dx=1.0
for l in range(0,j+i):
dx=dx*xa[k]
tx+=dx
matA1.append(tx)
matA.append(matA1)
#print(len(xa))
#print(matA[0][0])
matA=numpy.array(matA)
matB=[]
for i in range(0,order+1):
ty=0.0
for k in range(0,len(xa)):
dy=1.0
for l in range(0,i):
dy=dy*xa[k]
ty+=ya[k]*dy
matB.append(ty)
matB=numpy.array(matB)
matAA=numpy.linalg.solve(matA,matB)
print (matAA)
print (matA)
print (matB)
#画出拟合后的曲线
#print(matAA)
xxa= numpy.arange(-1,1.06,0.01)
yya=[]
for i in range(0,len(xxa)):
yy=0.0
for j in range(0,order+1):
dy=1.0
for k in range(0,j):
dy*=xxa[i]
dy*=matAA[j]
yy+=dy
yya.append(yy)
ax.plot(xxa,yya,color='g',linestyle='-',marker='')
ax.legend("aw")
plt.show()
python的结果
三，下面是c++实现。
c++第一部分：这里用了一个自己写的Matrix 类，作为头文件Matrix.h
#include<iostream>
#include <fstream> // std::ifstream
#include <stdlib.h>
#include <cmath>
using namespace std;
class Matrix
{
private:
unsigned row, col, size;
double *pmm;//数组指针
public:
Matrix(unsigned r, unsigned c) :row(r), col(c)//非方阵构造
{
size = r*c;
if (size>0)
{
pmm = new double[size];
for (unsigned j = 0; j<size; j++) //init
{
pmm[j] = 0.0;
}
}
else
pmm = NULL;
};
Matrix(unsigned n) :row(n), col(n)//方阵构造
{
size = n*n;
if (size>0)
{
pmm = new double[size];
for (unsigned j = 0; j<size; j++) //init
{
pmm[j] = 0.0;
}
}
else
pmm = NULL;
};
Matrix(const Matrix &rhs)//拷贝构造
{
row = rhs.row;
col = rhs.col;
size = rhs.size;
pmm = new double[size];
for (unsigned i = 0; i<size; i++)
pmm[i] = rhs.pmm[i];
}
~Matrix()//析构
{
if (pmm != NULL)
{
delete[]pmm;
pmm = NULL;
}
}
Matrix &operator=(const Matrix&); //如果类成员有指针必须重写赋值运算符，必须是成员
friend istream &operator>>(istream&, Matrix&);
friend ofstream &operator<<(ofstream &out, Matrix &obj);
friend ostream &operator<<(ostream&, Matrix&);
friend Matrix operator+(const Matrix&, const Matrix&);
friend Matrix operator*(const Matrix&, const Matrix&);
Matrix cov(_In_opt_ bool flag = true); //协方差阵 或者样本方差
double det(); //行列式
Matrix solveAb(Matrix &obj); // b是行向量或者列向量
Matrix diag(); //返回对角线元素
//Matrix asigndiag(); //对角线元素
Matrix T()const; //转置
void sort(bool);//true为从小到大
Matrix adjoint();
Matrix inverse();
void QR(_Out_ Matrix&, _Out_ Matrix&)const;
Matrix eig_val(_In_opt_ unsigned _iters = 1000);
Matrix eig_vect(_In_opt_ unsigned _iters = 1000);
unsigned Row()const{ return row; }
unsigned Col()const{ return col; }
double norm1();//1范数
double norm2();//2范数
double*operator[](int i){ return pmm + i*col; }//注意this加括号， (*this)[i][j]
void zeromean(_In_opt_ bool flag = true);//默认参数为true计算列
void normalize(_In_opt_ bool flag = true);//默认参数为true计算列
Matrix exponent(double x);//每个元素x次幂
Matrix eye();//对角阵
void maxlimit(double max,double set=0);//对角阵
};
//友元仅仅是指定了访问权限，不是一般意义的函数声明，最好在类外再进行一次函数声明。
//istream &operator>>(istream &is, Matrix &obj);
//ostream &operator<<(ostream &is, Matrix &obj);
//对称性运算符，如算术，相等，应该是普通非成员函数。
//Matrix operator*( const Matrix&,const Matrix& );
//Matrix operator+( const Matrix&,const Matrix& );
//dets递归调用
double dets(int n, double *&aa)
{
if (n == 1)
return aa[0];
double *bb = new double[(n - 1)*(n - 1)];//创建n-1阶的代数余子式阵bb
int mov = 0;//判断行是否移动
double sum = 0.0;//sum为行列式的值
for (int arow = 0; arow<n; arow++) // a的行数把矩阵a(nn)赋值到b(n-1)
{
for (int brow = 0; brow<n - 1; brow++)//把aa阵第一列各元素的代数余子式存到bb
{
mov = arow > brow ? 0 : 1; //bb中小于arow的行，同行赋值，等于的错过，大于的加一
for (int j = 0; j<n - 1; j++) //从aa的第二列赋值到第n列
{
bb[brow*(n - 1) + j] = aa[(brow + mov)*n + j + 1];
}
}
int flag = (arow % 2 == 0 ? 1 : -1);//因为列数为0，所以行数是偶数时候，代数余子式为1.
sum += flag* aa[arow*n] * dets(n - 1, bb);//aa第一列各元素与其代数余子式积的和即为行列式
}
delete[]bb;
return sum;
}
Matrix Matrix::solveAb(Matrix &obj)
{
Matrix ret(row, 1);
if (size == 0 || obj.size == 0)
{
cout << "solveAb(Matrix &obj):this or obj is null" << endl;
return ret;
}
if (row != obj.size)
{
cout << "solveAb(Matrix &obj):the row of two matrix is not equal!" << endl;
return ret;
}
double *Dx = new double[row*row];
for (int i = 0; i<row; i++)
{
for (int j = 0; j<row; j++)
{
Dx[i*row + j] = pmm[i*row + j];
}
}
double D = dets(row, Dx);
if (D == 0)
{
cout << "Cramer法则只能计算系数矩阵为满秩的矩阵" << endl;
return ret;
}
for (int j = 0; j<row; j++)
{
for (int i = 0; i<row; i++)
{
for (int j = 0; j<row; j++)
{
Dx[i*row + j] = pmm[i*row + j];
}
}
for (int i = 0; i<row; i++)
{
Dx[i*row + j] = obj.pmm[i]; //obj赋值给第j列
}
//for( int i=0;i<row;i++) //print
//{
// for(int j=0; j<row;j++)
// {
// cout<< Dx[i*row+j]<<"\t";
// }
// cout<<endl;
//}
ret[j][0] = dets(row, Dx) / D;
}
delete[]Dx;
return ret;
}
Matrix Matrix::exponent(double x)//每个元素x次幂
{
Matrix ret(row, col);
double a;
for (unsigned i = 0; i< row; i++)
{
for (unsigned j = 0; j < col; j++)
{
a=ret[i][j]= pow(pmm[i*col + j],x);
}
}
return ret;
}
void Matrix::maxlimit(double max, double set)//每个元素x次幂
{
for (unsigned i = 0; i< row; i++)
{
for (unsigned j = 0; j < col; j++)
{
pmm[i*col + j] = pmm[i*col + j]>max ? 0 : pmm[i*col + j];
}
}
}
Matrix Matrix::eye()//对角阵
{
for (unsigned i = 0; i< row; i++)
{
for (unsigned j = 0; j < col; j++)
{
if (i == j)
{
pmm[i*col + j] = 1.0;
}
}
}
return *this;
}
void Matrix::zeromean(_In_opt_ bool flag)
{
if (flag == true) //计算列均值
{
double *mean = new double[col];
for (unsigned j = 0; j < col; j++)
{
mean[j] = 0.0;
for (unsigned i = 0; i < row; i++)
{
mean[j] += pmm[i*col + j];
}
mean[j] /= row;
}
for (unsigned j = 0; j < col; j++)
{
for (unsigned i = 0; i < row; i++)
{
pmm[i*col + j] -= mean[j];
}
}
delete[]mean;
}
else //计算行均值
{
double *mean = new double[row];
for (unsigned i = 0; i< row; i++)
{
mean[i] = 0.0;
for (unsigned j = 0; j < col; j++)
{
mean[i] += pmm[i*col + j];
}
mean[i] /= col;
}
for (unsigned i = 0; i < row; i++)
{
for (unsigned j = 0; j < col; j++)
{
pmm[i*col + j] -= mean[i];
}
}
delete[]mean;
}
}
void Matrix::normalize(_In_opt_ bool flag)
{
if (flag == true) //计算列均值
{
double *mean = new double[col];
for (unsigned j = 0; j < col; j++)
{
mean[j] = 0.0;
for (unsigned i = 0; i < row; i++)
{
mean[j] += pmm[i*col + j];
}
mean[j] /= row;
}
for (unsigned j = 0; j < col; j++)
{
for (unsigned i = 0; i < row; i++)
{
pmm[i*col + j] -= mean[j];
}
}
///计算标准差
for (unsigned j = 0; j < col; j++)
{
mean[j] = 0;
for (unsigned i = 0; i < row; i++)
{
mean[j] += pow(pmm[i*col + j],2);//列平方和
}
mean[j] = sqrt(mean[j] / row); // 开方
}
for (unsigned j = 0; j < col; j++)
{
for (unsigned i = 0; i < row; i++)
{
pmm[i*col + j] /= mean[j];//列平方和
}
}
delete[]mean;
}
else //计算行均值
{
double *mean = new double[row];
for (unsigned i = 0; i< row; i++)
{
mean[i] = 0.0;
for (unsigned j = 0; j < col; j++)
{
mean[i] += pmm[i*col + j];
}
mean[i] /= col;
}
for (unsigned i = 0; i < row; i++)
{
for (unsigned j = 0; j < col; j++)
{
pmm[i*col + j] -= mean[i];
}
}
///计算标准差
for (unsigned i = 0; i< row; i++)
{
mean[i] = 0.0;
for (unsigned j = 0; j < col; j++)
{
mean[i] += pow(pmm[i*col + j], 2);//列平方和
}
mean[i] = sqrt(mean[i] / col); // 开方
}
for (unsigned i = 0; i < row; i++)
{
for (unsigned j = 0; j < col; j++)
{
pmm[i*col + j] /= mean[i];
}
}
delete[]mean;
}
}
double Matrix::det()
{
if (col == row)
return dets(row, pmm);
else
{
cout << ("行列不相等无法计算") << endl;
return 0;
}
}
/////////////////////////////////////////////////////////////////////
istream &operator>>(istream &is, Matrix &obj)
{
for (unsigned i = 0; i<obj.size; i++)
{
is >> obj.pmm[i];
}
return is;
}
ostream &operator<<(ostream &out, Matrix &obj)
{
for (unsigned i = 0; i < obj.row; i++) //打印逆矩阵
{
for (unsigned j = 0; j < obj.col; j++)
{
out << (obj[i][j]) << "\t";
}
out << endl;
}
return out;
}
ofstream &operator<<(ofstream &out, Matrix &obj)//打印逆矩阵到文件
{
for (unsigned i = 0; i < obj.row; i++)
{
for (unsigned j = 0; j < obj.col; j++)
{
out << (obj[i][j]) << "\t";
}
out << endl;
}
return out;
}
Matrix operator+(const Matrix& lm, const Matrix& rm)
{
Matrix ret(lm.row, lm.col);
for (unsigned i = 0; i<ret.size; i++)
{
ret.pmm[i] = lm.pmm[i] + rm.pmm[i];
}
return ret;
}
Matrix operator*(const Matrix& lm, const Matrix& rm)
{
if (lm.size == 0 || rm.size == 0 || lm.col != rm.row)
{
Matrix temp(0, 0);
temp.pmm = NULL;
return temp; //数据不合法时候，返回空矩阵
}
Matrix ret(lm.row, rm.col);
for (unsigned i = 0; i<lm.row; i++)
{
for (unsigned j = 0; j< rm.col; j++)
{
for (unsigned k = 0; k< lm.col; k++)//lm.col == rm.row
{
ret.pmm[i*rm.col + j] += lm.pmm[i*lm.col + k] * rm.pmm[k*rm.col + j];
}
}
}
return ret;
}
Matrix& Matrix::operator=(const Matrix& rhs)
{
if (this != &rhs)
{
row = rhs.row;
col = rhs.col;
size = rhs.size;
if (pmm != NULL)
delete[] pmm;
pmm = new double[size];
for (unsigned i = 0; i<size; i++)
{
pmm[i] = rhs.pmm[i];
}
}
return *this;
}
//||matrix||_2 求A矩阵的2范数
double Matrix::norm2()
{
double norm = 0;
for (unsigned i = 0; i < size; ++i)
{
norm += pmm[i] * pmm[i];
}
return (double)sqrt(norm);
}
double Matrix::norm1()
{
double sum = 0;
for (unsigned i = 0; i < size; ++i)
{
sum += abs(pmm[i]);
}
return sum;
}
void Matrix::sort(bool flag)
{
double tem;
for (unsigned i = 0; i<size; i++)
{
for (unsigned j = i + 1; j<size; j++)
{
if (flag == true)
{
if (pmm[i]>pmm[j])
{
tem = pmm[i];
pmm[i] = pmm[j];
pmm[j] = tem;
}
}
else
{
if (pmm[i]<pmm[j])
{
tem = pmm[i];
pmm[i] = pmm[j];
pmm[j] = tem;
}
}
}
}
}
Matrix Matrix::diag()
{
if (row != col)
{
Matrix m(0);
cout << "diag():row != col" << endl;
return m;
}
Matrix m(row);
for (unsigned i = 0; i<row; i++)
{
m.pmm[i*row + i] = pmm[i*row + i];
}
return m;
}
Matrix Matrix::T()const
{
Matrix tem(col, row);
for (unsigned i = 0; i<row; i++)
{
for (unsigned j = 0; j<col; j++)
{
tem[j][i] = pmm[i*col + j];// (*this)[i][j]
}
}
return tem;
}
void Matrix::QR(Matrix &Q, Matrix &R) const
{
//如果A不是一个二维方阵，则提示错误，函数计算结束
if (row != col)
{
printf("ERROE: QR() parameter A is not a square matrix!\n");
return;
}
const unsigned N = row;
double *a = new double[N];
double *b = new double[N];
for (unsigned j = 0; j < N; ++j) //(Gram-Schmidt) 正交化方法
{
for (unsigned i = 0; i < N; ++i) //第j列的数据存到a，b
a[i] = b[i] = pmm[i * N + j];
for (unsigned i = 0; i<j; ++i) //第j列之前的列
{
R.pmm[i * N + j] = 0; //
for (unsigned m = 0; m < N; ++m)
{
R.pmm[i * N + j] += a[m] * Q.pmm[m *N + i]; //R[i,j]值为Q第i列与A的j列的内积
}
for (unsigned m = 0; m < N; ++m)
{
b[m] -= R.pmm[i * N + j] * Q.pmm[m * N + i]; //
}
}
double norm = 0;
for (unsigned i = 0; i < N; ++i)
{
norm += b[i] * b[i];
}
norm = (double)sqrt(norm);
R.pmm[j*N + j] = norm; //向量b[]的2范数存到R[j,j]
for (unsigned i = 0; i < N; ++i)
{
Q.pmm[i * N + j] = b[i] / norm; //Q 阵的第j列为单位化的b[]
}
}
delete[]a;
delete[]b;
}
Matrix Matrix::eig_val(_In_opt_ unsigned _iters)
{
if (size == 0 || row != col)
{
cout << "矩阵为空或者非方阵！" << endl;
Matrix rets(0);
return rets;
}
//if (det() == 0)
//{
// cout << "非满秩矩阵没法用QR分解计算特征值！" << endl;
// Matrix rets(0);
// return rets;
//}
const unsigned N = row;
Matrix matcopy(*this);//备份矩阵
Matrix Q(N), R(N);
/*当迭代次数足够多时,A 趋于上三角矩阵，上三角矩阵的对角元就是A的全部特征值。*/
for (unsigned k = 0; k < _iters; ++k)
{
//cout<<"this:\n"<<*this<<endl;
QR(Q, R);
*this = R*Q;
/* cout<<"Q:\n"<<Q<<endl;
cout<<"R:\n"<<R<<endl; */
}
Matrix val = diag();
*this = matcopy;//恢复原始矩阵；
return val;
}
Matrix Matrix::eig_vect(_In_opt_ unsigned _iters)
{
if (size == 0 || row != col)
{
cout << "矩阵为空或者非方阵！" << endl;
Matrix rets(0);
return rets;
}
if (det() == 0)
{
cout << "非满秩矩阵没法用QR分解计算特征向量！" << endl;
Matrix rets(0);
return rets;
}
Matrix matcopy(*this);//备份矩阵
Matrix eigenValue = eig_val(_iters);
Matrix ret(row);
const unsigned NUM = col;
double eValue;
double sum, midSum, diag;
Matrix copym(*this);
for (unsigned count = 0; count < NUM; ++count)
{
//计算特征值为eValue，求解特征向量时的系数矩阵
*this = copym;
eValue = eigenValue[count][count];
for (unsigned i = 0; i < col; ++i)//A-lambda*I
{
pmm[i * col + i] -= eValue;
}
//cout<<*this<<endl;
//将 this为阶梯型的上三角矩阵
for (unsigned i = 0; i < row - 1; ++i)
{
diag = pmm[i*col + i]; //提取对角元素
for (unsigned j = i; j < col; ++j)
{
pmm[i*col + j] /= diag; //【i,i】元素变为1
}
for (unsigned j = i + 1; j<row; ++j)
{
diag = pmm[j * col + i];
for (unsigned q = i; q < col; ++q)//消去第i+1行的第i个元素
{
pmm[j*col + q] -= diag*pmm[i*col + q];
}
}
}
//cout<<*this<<endl;
//特征向量最后一行元素置为1
midSum = ret.pmm[(ret.row - 1) * ret.col + count] = 1;
for (int m = row - 2; m >= 0; --m)
{
sum = 0;
for (unsigned j = m + 1; j < col; ++j)
{
sum += pmm[m * col + j] * ret.pmm[j * ret.col + count];
}
sum = -sum / pmm[m * col + m];
midSum += sum * sum;
ret.pmm[m * ret.col + count] = sum;
}
midSum = sqrt(midSum);
for (unsigned i = 0; i < ret.row; ++i)
{
ret.pmm[i * ret.col + count] /= midSum; //每次求出一个列向量
}
}
*this = matcopy;//恢复原始矩阵；
return ret;
}
Matrix Matrix::cov(bool flag)
{
//row 样本数，column 变量数
if (col == 0)
{
Matrix m(0);
return m;
}
double *mean = new double[col]; //均值向量
for (unsigned j = 0; j<col; j++) //init
{
mean[j] = 0.0;
}
Matrix ret(col);
for (unsigned j = 0; j<col; j++) //mean
{
for (unsigned i = 0; i<row; i++)
{
mean[j] += pmm[i*col + j];
}
mean[j] /= row;
}
unsigned i, k, j;
for (i = 0; i<col; i++) //第一个变量
{
for (j = i; j<col; j++) //第二个变量
{
for (k = 0; k<row; k++) //计算
{
ret[i][j] += (pmm[k*col + i] - mean[i])*(pmm[k*col + j] - mean[j]);
}
if (flag == true)
{
ret[i][j] /= (row-1);
}
else
{
ret[i][j] /= (row);
}
/*temp.Format("cov=%f,column=%d mean(%d)=%f,mean(%d)=%f",cov[i*column+j],row,i,mean[i],j,mean[j]);
MessageBox(temp);*/
}
}
for (i = 0; i<col; i++) //补全对应面
{
for (j = 0; j<i; j++)
{
ret[i][j] = ret[j][i];
}
}
return ret;
}
Matrix Matrix::adjoint()
{
//调动之前，检查时候方阵，这里默认为aa为方阵
//确定本函数是否修改传入的数据 ：no
//调用函数内删除内存delete []padjoint;
if (row != col)
{
Matrix ret(0);
return ret;
}
const int n = row;
Matrix ret(row);
double *bb = new double[(n - 1)*(n - 1)];//创建n-1阶的代数余子式阵bb
int pi, pj, q;
for (int ai = 0; ai<n; ai++) // a的行数把矩阵a(nn)赋值到b(n-1)
{
for (int aj = 0; aj<n; aj++)
{
for (int bi = 0; bi<n - 1; bi++)//把元素aa[ai][0]余子式存到bb[][]
{
for (int bj = 0; bj<n - 1; bj++)//把元素aa[ai][0]代数余子式存到bb[][]
{
if (ai>bi) //ai行的代数余子式是：小于ai的行，aa与bb阵，同行赋值
pi = 0;
else
pi = 1; //大于等于ai的行，取aa阵的ai+1行赋值给阵bb的bi行
if (aj>bj) //ai行的代数余子式是：小于ai的行，aa与bb阵，同行赋值
pj = 0;
else
pj = 1; //大于等于ai的行，取aa阵的ai+1行赋值给阵bb的bi行
bb[bi*(n - 1) + bj] = pmm[(bi + pi)*n + bj + pj];
}
}
if ((ai + aj) % 2 == 0) q = 1;//因为列数为0，所以行数是偶数时候，代数余子式为-1.
else q = (-1);
ret.pmm[ai*n + aj] = q*dets(n - 1, bb); //加符号变为代数余子式
}
}
delete[]bb;
return ret;
}
Matrix Matrix::inverse()
{
double det_aa = det();
if (det_aa == 0)
{
cout << "行列式为0 ，不能计算逆矩阵。" << endl;
Matrix rets(0);
return rets;
}
Matrix adj = adjoint();
Matrix ret(row);
for (unsigned i = 0; i<row; i++) //print
{
for (unsigned j = 0; j<col; j++)
{
ret.pmm[i*col + j] = adj.pmm[i*col + j] / det_aa;
}
}
return ret;
}
c++第二部分：主函数部分
#include "matrix.h"
#include <math.h>
#include<iostream>
#include<fstream>
const int N = 100;
double fun(double x)
{
return (pow((x*x - 1), 3.0) + 0.5)*sin(x * 2);
}
int main()
{
Matrix x(1, N);
Matrix y(1, N);
ofstream ofs;
ofs.open("pydata.txt", ofstream::out);
double src = -1.0;
for (int i = 0; i < N; i++)
{
src += 0.02;
x[0][i] = src;
y[0][i] = fun(src);
}
Matrix sx(x);//加入随机因素前的x
for (int i = 0; i < N; i++)
{
double ran = rand() % 600 / 1000.0 + 0.7; // 0.7~1.3 倍的随机
x[0][i] = x[0][i]*ran;
y[0][i] = y[0][i] * ran;
}
//构建矩阵
int k = 8; //次方数
Matrix xx(k + 1, k + 1); //累加矩阵k+1阶
for (size_t r = 0; r < k+1; r++)
{
for (size_t c = r; c < k+1; c++)
{
double sum = 0;
for (size_t i = 0; i < N; i++)
{
sum += pow(x[0][i], r+c);
}
xx[c][r]=xx[r][c] = sum;
}
}
cout << "xx:\n"<<xx;
Matrix yy(k+1, 1); //累加 y 向量yy
for (size_t r = 0; r < k + 1; r++)
{
double sum = 0;
for (size_t i = 0; i < N; i++)
{
sum += pow(x[0][i], r )* y[0][i];
}
yy[r][0] = sum;
}
cout << "yy:\n" << yy;
Matrix aa= xx.solveAb(yy); // 克拉默法则求 xx*aa = yy 中的aa
cout << "aa:\n" << aa << endl;
Matrix py(1, 100); //预测值 py
for (size_t i = 0; i < N ; i++) //对N个元素预测
{
double sum = 0;
for (size_t a = 0; a < k+1; a++)
{
sum += pow(sx[0][i], a)* aa[a][0];
}
py[0][i] = sum;
ofs << x[0][i] << " " << y[0][i] << " " << py[0][i]<<endl;
}
ofs.close();
system("pause");
}
c++第三部分： 根据c++的输出 pydata.txt 用python画图
import matplotlib.pyplot as plt
import numpy
fin = open("pydata.txt", "r")
line = fin.readline()
x=[]
y=[]
py=[]
while line:
#print(line)
line=line.strip()
line=line.split()
x.append(float(line[0]))
y.append(float(line[1]))
py.append(float(line[2]))
line = fin.readline()
fin.close()
ax = plt.subplot(111)
xa= numpy.arange(-1,1,0.02)
ax.plot(xa,y,color='m',linestyle='',marker='.')
ax.plot(xa,py,color='g',linestyle='-',marker='')
plt.show()
画图的结果
