Calculate the n-th discrete difference in Python
In Python, we can easily calculate the discrete difference of a sequence. The discrete difference of a sequence is simply the difference between each adjacent element in the sequence. But what if we want to calculate the difference multiple times? This is where the n-th discrete difference comes in. In this article, we will learn how to calculate the n-th discrete difference in Python.
Calculating the first discrete difference
Before we dive into calculating the n-th discrete difference, let’s first learn how to calculate the first discrete difference of a sequence. Here’s some sample code that calculates the first discrete difference of a list of numbers:
def first_difference(seq):
return [n - p for n, p in zip(seq[1:], seq[:-1])]
In this code, we use the zip
function to combine each adjacent pair of elements in the sequence. We then use a list comprehension to calculate the difference between each pair of elements.
Let’s see this code in action by calculating the first discrete difference of the sequence [1, 3, 5, 7, 9]
:
>>> seq = [1, 3, 5, 7, 9]
>>> first_difference(seq)
[2, 2, 2, 2]
The first discrete difference of this sequence is [2, 2, 2, 2]
.
Calculating the n-th discrete difference
Now that we know how to calculate the first discrete difference of a sequence, let’s learn how to calculate the n-th discrete difference. Here’s some sample code that calculates the n-th discrete difference of a sequence:
def nth_difference(seq, n):
if n == 0:
return seq
return nth_difference(first_difference(seq), n - 1)
In this code, we recursively call first_difference
on the sequence until we have calculated the desired n-th discrete difference. We use a base case of n == 0
, which simply returns the original sequence.
Let’s see this code in action by calculating the second discrete difference of the sequence [1, 3, 5, 7, 9]
:
>>> seq = [1, 3, 5, 7, 9]
>>> nth_difference(seq, 2)
[0, 0, 0]
The second discrete difference of this sequence is [0, 0, 0]
.
Conclusion
In this article, we learned how to calculate the n-th discrete difference of a sequence in Python. We started by learning how to calculate the first discrete difference, which is simply the difference between each adjacent element in the sequence. We then learned how to calculate the n-th discrete difference by recursively calling first_difference
on the sequence until we have calculated the desired n-th discrete difference.