Assignment 1 Problem 2

Exercise 2

---

1. Write an iterative function to compute the factorial of a natural number
2. Write a recursive function to compute the factorial of a natural number.
3. Write a function to compute $\frac{x^n}{n!}$ given a float $x$ and natural number $n$ as arguments
4. Write a function to iteratively sum up the value $\frac{x^n}{n!}$ from $n = 1$ to a given $N$ for a given $x$, i.e., $$ F(x, N) = 1 + \sum_{i = 1}^N \dfrac{x^n}{n!}$$
5. Write a function to iteratively sum up the value $\frac{x^n}{n!}$ from $n = 1$ to a chosen value of $N'$ such that $$F(x, N') - F(x, N' - 1) < \epsilon$$ for given real number $x$ and positive small number $\epsilon$
6. Choose two real numbers $p$ and $q$ and compute the following two values $$ v_1 = F(p, 100) * F(q, 100) $$ $$ v_2 = F(p + q, 100) $$ Compute the difference $v_1 - v_2$ and comment on what you see.

Solution to Problem 2

def factorial(n):
    product = 1
    for i in range(n):
        # print(i, product)
        product *= (i + 1)
    return product

print(factorial(3))

6

def factorial_recursive(n):
    if n > 1:
        return n * factorial_recursive(n - 1)
    else:
        return 1

print(factorial_recursive(3))

6

def compute_ratio(x, n):
    ratio = x**n / factorial(n)
    return ratio

compute_ratio(2, 3)

1.3333333333333333

def compute_sum(x, N):
    sum = 1
    for i in range(N):
        sum += compute_ratio(x, i + 1)
    return sum

print(compute_sum(2, 3))

6.333333333333333

def compute_sum_epsilon(x, epsilon):
    sum = 1
    var = epsilon
    i = 1
    while var >= epsilon:
        var = compute_ratio(x, i)
        sum += var
        i += 1
    return sum

print(compute_sum_epsilon(2, 0.01))

7.387301587301587

print(compute_sum(2, 100))

7.389056098930649

def compute_sum_epsilon(x, epsilon):
    sum = 1
    i = 1
    while True: 
        var = compute_ratio(x, i)
        sum += var
        i += 1
        if var < epsilon:
            break
    return sum

print(compute_sum_epsilon(2, 0.01))

7.387301587301587

p = -1.5
q = 7.1

v1 = compute_sum(p, 100) * compute_sum(q, 100)
v2 = compute_sum(p + q, 100)

print(v1, v2, v2 - v1)

270.4264074261525 270.4264074261526 1.1368683772161603e-13

for i in range(10):
    print(i, compute_sum(i, 100))

0 1.0
1 2.7182818284590455
2 7.389056098930649
3 20.08553692318766
4 54.598150033144265
5 148.41315910257657
6 403.4287934927351
7 1096.6331584284578
8 2980.957987041728
9 8103.083927575384