Integral of a Polynomial in Python

Question

How would we write a function in python that returns the definite integral of a polynomial between two points (X_1 and X_2)?

The function takes 3 arguments:

a list A of polynomial coefficients (i.e. for polynomial f(x)=5x^4−2x+1, this list becomes A=[5,0,0,−2,1])
a real number X_1
a real number X_2

We are given the formula for the definite integral of a polynomial such as

My attempt at this function is below, however the output returns 0.2

    def take_integral(A, X_1, X_2):

            integral = 0

            for i in range(len(A)):
                integral = A*(X_2**(i+1) - X_1**(i+1))/(i+1)
            return integral

    print(take_integral([1, 2, 1], 0, 3))

The expected result from the function should be:


    print(take_integral([1, 2, 1], 0, 3))
    21.0
    print(take_integral([5, 0, 0, -2, 1], 0, 1))
    1.0

Asked By: codemachine98

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Source

Answer 1

In your function you ignore X_1 and X_2 and set them both to 0. Hence the result is always 0

Answered By: Paul

Answer 2

A few points here:

There’s a serious problem with multiplying A with the whole bunch of stuff afterwards. A is a list, where the rest gives a float. The computer doesn’t necessarily know what to do with that. Imagine if I told you to multiply a set of crayons by 4. You can guess what I want, but ultimately, it doesn’t really make sense. You want to multiply the contents of the list instead of the list itself.
With your given formula for computing a definite integral of a polynomial, I’m pretty sure there should be a sum where you add all the terms together. That’s what the definition of a polynomial is, right? Currently, you’re computing each term and erasing the previous term. So, you need to add them up instead.
Your attempt also kind of…"flips" the exponents. Since the list of coefficients A is sorted in descending order of power (A[0] being the coefficient of the highest power), doing X_1**(i+1) when i=0 in your sample input where A=[1,2,1] would be multiplying the highest power with the lowest exponent instead of the desired 3. So you want to iterate through the list from left to right, but your i when you’re raising your X‘s needs to decrease. Thankfully, len(A) comes in handy for solving that.
- If you do len(A)-i, you’ll get exactly the i+1 according to the formula. This is true since len(A) will give the one more than the highest power in your polynomial, and i starts from 0 to one less than the length of A, and as a result gives you i+1 as per the formula.

Here’s the code that will give you the right answers:

def take_integral(A, X_1, X_2):

        integral = 0

        for i in range(len(A)):
            integral += A[i]*(X_2**((len(A))-i) - X_1**((len(A))-i))/((len(A))-i)
        return integral

print(take_integral([1, 2, 1], 0, 3))
21
print(take_integral([5, 0, 0, -2, 1], 0, 1))
1

Answered By: corvelk

Answer 3

This can also be done more efficiently using numpy libraries.

import numpy as np 

def take_integral(A, X_1, X_2):
    powers = np.flipud(np.array(range(len(A)))) + 1 # Create an array
    A = A/ powers # adjust coefficients
    return np.sum(A * (X_2 ** powers - X_1 ** powers))

Answered By: user157545

Integral of a Polynomial in Python

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