# Exercise 2b

# Here we have a Python function which calculates the integral in two
# different ways: numerically and analytically.
#
# We want to check the precision of the numerical integration as a function of
# the number of steps (subintervals). To do this, we calculate and print the
# relative differences between the analytic result and the numerical result
# for different values of the number of steps.
#
# Steps:
# 0. Familizare yourselves with code below.
# 1. Implement the serial version using a for loop or map
# 2. Implement the parallel version using multiprocessing.Pool
# 3. Time both versions
# 4. What (if any) do you get?

def integrate(f, a, b, n):
    "Perform numerical integration of f in range [a, b], with n steps"
    s = []
    for i in range(n):
        dx = (b - a) / n
        x = a + (i + 0.5) * dx
        y = f(x)
        s = s + [y * dx]
    return sum(s)

def f(x):
    "A polynomial that we'll integrate"
    return x ** 4 - 3 * x

def F(x):
    "The analatic integral of f. (F' = f)"
    return 1 / 5 * x ** 5 - 3 / 2 * x ** 2

def compute_error(n):
    "Calculate the difference between the numerical and analytical integration results"
    a = -1.0
    b = +2.0
    F_analytical = F(b) - F(a)
    F_numerical = integrate(f, a, b, n)
    return abs((F_numerical - F_analytical) / F_analytical)

def main():
    ns = [10_000, 25_000, 50_000, 75_000]
    errors = ... # TODO: write a for loop, serial map, and parallel map here

    for n, e in zip(ns, errors):
        print(f'{n} {e:.8%}')

if __name__ == '__main__':
    main()

# Bonus steps, very optional:
# 6. Implement a parallel version with threads (using multiprocessing.pool.ThreadPool).
# 7. Time this version, and hypothetize about the result.