Monte Carlo simulation to find pi

Algorithm idea

basic idea of the code

Is to use the Monte Carlo method (Monte Carlo method) to estimate the $\pi$ . The Monte Carlo method is a numerical calculation method based on probability and statistics, and results or approximate values are obtained through random sampling. In this program, we generate a centered at the origin with radius $circle of r$ . Then we simulate throwing $n$ random points, if the point is inside the circle, it means that the current point can make a complete rotation around the circle; if the point is outside the circle, it means that there is no movement around the circle at this moment. Finally, the estimated number of points in the circle $p$ and total points $The ratio of n$ is multiplied by $4$ , an estimated value of pi can be obtained.

The code mainly includes the following parts:

monte_carlo_pi function

This function randomly throws n points inside a unit circle of radius r and returns the number of points thrown inside the circle. The process of throwing is actually to generate a binary point (x,y) according to the uniform distribution, and then judge whether the point is in the circle, that is, whether it satisfies $x^2+y^2\leqslant r ^2$ .

Main function main

This function uses the monte_carlo_pi function to simulate sampling with different random seeds for each sample. Then, divide the obtained sum of points in the circle by the total number of samples to obtain the ratio of the area of the circle to the area of the square $\frac{p}{n}$ Approximation $\hat{p}$ . Since the area of a circle is $\frac{\pi r^2}{4}$ , the area of the square is $r^2$ , so $\frac{p}{n} \approx \frac{\pi}{4}$ .

Finally, multiply by 4 to get the approximate $\pi$ and calculates the estimated bias from the sample standard deviation and confidence interval.

The usefulness of this code is to use the Monte Carlo method to estimate pi. This method can get more accurate results in a short time and is often used in numerical calculations.

full code

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>

/* 该函数模拟了在一个半径为r的圆内投掷n个点的情况，并返回样本中落在圆内的点的数量 */
int monte_carlo_pi(int n, float r)
{
    
    
    int i;
    float x, y, distance;
    int p = 0; /* 落在圆内的点的数量 */

    for (i = 0; i < n; i++) {
    
    
        /* 对每个点进行均匀随机分布，即随机生成一个二维点 (x, y)，且每个点被生成的概率相等 */
        x = (float) rand() / RAND_MAX * 2 * r - r;  /* 在[-r, r]范围内随机生成x坐标 */
        y = (float) rand() / RAND_MAX * 2 * r - r;  /* 在[-r, r]范围内随机生成y坐标 */

        /* 判断该点是否在圆内 */
        distance = sqrt(x*x + y*y);  /* 计算点到原点的距离 */
        if (distance <= r) {
    
       /* 如果距离小于等于r，则表示该点在圆内*/
            p++;  /* 投掷的点数加1 */
        }
    }

    return p;
}

/* 主函数，对输入的参数进行处理，调用monte_carlo_pi函数模拟多次投掷情况，并输出结果 */
int main(void)
{
    
    
    int i;  /* 循环计数器 */
    int n = 1000000;  /* 每次模拟中投掷的点的数量 */
    float r = 1.0;  /* 圆的半径 */
    double mu = 0, s = 0, p_mean, p_stddev, err;  /* p的均值、p的方差、样本均值、样本标准差、误差程度 */

    srand(time(NULL));  /* 设置随机数种子 */

    /* 利用不同的随机种子进行10次模拟 */
    for (i = 0; i < 10; i++) {
    
    
        /* 进行一次投掷，并累计落在圆内的点的数量 */
        int p = monte_carlo_pi(n, r);  /* 落在圆内的点的数量 */
        mu += p;  /* 各次投掷所得到的p的和，后面再除以10即为均值 */
        s += p * p;  /* 各次投掷所得到的p的平方的和，后面再除以10即为方差 */
    }

    /* 计算得到的p的平均值和标准差 */
    mu /= (n * 10); /* 根据大数定理，采样数量越多，样本均值越接近总体均值*/

    // 计算样本标准差
    s = s / (n * 10 - 1); // 样本方差：S^2=(∑(Xi-X)²)/(N-1)
    p_stddev = sqrt(s); // 样本标准差： S = √S^2

    err = 1.96 * p_stddev / sqrt(n * 10); /* 根据95%置信区间计算误差程度，1.96为正态分布的值 */

    /* 输出结果 */
    printf("样本数量：%d\n", n * 10);  /* 总共采样了多少个样本 */
    printf("圆周率的估计值：%lf\n", 4 * mu);  /* 各次投掷所得到的p的均值的均值 */
    printf("误差程度：+-%lf\n", 4 * err); /* 置信区间为平均值加减误差程度 */

    return 0;  /* 程序正常结束*/
}

operation result:

样本数量：10000000
圆周率的估计值：3.141101
误差程度：+-0.045014