General tangent function

I have an interesting problem in Python where I have two functions (arbitrary) and I would like to find a common tangent between them and points on the x axis where the common tangent touches each function. Ideally, there would be a function that gives all the coordinates (imagine a very seductive set of functions with several solutions).

So, I have something that works, but it is very rude. What I did puts each function in a matrix, so it M[h,0]contains the values ​​of x, M[h,1]- the values ​​of the function 1 y, M[h,2]- the values ​​of the function 2 y. Then I find the derivative and put it in a new matrix D[h,1]and D[h,2](the interval is one less than M[h,:]. My idea was to essentially "construct" the slopes along the x axis and intercepts by y and find all these points for the nearest pair, then give values.

Two issues here:

• the program does not know whether the next pair is a solution or not, and

• it is painfully slow (numb_of_points ^ 2 search). I understand that some optimization libraries may help, but I'm worried that they will hone one solution and ignore the others.

Does anyone think of a better way to do this? My "code" is here:

def common_tangent(T):
    x_points = 600
    x_start = 0.0001 
    x_end = 0.9999
    M = zeros(((x_points+1),5) )
    for h in range(M.shape[0]):
        M[h,0] = x_start + ((x_end-x_start)/x_points)*(h) # populate matrix
        """ Some function 1 """
        M[h,1] = T*M[h,0]**2 + 56 + log(M[h,0])
        """ Some function 2 """
        M[h,2] = 5*T*M[h,0]**3 + T*M[h,0]**2 - log(M[h,0])
    der1 = ediff1d(M[:,1])*x_points # derivative of the first function
    der2 = ediff1d(M[:,2])*x_points # derivative of the second function
    D = zeros(((x_points),9) )
    for h in range(D.shape[0]):
        D[h,0] = (M[h,0]+M[h+1,0])/2 # for the der matric, find the point between
        D[h,1] = der1[h] # slope m_1 at this point
        D[h,2] = der2[h] # slope m_2 at this point
        D[h,3] = (M[h,1]+M[h+1,1])/2# average y_1 here
        D[h,4] = (M[h,2]+M[h+1,2])/2# average y_2 here
        D[h,5] = D[h,3] - D[h,1]*D[h,0] # y-intercept 1
        D[h,6] = D[h,4] - D[h,2]*D[h,0] # y-intercept 2
    monitor_distance = 5000 # some starting number
    for h in range(D.shape[0]):
        for w in range(D.shape[0]):
            distance = sqrt( #in "slope intercept space" find distance
                (D[w,6] - D[h,5])**2 +
                (D[w,2] - D[h,1])**2
                )
            if distance < monitor_distance: # do until the closest is found
                monitor_distance = distance
                fraction1 = D[h,0]
                fraction2 = D[w,0]
                slope_02 = D[w,2]
                slope_01 = D[h,1]
                intercept_01 = D[h,5]
                intercept_02 = D[w,6]
    return (fraction1, fraction2)

It has many applications in materials science, in the search for common tangents between several Gibb functions for calculating phase diagrams. It would be nice to get a reliable feature for everyone to use ...