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

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* feat: This function uses the tail-recursive form of the Euclidean algorithm to calculate

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* reduced the number of characters per line in the comments

* removed unwanted code

* feat: Implemented a new function to swaap numbers without dummy variable

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* removed previos code

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* Update maths/fibonacci.py

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

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---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>
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ARNAV RAJ 2024-10-04 21:59:39 +05:30 committed by GitHub
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@ -7,6 +7,8 @@ the Binet's formula function because the Binet formula function uses floats
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 collections.abc import Iterator
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 @@ if __name__ == "__main__":
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