How does the Fibonacci sequence perform in algorithmic efficiency?

How does the Fibonacci sequence perform in algorithmic efficiency?

The Fibonacci sequence can be computed using various algorithms, each with different efficiencies. The naive recursive approach has exponential time complexity, while the iterative and dynamic programming methods can compute it in linear time.

What are the differences between the recursive and iterative methods?

The recursive method calls itself for each Fibonacci number, leading to repeated calculations, while the iterative method uses a loop to calculate each number sequentially, which is much more efficient.

Can you show me an example of both methods in code?

Sure! Here's a simple example in Python:

Recursive:

def fibonacci_recursive(n):
    if n <= 1:
        return n
    return fibonacci_recursive(n-1) + fibonacci_recursive(n-2)

Iterative:

def fibonacci_iterative(n):
    a, b = 0, 1
    for _ in range(n):
        a, b = b, a + b
    return a

Which method would you recommend for large values of n?

For large values of n, I recommend using the iterative method or dynamic programming, as they are much more efficient and avoid the exponential time complexity of the recursive method.