From 9a572dec2b6011e7c2c0d82f50989b3a404ea426 Mon Sep 17 00:00:00 2001 From: ARNAV RAJ <126798788+Acuspeedster@users.noreply.github.com> Date: Fri, 4 Oct 2024 21:59:39 +0530 Subject: [PATCH] feat: Implemented Matrix Exponentiation Method (#11747) * feat: add Matrix Exponentiation method docs: updated the header documentation and added new documentation for the new function. * feat: added new function matrix exponetiation method * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * feat: This function uses the tail-recursive form of the Euclidean algorithm to calculate * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * reduced the number of characters per line in the comments * removed unwanted code * feat: Implemented a new function to swaap numbers without dummy variable * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * removed previos code * Done with the required changes * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Done with the required changes * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Done with the required changes * Done with the required changes * Done with the required changes * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update maths/fibonacci.py Co-authored-by: Tianyi Zheng * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Done with the required changes * Done with the required changes * Done with the required changes * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: Tianyi Zheng --- maths/fibonacci.py | 88 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 88 insertions(+) diff --git a/maths/fibonacci.py b/maths/fibonacci.py index 927700b0418e..24b2d7ae449e 100644 --- a/maths/fibonacci.py +++ b/maths/fibonacci.py @@ -7,6 +7,8 @@ NOTE 2: the Binet's formula function is much more limited in the size of inputs that it can handle due to the size limitations of Python floats +NOTE 3: the matrix function is the fastest and most memory efficient for large n + See benchmark numbers in __main__ for performance comparisons/ https://en.wikipedia.org/wiki/Fibonacci_number for more information @@ -17,6 +19,9 @@ from math import sqrt from time import time +import numpy as np +from numpy import ndarray + def time_func(func, *args, **kwargs): """ @@ -230,6 +235,88 @@ def fib_binet(n: int) -> list[int]: return [round(phi**i / sqrt_5) for i in range(n + 1)] +def matrix_pow_np(m: ndarray, power: int) -> ndarray: + """ + Raises a matrix to the power of 'power' using binary exponentiation. + + Args: + m: Matrix as a numpy array. + power: The power to which the matrix is to be raised. + + Returns: + The matrix raised to the power. + + Raises: + ValueError: If power is negative. + + >>> m = np.array([[1, 1], [1, 0]], dtype=int) + >>> matrix_pow_np(m, 0) # Identity matrix when raised to the power of 0 + array([[1, 0], + [0, 1]]) + + >>> matrix_pow_np(m, 1) # Same matrix when raised to the power of 1 + array([[1, 1], + [1, 0]]) + + >>> matrix_pow_np(m, 5) + array([[8, 5], + [5, 3]]) + + >>> matrix_pow_np(m, -1) + Traceback (most recent call last): + ... + ValueError: power is negative + """ + result = np.array([[1, 0], [0, 1]], dtype=int) # Identity Matrix + base = m + if power < 0: # Negative power is not allowed + raise ValueError("power is negative") + while power: + if power % 2 == 1: + result = np.dot(result, base) + base = np.dot(base, base) + power //= 2 + return result + + +def fib_matrix_np(n: int) -> int: + """ + Calculates the n-th Fibonacci number using matrix exponentiation. + https://www.nayuki.io/page/fast-fibonacci-algorithms#:~:text= + Summary:%20The%20two%20fast%20Fibonacci%20algorithms%20are%20matrix + + Args: + n: Fibonacci sequence index + + Returns: + The n-th Fibonacci number. + + Raises: + ValueError: If n is negative. + + >>> fib_matrix_np(0) + 0 + >>> fib_matrix_np(1) + 1 + >>> fib_matrix_np(5) + 5 + >>> fib_matrix_np(10) + 55 + >>> fib_matrix_np(-1) + Traceback (most recent call last): + ... + ValueError: n is negative + """ + if n < 0: + raise ValueError("n is negative") + if n == 0: + return 0 + + m = np.array([[1, 1], [1, 0]], dtype=int) + result = matrix_pow_np(m, n - 1) + return int(result[0, 0]) + + if __name__ == "__main__": from doctest import testmod @@ -242,3 +329,4 @@ def fib_binet(n: int) -> list[int]: time_func(fib_memoization, num) # 0.0100 ms time_func(fib_recursive_cached, num) # 0.0153 ms time_func(fib_recursive, num) # 257.0910 ms + time_func(fib_matrix_np, num) # 0.0000 ms