"""
Contains several methods that solve Differential Equations.


"""

def euler(de, x, tstart, tend, steps):
    """
    Propagates x by solving diffeq using Eulers method.

    de is a funciton of t and x and return dxdt. 
    x is list with initial values.
    tstart starting time
    tend ending time
    steps number of steps
    """

    t=tstart
    dt=float(tend-tstart)/steps
    for i in range(steps):
       dx=de(t,x)
       x=[a+b*dt for (a,b) in zip(x,dx)]
       t+=dt
    return x


def second(de, x, tstart, tend, steps):
    """
    Propagates x by solving diffeq using Second Order method.

    de is a funciton of t and x and return dxdt. 
    x is list with initial values.
    tstart starting time
    tend ending time
    steps number of steps
    """

    t=tstart 
    dt=float(tend-tstart)/steps 
    for i in range(steps):
       dx=de(t,x) #derivative at initial point
       xt=[a+b*dt for (a,b) in zip(x,dx)] #step using euler
       t+=dt
       dx2=de(t,xt) # deriv at forward step
       x=[a+(b+c)*dt/2 for (a,b,c) in zip(x,dx,dx2)] #step forward with average
    return x



def runge(de, x, tstart, tend, steps):
    """
    Propagates x by solving diffeq using 4th order Runge-Kutta Method

    de is a funciton of t and x and return dxdt. 
    x is list with initial values.
    tstart starting time
    tend ending time
    steps number of steps
    """

    t=tstart
    dt=float(tend-tstart)/steps
    for i in range(steps):
# evaluate derivative at initial point to get k1
# half step forward with k1 and evaluate deriv to get k2
# half step forward with k2 and evaluate deriv to get k3
# full step forward with k3 and evaluate deriv to get k4
       k1=de(t,x)
       k2=de(t+dt/2,[a+b*dt/2.0 for a,b in zip(x,k1)])
       k3=de(t+dt/2,[a+b*dt/2.0 for a,b in zip(x,k2)])
       k4=de(t+dt,[a+b*dt for a,b in zip(x,k3)])
# full step forward with average of k1,k2,k3,k4 for new point
       x=[a+(b1+2*b2+2*b3+b4)/6.0*dt for (a,b1,b2,b3,b4) in zip(x,k1,k2,k3,k4)]
       t+=dt
    return x