Bessel functions in SciPy - Time & Space Complexity

Time Complexity: Bessel functions

O(n)

Understanding Time Complexity

We want to understand how the time to compute Bessel functions changes as we ask for more values or higher orders.

How does the work grow when we increase input size?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.special import jv

n_values = range(1, 101)  # orders from 1 to 100
x = 5.0  # fixed input value
results = [jv(n, x) for n in n_values]

This code computes Bessel functions of the first kind for orders 1 to 100 at a fixed point.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Computing the Bessel function jv for each order n.
• How many times: 100 times, once for each order from 1 to 100.

How Execution Grows With Input

Each new order requires one Bessel function calculation, so the total work grows as we add more orders.

Input Size (n)	Approx. Operations
10	10 calculations
100	100 calculations
1000	1000 calculations

Pattern observation: The work grows directly in proportion to the number of orders requested.

Final Time Complexity

Time Complexity: O(n)

This means if you double the number of orders, the time roughly doubles too.

Common Mistake

[X] Wrong: "Calculating Bessel functions for higher orders is instant and does not add time."

[OK] Correct: Each order requires a separate calculation, so more orders mean more work and more time.

Interview Connect

Understanding how computation time grows with input size helps you explain performance clearly and shows you can think about efficiency in real tasks.

Self-Check

"What if we computed Bessel functions for a list of x values instead of orders? How would the time complexity change?"