import numpy as np
import matplotlib.pyplot as plt

def ydot(t,y):
    # right-hand side of differential equation
    #z =  t+y # oppgave (a)
    z =  t-y # oppgave (b)
    #z = 4*t-2*y # oppgave (c)
    return z
    
def eulerstep(t,y,h):
    # one step of Euler's Method
    # Input: current time t, current value y,  stepsize h
    # Output: approximate solution value at time t+h
    y=y+h*ydot(t,y)
    return y
    
def oppgave_6_1_5(y_exact):

    k = range(5)    
    H = []
    E = []
    for k in range(10):
        h = 0.1*2**(-k)
        n = round(1/h)
        
        t = 0 
        y = 0
        for i in range(n):
            y = eulerstep(t, y, h)
            t = t + h
        H.append(h)
        E.append(abs(y - y_exact(1)))

    # plot the error vs h
    plt.figure(0)
    plt.clf()
    fs = 12 # font size
    plt.loglog(H,E,'x-',lw=3, markersize=10)
    plt.xlabel('h', fontsize=fs)
    plt.ylabel('Error',fontsize=fs)
    plt.grid(True)
    plt.show()
    
if __name__ == "__main__":
        # Exact solution:
    # (a) 
    #y_exact = lambda t: -t-1+np.exp(t)
    # (b) 
    y_exact = lambda t: t-1+np.exp(-t)
    oppgave_6_1_5(y_exact)