Best step size h for Richardson Extrapolation using Python 3

Question

I'm trying to use a while loop to determine the best value of h to minimise error in a Richardson extrapolation scheme, but I'm having trouble with the code. My script always outputs half the value of h that I input and displays no change in the relative error. The test function is: $$(x+2)e^x$$

Here is my script thus far:

f=(2+x)*np.exp(x)
x=2
h=0.4
n=4

def richardson(f,x,h,n): 
    '''Richardson extrapolation method for numerical calculation of second derivative '''
    n=4
    L=zeros(n,n)
    for I in range(n):
        L[I][0]=central_sec(function,x, 1/(2**(I+1)))
    for j in range(1,n):
        for i in range(n-j):
            L[i][j]=((4**(j))*L[i+1][j-1]-L[i][j-1])/(4**(j)-1)

    return L[0][n-1]

print('Approximation using Richardson Extrapolation is: ' + str(richardson(f, x, h, n)))

tol=1e-5
error=0.1
n=4

'''Calculating minimum error, relative error and best step size h'''

while(abs(error)>abs(tol)):
    F=richardson(f, x, h, n)
    h=h/2
    error=abs((F-analytic(x))/analytic(x))
    n=n+1

print("Function value is:", F, "Best h value is:", h, "Number of iterations:", n)

def relerror_richardson(f, x, h, n):
    f=abs((richardson(f, x, h, n)-analytic(2))/analytic(2))
    return f

print('Relative error is: ' + str(relerror_richardson(f, x, h, n)))


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