Revision history [back]

It seems like this is just ridiculously slow, but I'm not sure why. I tried tweaking your code in a variety of ways and nothing helped. To try to figure out what was going on I wrote the following Runge-Kutta 4th order code in Cython; its pretty hackish but it makes it clear that the solution isn't doing anything very strange. The version below is hard-coded for your ll=2.

%cython

cdef double ll = 2.0

cpdef double leg_f1(double t, double y, double yp):
    return yp

cpdef double leg_f2(double t, double y, double yp):
    return ((ll**2 + ll)*y - 2.0*yp*t)/(t**2 - 1.0)

cpdef RK4_2d(f1, f2, double t_start, double y_start, double yp_start, double t_end, int steps):
    cdef double step_size = (t_end - t_start)/steps
    cdef double t_current = t_start
    cdef double y_current = y_start
    cdef double yp_current = yp_start
    cdef list answer_table = []
    cdef int j
    answer_table.append([t_current,y_current, yp_current])
    for j in range(0,steps):
        k1=f1(t_current, y_current, yp_current)
        k2=f1(t_current+step_size/2, y_current + k1*step_size/2, yp_current + k1*step_size/2)
        k3=f1(t_current+step_size/2, y_current + k2*step_size/2, yp_current + k1*step_size/2)
        k4=f1(t_current+step_size, y_current + k3*step_size, yp_current + k1*step_size/2)
        y_current = y_current + (step_size/6)*(k1+2*k2+2*k3+k4)
        k1=f2(t_current, y_current, yp_current)
        k2=f2(t_current+step_size/2, y_current + k1*step_size/2, yp_current + k1*step_size/2)
        k3=f2(t_current+step_size/2, y_current + k2*step_size/2, yp_current + k1*step_size/2)
        k4=f2(t_current+step_size, y_current + k3*step_size, yp_current + k1*step_size/2)
        yp_current = yp_current + (step_size/6)*(k1+2*k2+2*k3+k4)
        t_current += step_size
        answer_table.append([t_current,y_current,yp_current])
    return answer_table

and then you can get a plot with:

ans = RK4_2d(leg_f1, leg_f2, 0.0, 0.0, 1.0, 1.0, 40)
list_plot([[x[0],x[1]] for x in ans], plotjoined = True)

Anyway for me it seems easier to implement such things directly than to wrestle with the GSL wrapping.