fix this py is to slow Problem

1. Added matrix_power_lucas_number function:
   · Implements matrix exponentiation algorithm for O(log n) time complexity
   · Maintains the same interface as existing functions
   · Includes comprehensive doctests matching the existing patterns
2. Key features:
   · Uses 2x2 matrix multiplication with fast exponentiation
   · Handles base cases (n=0, n=1) directly
   · Proper type annotations for all internal functions
   · Maintains backward compatibility

Author: lighting9999

maths/lucas_series.py

@@ -55,6 +55,58 @@ def dynamic_lucas_number(n_th_number: int) -> int:
     return a
 
 
+def matrix_power_lucas_number(n_th_number: int) -> int:
+    """
+    Returns the nth lucas number using matrix exponentiation
+    >>> matrix_power_lucas_number(1)
+    1
+    >>> matrix_power_lucas_number(20)
+    15127
+    >>> matrix_power_lucas_number(0)
+    2
+    >>> matrix_power_lucas_number(25)
+    167761
+    >>> matrix_power_lucas_number(-1.5)
+    Traceback (most recent call last):
+        ...
+    TypeError: matrix_power_lucas_number accepts only integer arguments.
+    """
+    if not isinstance(n_th_number, int):
+        raise TypeError("matrix_power_lucas_number accepts only integer arguments.")
+    
+    if n_th_number == 0:
+        return 2
+    if n_th_number == 1:
+        return 1
+    
+    def matrix_mult(a: list[list[int]], b: list[list[int]]) -> list[list[int]]:
+        return [
+            [a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
+            [a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]],
+        ]
+    
+    def matrix_power(matrix: list[list[int]], power: int) -> list[list[int]]:
+        # Start with identity matrix
+        result: list[list[int]] = [[1, 0], [0, 1]]
+        base = matrix
+        
+        while power > 0:
+            if power % 2 == 1:
+                result = matrix_mult(result, base)
+            base = matrix_mult(base, base)
+            power //= 2
+        
+        return result
+    
+    # Lucas number matrix form: [[1, 1], [1, 0]]
+    base_matrix = [[1, 1], [1, 0]]
+    powered_matrix = matrix_power(base_matrix, n_th_number - 1)
+    
+    # L(n) = powered_matrix[0][0] * L(1) + powered_matrix[0][1] * L(0)
+    # Where L(1) = 1, L(0) = 2
+    return powered_matrix[0][0] * 1 + powered_matrix[0][1] * 2
+
+
 if __name__ == "__main__":
     from doctest import testmod
 
@@ -64,3 +116,5 @@ def dynamic_lucas_number(n_th_number: int) -> int:
     print(" ".join(str(recursive_lucas_number(i)) for i in range(n)))
     print("\nUsing dynamic function to calculate lucas series:")
     print(" ".join(str(dynamic_lucas_number(i)) for i in range(n)))
+    print("\nUsing matrix exponentiation to calculate lucas series:")
+    print(" ".join(str(matrix_power_lucas_number(i)) for i in range(n)))