NumPy - Fibonacci Series using Binet Formula

NumPy - Fibonacci Series using Binet Formula

The Fibonacci sequence can be computed using Binet's formula, which is a closed-form expression. While this formula allows for the direct computation of any term in the sequence, it's worth noting that due to floating-point arithmetic limitations, it may not give exact results for large indices in the sequence.

Binet's formula is:

F(n)=5​(ϕn−(1−ϕ)n)​

where ϕ (phi) is the golden ratio:

ϕ=21+5​​

Here's how you can implement this in NumPy to calculate the first n Fibonacci numbers:

import numpy as np # Define the golden ratio phi = (1 + np.sqrt(5)) / 2 # Binet's formula to calculate the nth Fibonacci number def fibonacci(n): return np.round((phi**n - (-1/phi)**n) / np.sqrt(5)) # Number of terms n_terms = 10 # Calculate the first n_terms of the Fibonacci sequence fib_sequence = [fibonacci(n) for n in range(n_terms)] print(fib_sequence)

Please note the use of np.round in the function. This is to correct the floating-point errors that occur when the numbers get large.

Keep in mind that this implementation is not the most efficient for large n because floating-point numbers have limited precision, and the results of Binet's formula become increasingly inaccurate for larger n. For very large values of n, other methods such as matrix exponentiation or simple iterative methods may be more appropriate.

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