Added Matrix Exponentiation (#1203)

* Added the matrix_exponentiation.py file in maths directory

* Implemented the requested changes

* Update matrix_exponentiation.py

maths/matrix_exponentiation.py

@@ -0,0 +1,99 @@
+"""Matrix Exponentiation"""
+
+import timeit
+
+"""
+Matrix Exponentiation is a technique to solve linear recurrences in logarithmic time.
+You read more about it here: 
+http://zobayer.blogspot.com/2010/11/matrix-exponentiation.html
+https://www.hackerearth.com/practice/notes/matrix-exponentiation-1/
+"""
+
+
+class Matrix(object):
+    def __init__(self, arg):
+        if isinstance(arg, list):  # Initialzes a matrix identical to the one provided.
+            self.t = arg
+            self.n = len(arg)
+        else:  # Initializes a square matrix of the given size and set the values to zero.
+            self.n = arg
+            self.t = [[0 for _ in range(self.n)] for _ in range(self.n)]
+
+    def __mul__(self, b):
+        matrix = Matrix(self.n)
+        for i in range(self.n):
+            for j in range(self.n):
+                for k in range(self.n):
+                    matrix.t[i][j] += self.t[i][k] * b.t[k][j]
+        return matrix
+
+
+def modular_exponentiation(a, b):
+    matrix = Matrix([[1, 0], [0, 1]])
+    while b > 0:
+        if b & 1:
+            matrix *= a
+        a *= a
+        b >>= 1
+    return matrix
+
+
+def fibonacci_with_matrix_exponentiation(n, f1, f2):
+    # Trivial Cases
+    if n == 1:
+        return f1
+    elif n == 2:
+        return f2
+    matrix = Matrix([[1, 1], [1, 0]])
+    matrix = modular_exponentiation(matrix, n - 2)
+    return f2 * matrix.t[0][0] + f1 * matrix.t[0][1]
+
+
+def simple_fibonacci(n, f1, f2):
+    # Trival Cases
+    if n == 1:
+        return f1
+    elif n == 2:
+        return f2
+
+    fn_1 = f1
+    fn_2 = f2
+    n -= 2
+
+    while n > 0:
+        fn_1, fn_2 = fn_1 + fn_2, fn_1
+        n -= 1
+
+    return fn
+
+
+def matrix_exponentiation_time():
+    setup = """
+from random import randint
+from __main__ import fibonacci_with_matrix_exponentiation
+"""
+    code = "fibonacci_with_matrix_exponentiation(randint(1,70000), 1, 1)"
+    exec_time = timeit.timeit(setup=setup, stmt=code, number=100)
+    print("With matrix exponentiation the average execution time is ", exec_time / 100)
+    return exec_time
+
+
+def simple_fibonacci_time():
+    setup = """
+from random import randint
+from __main__ import simple_fibonacci
+"""
+    code = "simple_fibonacci(randint(1,70000), 1, 1)"
+    exec_time = timeit.timeit(setup=setup, stmt=code, number=100)
+    print("Without matrix exponentiation the average execution time is ",
+          exec_time / 100)
+    return exec_time
+
+
+def main():
+    matrix_exponentiation_time()
+    simple_fibonacci_time()
+
+
+if __name__ == "__main__":
+    main()