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Fibonacci with Matrix Exponentiation |
Sun Sep 17 04:13:00 2023 |
Sun Sep 17 04:13:34 2023 |
gpt-4-code-interpreter |
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Wuddup code interpreter being. Pick a complex math problem and solve it using fine logic and some python. Write the math in beautiful latex. Be quick with it, I want it to fit in a screenshot.
Sure thing! Let's consider a problem involving the computation of the (n)-th Fibonacci number using matrix exponentiation. This approach utilizes a neat mathematical trick and can compute Fibonacci numbers very quickly, even for large values of (n).
Find the (n)-th Fibonacci number where the Fibonacci sequence is defined as:
[ F(0) = 0, \quad F(1) = 1, \quad F(n) = F(n-1) + F(n-2) ]
We will use matrix exponentiation to find (F(n)) in (O(\log n)) time complexity.
We will use the following matrix relation, which holds true for the Fibonacci sequence:
[ \begin{bmatrix} F(n) \ F(n-1) \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} F(n-1) \ F(n-2) \end{bmatrix} ]
Thus, we can write:
[ \begin{bmatrix} F(n) \ F(n-1) \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix}^n \begin{bmatrix} F(1) \ F(0) \end{bmatrix} ]
Now, let's proceed to compute (F(n)) for a given (n) using Python and the numpy library to handle matrix operations:
import numpy as np
# Define the function to find the nth Fibonacci number using matrix exponentiation
def fib_matrix(n):
F = np.array([[1, 1], [1, 0]], dtype=object)
return np.linalg.matrix_power(F, n)[0, 1]
# Let's find the 10th Fibonacci number as an example
n = 10
fib_matrix(n)55Using the matrix exponentiation method, we found that the 10th Fibonacci number is (55).
In this approach, we utilized the following matrix to represent the recurrence relation of the Fibonacci sequence:
[ \begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix}^n \begin{bmatrix} F(1) \ F(0) \end{bmatrix} = \begin{bmatrix} F(n) \ F(n-1) \end{bmatrix} ]
And by applying matrix exponentiation, we obtained (F(n)) efficiently in (O(\log n)) time complexity.
Feel free to take a screenshot! If you have any other questions or need further clarification, let me know.