While 循环比矢量化函数更有效

问：

我正在尝试使用随机数估计π值。最初，我使用了一种方法，即生成 1,000,000 个随机数，并计算其中有多少落在单位圆内以估计π。但是，当使用 while 循环时，我可以通过较少的绘制次数获得更精确的π估计值。为什么与具有固定次数的初始方法相比，这种循环方法在较少的绘制次数下产生更准确的结果？

1,000,000 次抽奖的初始代码：

def pi_estimation1(n):
    # Create a random number generator
    rng = np.random.default_rng()

    # Generate x and y coordinates simultaneously
    coordinates = rng.random((n, 2))

    # Use vectorized operations (UFunc) to filter points inside and outside the circle
    x_in_circle = coordinates[coordinates[:, 0]**2 + coordinates[:, 1]**2 <= 1]
    x_out_circle = coordinates[coordinates[:, 0]**2 + coordinates[:, 1]**2 > 1]

    

    # Calculate the value of pi based on the ratio of points inside the circle
    pi = 4 * len(x_in_circle) / n
    return pi, x_in_circle, x_out_circle,n

# Example of usage
estimated_pi, x_in_circle, x_out_circle,n = pi_estimation1(1000000)
# # Plot points inside and outside the circle
plt.figure(figsize=(5, 5))
plt.scatter(x_in_circle[:, 0], x_in_circle[:, 1], color='blue')
plt.scatter(x_out_circle[:, 0], x_out_circle[:, 1], color='red')
plt.show()

print("The estimated value of pi is ", estimated_pi)
print("The exact value of pi is ", np.pi)
print("The error relative to the exact value is ", abs(estimated_pi - np.pi) / np.pi)
print("Number of points ",n)

结果：

The estimated value of pi is  3.140236
The exact value of pi is  3.141592653589793
The error relative to the exact value is  0.0004318362497579398
Number of points  1000000

带有 while 循环的代码：

def pi_estimation_while_loop(deviation):
    
    # Initialize variables
    pi = 0
    n = 0
    dev=1
    in_circle = 0
    
    rng=np.random.default_rng()
    
    # Loop until the deviation is less than 0.0001
    while deviation < dev :
        coordinate = rng.random((1, 2))
        if coordinate[0,0]**2 + coordinate[0,1]**2 <= 1:
            in_circle += 1
        
    
        n += 1
        pi = 4 * in_circle / n
        dev=(abs(pi-np.pi)/np.pi)
        #
        
        
        
    return pi, n, dev

pi, n, dev = pi_estimation_while_loop(1e-9)
print("The estimated value of pi is ", pi)
print("The exact value of pi is ", np.pi)
print("The error relative to the exact value is ", abs(pi - np.pi) / np.pi)
print("The number of iterations is ", n)

结果：

The estimated value of pi is  3.141592654644592
The exact value of pi is  3.141592653589793
The error relative to the exact value is  3.3575292254192973e-10
The number of iterations is  499581

我想知道这两种方法之间准确性差异的可能原因，以及如何以最少的平局次数优化π的估计。任何帮助或解释将不胜感激。谢谢！