【C++类的继承、父子类】牛顿插值公式求近似值

news2024/12/28 5:45:55

/**
 * @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

$i$	$x_i$	$Fx_i$	一阶	二阶	三阶
0	121	11
1	144	12	$\frac{1}{23}$
2	169	13	$\frac{1}{25}$	$-\frac{1}{13800}$
3	196	14	$\frac{1}{27}$	$-\frac{1}{17550}$	≈ $2.06449*10^{-7}$

公式

一阶差商 $F[x_{i-1},x_i]=\frac{Fx_{i-1}-Fx_i}{x_{i-1}-x_i}$

$F[x_0,x_1)]=\frac{Fx_0-Fx_1}{x_0-x_1}=\frac{11-12}{121-144}=\frac{-1}{-23}=\frac{1}{23}$

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

在这里插入图片描述

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;
}