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pythonmathmathematical-expressions
Maria Evans
Maria Evans

Reputation: 25

Interpreting formulas with ΣΣ (two sigma) in Python

Consider the function f(x,y) that equals two sigma (ΣΣ) where i ranges from 1 to 10 and (first sigma) and j ranges from 1 to 10 (second sigma) of the quantity {ix^2 + jy^3)

I believe the first sigma would be an inner loop and the second sigma an outer loop, but I am having trouble rewriting this into Python.

How can I convert this into Python?

Please click to see the mathematical expression. I cannot embed

Upvotes: 1

Views: 706

Answers (3)

trincot
trincot

Reputation: 350310

The mathematical formula can be rewritten without summation, leading to this simple function:

def f(x, y):
    return 550 * (x * x  + y * y * y)

Here is how it is derived:

          ∑𝑖=1..10∑𝑗=1..10𝑖𝑥² + 𝑗𝑦³

      = 10∑𝑖=1..10𝑖𝑥² + 10∑𝑗=1..10𝑗𝑦³

      = 10𝑥²∑𝑖=1..10𝑖 + 10𝑦³∑𝑗=1..10𝑗

Using triangular number formula:

      = 10𝑥²(10⋅11)/2 + 10𝑦³(10⋅11)/2

      = 550(𝑥² + 𝑦³)

Upvotes: 1

Souperman
Souperman

Reputation: 9866

The first answer's approaches all work. Often the "pythonic" approach is to define a list comprehension before using sum so another approach would be:

def f(x, y):
    return sum([(i * x ** 2 + j * y ** 3) for i in range(1,11) for j in range(1, 11)]);

Upvotes: 0

AKX
AKX

Reputation: 169051

I'm no mathematician, but as far as I could tell, that would translate to

def f(x, y):
    return sum(
        sum(
            i * x ** 2 + j * y ** 3
            for j in range(1, 11)
        )
        for i in range(1, 11)
    )

or written out as for loops,

def f(x, y):
    value = 0
    for i in range(1, 11):
        for j in range(1, 11):
            value += i * x ** 2 + j * y ** 3
    return value

Upvotes: 3

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