import numpy as np
import matplotlib.pyplot as plt
from math import factorial


#a)
"""
s[0] = 0
a[0] = x
s[j] = s[j-1] + a[j-1]
a[j] = -x**2/((2*j + 1)*2*j) * a[j-1]
"""

#b)
def sin_Taylor(x, n):
    a0 = x
    s0 = 0
    a = np.zeros(n+2)
    s = np.zeros(n+2)
    a[0] = a0

    for j in range(1,n+2):
        s[j] = s[j-1] + a[j-1]
        a[j] = -x**2/((2*j + 1)*2*j) * a[j-1]
    
    return s[-1], abs(a[-1])
        
print(sin_Taylor(np.pi/2, 20))

#c)
"""
Taylor-rekke for h?nd:
s[n+1] => s[3] = a[0] + a[1] + a[2]
a[0] = x
a[1] = -x**3 / 3!
a[2] = x**5 / 5! 
"""

def test_sin_Taylor():
    tol = 1e-10
    x = 0.5
    expected = x - x**3/(factorial(3)) + x**5 / factorial(5)
    computed = sin_Taylor(x, 2)[0]
    assert abs(expected - computed) < tol

test_sin_Taylor()


#d)
x_max = 4*np.pi

"""
x and y must be arrays, and we can use
a nested loop;
for every n value we loop through x, compute the Taylor series 
for each element in x, and fill the result into y. Then 
we plot y as a function of x for each n value. 
"""
x = np.linspace(0,x_max,1001)
y = np.zeros_like(x)

for n in range(10):
    for i in range(len(x)):
        y[i] = sin_Taylor(x[i],n)[0]
    plt.plot(x,y,label=f'n={n}')

plt.plot(x,np.sin(x),label='Exact')
plt.axis([0,x_max,-2,2])
plt.legend()
plt.show()