牛頓法範例

newton.cpp

// 比較可以重複利用牛頓法的寫法（使用 function pointer）

#include <iostream>
using namespace std;

typedef double (*Equation)(double x); // 把 function pointer 命名為 Equation

double diff(Equation f, double x, double h=1e-6) {
	return (f(x + h) - f(x)) / h;
}

// The format should be the left part of the equation f(x) = 0
double f1(double x) {
	// You can change the function by yourself
	return x * x - 2;
}

// Another example of the equation
double f2(double x) {
	// to solve the equation x^2 -5x + 6 = 0, you will get different answer for different initial values
	// You will get x = 2 if the initial value is for example, 0
	// You will get x = 3 if the initial value is for example, 5
	// You can try different initial values!
	return x*x - 5*x + 6;
}

// Another example of the equation
double f3(double x) {
	// to solve the equation x^3 - 6x^2 + 11x - 6 = 0
	return x*x*x - 6*x*x + 11*x - 6;
}

double newton(Equation f, double x_init, int N) {
	double x = x_init;
	for (int i = 0; i < N; i++) {
		x = x - (f(x) / diff(f, x));
	}
	return x;
}

int main() {
	double x;
	int N;
	cout << "Please enter a initial point: ";
	cin >> x;
	cout << "Please enter the times to apply the Newton method (such as 100): ";
	cin >> N;
	cout << "We have the root: " << newton(f3, x, N) << endl;
	return 0;
}

newton_sqrt2.cpp

// 用牛頓法計算根號2的範例
#include <iostream>
using namespace std;

double f(double x) {
	return x * x - 2; // f(x) = x^2 - 2 = 0, to find x^2=2, x = ?
}

double f_diff(double x, double h=1e-6) {
	return (f(x + h) - f(x)) / h;
}

double newton(double x_init, int N) {
	double x = x_init;
	for (int i = 0; i < N; i++) {
		double x_next = x - (f(x) / f_diff(x));
		x = x_next;
	}
	return x;
}

int main() {
	double x;
	int N;
	cout << "Please enter a initial point: ";
	cin >> x;
	cout << "Please enter the times to apply the Newton method: ";
	cin >> N;
	cout << "We have the root: " << newton(x, N) << endl;
	return 0;
}