human
I am working on the file "FEEG6016 Simulation and Modelling/01-Monte-Carlo-Lab-1.ipynb".
The first task is:
The convergence rate is the same (only roughly, typically), but the Mean Value method is expected to be better in terms of its absolute error.
Dimensionality
Compared to standard integration methods (Gauss quadrature, Simpson's rule, etc) the convergence rate for Monte Carlo methods is very slow. However, there is one crucial advantage: as you change dimension, the amount of calculation required is unchanged, whereas for standard methods it grows geometrically with the dimension.
Try to compute the volume of an $n$-dimensional unit hypersphere, which is the object in $\mathbb{R}^n$ such that
$$
\begin{equation}
  \sum_{i=1}^n x_i^2 \le 1.
\end{equation}
$$
The volume of the hypersphere can be found in closed form, but can rapidly be computed using the Monte Carlo method above, by counting the $k$ points that randomly fall within the hypersphere and using the standard formula $I \simeq V k / N$.
Can you write Python program for it?
 ... Write me a Python program to compute the volume of an $n$-dimensional unit hypersphere, which is the object in $\mathbb{R}^n$ such that?
assistant
#Here is Python program to compute the volume of an $n$-dimensional unit hypersphere, which is the object in $\mathbb{R}^n$ such that
.

def mc_integrate_multid(f, domain, N = 10000):
    """
    Monte Carlo integration in arbitrary dimensions (read from the size of the domain): to be completed
    """
    import numpy.random
    dimensionality = len(domain) #dimensionality of the hypersphere - determined by the length of the domain list
    
    #initialisation of lists/arrays
    domain_range = numpy.zeros(shape=(dimensionality, 1))
    funcValuesArray = numpy.zeros(N)

    for i in range(0, dimensionality):
        domain_range[i] = (domain[i][1]-domain[i][0])  #calculates domain range for each dimension

    for i in range(0, N):
        XYZ = []
        for j in range(0, dimensionality):
            XYZ.append(float(numpy.random.rand() * (domain_range[j] + domain[j][0]))) 
        #print XYZ
        funcValuesArray[i] = f(numpy.array(XYZ)) #creates an N element list of function values at random points (x, y, z ...) 
        #print(funcValuesArray[i])
    #print(funcValuesArray)

    
    zeroes = numpy.zeros(N)
    boolArray = funcValuesArray <= zeroes  #compares the array of function values with an of array of zeroes
    Hits = sum(boolArray) #sums the number of hits (i.e. points inside or on the surface of the hypersphere)
    hyperVolume = 1 #initialises hypervolume
    for i in range(0, dimensionality):
        hyperVolume *= domain_range[i] #calculates hypervolume of hypercuboid domain
        
    I = (Hits*hyperVolume)/N #calculates integral's value
    return float(I)#*2**dimensionality

def MakeHyperSphereOfRadiusR(R):
    def HyperSphere(ArrayOfDimensionalVariables):
        """defines a hypersphere of radius R with variables in the Array e.g. for a 3d
        sphere with radius R pass an array [x, y, z] and the radius R. The function returns a value
        which is <= 0 if the value is inside the function and >0 if it's outside the function"""
        #print(ArrayOfDimensionalVariables)
        return sum(ArrayOfDimensionalVariables**2)-R**2  #e.g. for 3d sphere for points inside or on the surface of the sphere...