@[拉格朗日插值、牛顿插值C++实现]

1、拉格朗日插值

1.1、线性插值

1.2、二次插值

1.3、N次拉格朗日插值多项式

2、牛顿差值

2.1、插商

2.1.1、一阶插商

2.1.2、二阶插商

2.1.3、K阶插商

2.2、线性差值

2.3、二次插值多项式

2.4、N次插值多项式

3、代码实现

#include<iostream>
using namespace std;
double n, x[100], y[100], l[100][100];

void input()
{
	for (int i = 0; i < n; i++)
		cin >> x[i] >> y[i];
}

//拉格朗日插值

double LagrangeInsert(double test)
{
	double temp1 = 1, temp2 = 1, result = 0;
	for (int m = 0; m < n; m++)
	{
		for (int i = 0; i < n; i++)
		{

			if (i != m)
			{   //求分子
				temp1 *= (test -x[i]);
				//求分母
				temp2 *= (x[m] - x[i]);
			}
		}
		//得出Lm（x）的结果
		result += ((temp1 / temp2) * y[m]);
		temp1 = 1;
		temp2 = 1;
	}
	return result;
}

//牛顿插值
double newton(double test)
{
	double re = 0;
	for (int i = 0; i < 100; i++) l[i][0] = y[i];
	for (int j = 1; j < n; j++)
	{
		for (int i = j; i < n; i++)
		{

			l[i][j] = (l[i][j - 1] - l[i - 1][j - 1]) / (x[i] - x[i - j]);
		}
	}
	double g[100];
	for (int i = 0; i < 100; i++) g[i] = 1.0;
	for (int i = 0; i < n; i++)
	{
		for (int j = 0; j < i; j++)
		{
			g[i] *= (test - x[j]);
		}
		re += g[i] * l[i][i];
	}
	return re;
}


int main()
{
	cout << "输入多少个点" << endl;
	cin >> n;
	cout << "输入插值点：" << endl;
	input();
	double test;
	while (1)
	{
		cout << "输入要计算的值或输入 -1 退出" << endl;
		cin >> test;
		if (test == -1) break;
		double re1 = newton(test);
		double re2 = LagrangeInsert(test);
		cout << "拉格朗日差值result: " << re1 << endl;
		cout << "牛顿差值result: " << re2 << endl;
	}
	return 0;
}

待博主有空后，再将上面的基础内容补齐。