Fibonacci Dialog, ChatGPT 4.0

python
def fibonacci(n):
	"""Calculate the nth Fibonacci number using matrix 
		exponentiation."""
	F = [[1, 1], [1, 0]]
	if n == 0:
		return 0
	power(F, n - 1)
	return F[0][0]
	
def multiply(F, M):
	"""Helper function to multiply Fibonacci matrices."""
	x = F[0][0] * M[0][0] + F[0][1] * M[1][0]
	y = F[0][0] * M[0][1] + F[0][1] * M[1][1]
	z = F[1][0] * M[0][0] + F[1][1] * M[1][0]
	w = F[1][0] * M[0][1] + F[1][1] * M[1][1]

	F[0][0] = x
	F[0][1] = y
	F[1][0] = z
	F[1][1] = w
	
def power(F, n):
	"""Helper function to exponentiate the Fibonacci matrix."""
	M = [[1, 1], [1, 0]]
	for _ in range(2, n+1):
		multiply(F, M)

# Calculate the 100th Fibonacci number.
fibonacci(100)

I apologize for the error. It seems there was a problem with the execution of the code. Let me try a different approach. I'll use the closed-form expression for the nth Fibonacci number, which is known as Binet's formula. This formula can be used to compute Fibonacci numbers directly:

F(n) = \frac{{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}}{{2^n \sqrt{5}}}

However, for large values of $n$ , this formula can suffer from rounding errors due to the finite precision of floating-point arithmetic. As an alternative, we can use an iterative approach to calculate large Fibonacci numbers.

Let's try the iterative approach.

python
def fibonacci(n):
	"""Calculate the nth Fibonacci number iteratively."""
	a, b = 0, 1
	for _ in range(n):
		a, b = b, a + b
	return a
	
	# Calculate the 100th Fibonacci number.
	fibonacci(100)

RESULT

354224848179261915075

The 100th Fibonacci number is 354224848179261915075.