/*** @file            * @author			jUicE_g2R(qq:3406291309)————彬(bin-必应)*						通信与信息专业大二在读	* * @brief			Microsoft 源代码注释语言 SAL* * @copyright		2023.10* @COPYRIGHT			 原创技术笔记：转载需获得博主本人同意，且需标明转载源** @language        C/C++** @IDE			   Base on Microsoft Visual Studio 2022*/

数据

公式

一阶差商 $F[x_{i-1},x_i]=\frac{F{x}_{i-1}-F{x}_i}{x_{i-1}-x_i}$

$F[x_0,x_1)]=\frac{F{x}_0-F{x}_1}{x_0-x_1}=\frac{11-12}{121-144}=\frac{-1}{-23}=\frac{1}{23}$

二阶差商 $F[x_{i-2},x_{i-1},x_i]=\frac{F[x_{i-2},x_{i-1}]-F[x_{i-1},x_i]}{x_{i-2}-x_i]}$

$F[x_0,x_1,x_2]=\frac{F[x_0,x_1]-F[x_1,x_2]}{x_0-x_2}=\frac{\frac{1}{23}-\frac{1}{25}}{121-169}=\frac{\frac{2}{575}}{-48}=-\frac{1}{13800}$

三阶差商 $F[x_{i-3},x_{i-2},x_{i-1},x_i]=\frac{F[x_{i-3},x_{i-2},x_{i-1}]-F[x_{i-2},x_{i-1},x_i]}{x_{i-3}-x_i]}$

牛顿插值公式

C++代码实现

//牛顿插值
#include <bits/stdc++.h>
using namespace std;#define NUMSIZE 10
double FX[NUMSIZE][2] = { {121,11},{144,12},{169,13},{196,14} };						//第一个存x，第二个存y；fx=√x
bool AddFlag;
void Add(int x, int y, int loc) { FX[loc][0] = x;		FX[loc][1] = y; }class InEquality {																		//求均差
public:double ValidDif[NUMSIZE];															//返回公式要用的值double Valid[NUMSIZE];void RetDev(int xNum) {_xNum = xNum;if (!AddFlag) {for (int i = 1; i < xNum; i++) {											//从一阶差商（均差）开始_order = i;GetVal();}}else {_order = xNum;GetVal();}}
private:int _xNum, _order;double dif[NUMSIZE][NUMSIZE];														//第一个存阶乘数。第二个存均差void GetVal() {int st_i = _order;if (_order == 1) {																//求一阶差商for (int i = st_i; i < _xNum; i++) {dif[_order][i] = (FX[i - 1][1] - FX[i][1]) / (FX[i - 1][0] - FX[i][0]);	//(Fx_i - Fx_i+1)/(x_i - x_i+1)}}else {for (int i = st_i; i < _xNum; i++) {										//求非一阶差商dif[_order][i] = (dif[_order - 1][i - 1] - dif[_order - 1][i]) / (FX[i - _order][0] - FX[i][0]);}}ValidDif[_order] = dif[_order][_order];}
};class DeltaRide {
public:double DR[NUMSIZE] = {1};void RetRes(int xNum, int x) {_xNum = xNum;		_x = x;GetVal();}
private:int _xNum, _x;void GetVal(void) {for (int i = 1; i < _xNum; i++) {DR[i] = DR[i - 1] * (_x - FX[i - 1][0]);}}
};class Newton :public InEquality, public DeltaRide {										//继承 两个父类 中 public范围 的 数据以及函数
public:double RetNWT(int xNum) {_xNum = xNum;return Merge();}
private:double _Nx;int _xNum;double Merge(void){_Nx = DR[0] * FX[0][1];for (int i = 1; i < _xNum; i++) {_Nx += DR[i] * ValidDif[i];}return _Nx;}
};void Display(int n) {cout << "====================" << endl;cout << "编号	x     y" << endl;for (int i = 0; i < n; i++) {cout << i << "     " << FX[i][0] << "     " << FX[i][1] << endl;}cout << "====================" << endl;
}int main(int* argc, char* argv[]) {int n = 4;AddFlag = false;Display(n);int x;		cout << "请输入数据："; cin >> x;Newton Nx;Nx.RetDev(n);Nx.RetRes(n, x);cout << "√" << x << "=" << Nx.RetNWT(n);//...AddFlag=true，向数据模型继续补充，使误差更小return 0;
}

结果展示

分析

这样数据已经非常逼近了，可以继续Add数据，使模型更贴合√x函数曲线，减小计算得到值的误差