Modified midpoint method for solving the initial value problem in Python，midpointsolving,''' yStop =

Modified midpoint method for solving the initial value problem in Python，midpointsolving,''' yStop =

''' yStop = integrate (F,x,y,xStop,tol=1.0e-6)    Modified midpoint method for solving the    initial value problem y' = F(x,y}.    x,y   = initial conditions    xStop = terminal value of x    yStop = y(xStop)    F     = user-supplied function that returns the            array F(x,y) = {y'[0],y'[1],...,y'[n-1]}.'''from numpy import zeros,float,sumfrom math import sqrtdef integrate(F,x,y,xStop,tol):    def midpoint(F,x,y,xStop,nSteps):  # Midpoint formulas        h = (xStop - x)/nSteps        y0 = y        y1 = y0 + h*F(x,y0)        for i in range(nSteps-1):            x = x + h            y2 = y0 + 2.0*h*F(x,y1)            y0 = y1            y1 = y2        return 0.5*(y1 + y0 + h*F(x,y2))    def richardson(r,k):  # Richardson's extrapolation              for j in range(k-1,0,-1):            const = (k/(k - 1.0))**(2.0*(k-j))            r[j] = (const*r[j+1] - r[j])/(const - 1.0)        return    kMax = 51    n = len(y)    r = zeros((kMax,n),dtype=float)  # Start with two integration steps    nSteps = 2    r[1] = midpoint(F,x,y,xStop,nSteps)    r_old = r[1].copy()  # Increase the number of integration points by 2  # and refine result by Richardson extrapolation    for k in range(2,kMax):        nSteps = 2*k        r[k] = midpoint(F,x,y,xStop,nSteps)        richardson(r,k)      # Compute RMS change in solution        e = sqrt(sum((r[1] - r_old)**2)/n)      # Check for convergence        if e < tol: return r[1]        r_old = r[1].copy()    print "Midpoint method did not converge"