from scipy import *
from scipy import integrate, optimize
from pylab import *

def Schrod_deriv(y, r, l, E, Z):
    "Given y=[u,u'] returns dy/dr=[u',u'']"
    du2 = (l*(l+1)/r**2-2*Z/r-E)*y[0]
    return [y[1],du2]

def Shoot(E,R,l,Z):
    y0=[0.0,-1e-7]
    Rb=R[::-1]
    y = integrate.odeint(Schrod_deriv, y0, Rb, args=(l,E ,Z))[:,0]
    norm = integrate.simps(Rb, y**2)
    f0 = y[-1]/sqrt(norm)
    f1 = y[-2]/sqrt(norm)
    final = f0 + (f1-f0)*(0.0-Rb[-1])/(Rb[-2]-Rb[-1])
    #print 'f0=', f0, 'f1=', f1, 'extrapolated=', final
    return final
    

(l, E, Z) = (0, -1., 1.)



R = logspace(-7,2.2,500)
E0 = -1.2
dE = 0.4
print optimize.brentq(Shoot, E0, E0+dE, args=(R,l,Z))