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>

maths/fibonacci.py

@@ -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