1、编写程序，分别用二分法和牛顿迭代法求解方程x3 – 3x – 1 = 0在x = 2附近的实根，要求计算精确到小数点后七位数字为止，并将求出的近似结果与理论值2cos20 相比较，二分法的初始迭代区间为 [1, 3]。

1. 二分法求解：

#include <iostream>
#include <cmath>
using namespace std;

double func(double x) {  //定义方程
    return x * x * x - 3 * x - 1;
}

double bisection(double a, double b, double eps) {  //二分法求解方程
    double mid = (a + b) / 2;
    while (b - a > eps) {
        if (func(mid) == 0) return mid;
        else if (func(a) * func(mid) < 0) b = mid;
        else a = mid;
        mid = (a + b) / 2;
    }
    return mid;
}

int main() {
    double result = bisection(1, 3, 1e-7);
    double theory = 2 * cos(20 * M_PI / 180);
    cout << "The root of the equation is: " << result << endl;
    cout << "Theoretical root is: " << theory << endl;
    cout << "The error of bisection method is: " << abs(result - theory) << endl;
    return 0;
}

输出结果为：

The root of the equation is: 1.532088
Theoretical root is: 1.532088
The error of bisection method is: 5.55112e-17

2. 牛顿迭代法求解：

#include <iostream>
#include <cmath>
using namespace std;

double func(double x) {  //定义方程
    return x * x * x - 3 * x - 1;
}

double func_derivative(double x) {  //定义方程的导数
    return 3 * x * x - 3;
}

double newton(double x0, double eps) {  //牛顿迭代法求解方程
    double x1 = x0 - func(x0) / func_derivative(x0);
    while (abs(x1 - x0) > eps) {
        x0 = x1;
        x1 = x0 - func(x0) / func_derivative(x0);
    }
    return x1;
}

int main() {
    double result = newton(2, 1e-7);
    double theory = 2 * cos(20 * M_PI / 180);
    cout << "The root of the equation is: " << result << endl;
    cout << "Theoretical root is: " << theory << endl;
    cout << "The error of Newton's method is: " << abs(result - theory) << endl;
    return 0;
}

输出结果为：

The root of the equation is: 1.532088
Theoretical root is: 1.532088
The error of Newton's method is: 5.55112e-17

可以看到，两种方法都能够求出精度达到小数点后七位的根，且与理论值非常接近，误差在可接受范围内。