Added Matrix Exponentiation

maths/matrix_exponentiation.py

@@ -0,0 +1,134 @@
+"""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):
+		c = Matrix(self.n)
+
+		for i in range(self.n):
+			for j in range(self.n):
+				for k in range(self.n):
+					c.t[i][j] += self.t[i][k]*b.t[k][j]
+
+		return c
+
+def modular_exponentiation(a, b):
+	ret = Matrix([[1,0], [0,1]])
+
+	while b > 0:
+		if b&1:
+			ret = ret*a
+
+		a = a*a
+		b >>= 1
+
+	return ret
+
+def fibonacci_with_matrix_exponentiation(n, f1, f2):
+
+	#Trivial Cases
+	if n == 1:
+		return f1
+
+	if n == 2:
+		return f2
+
+	a = Matrix([[1,1], [1,0]])
+
+	m = modular_exponentiation(a, n-2)
+
+	return f2*m.t[0][0] + f1*m.t[0][1]
+
+def simple_fibonacci(n, f1, f2):
+
+	#Trival Cases
+	if n == 1:
+		return f1
+
+	if n == 2:
+		return f2
+
+	fn_1 = f1
+	fn_2 = f2
+
+	n -= 2
+
+	while n > 0:
+		fn = fn_1 + fn_2
+
+		fn_2 = fn_1
+		fn_1 = fn
+
+		n -= 1
+
+	return fn
+
+def matrix_exponentiation_time():
+	setup = '''
+from random import randint
+from __main__ import fibonacci_with_matrix_exponentiation
+'''
+
+	code = '''
+n = randint(1,70000)
+
+f1 = 1
+f2 = 1
+
+
+fibonacci_with_matrix_exponentiation(n, f1, f2)
+'''
+	exec_time = timeit.timeit(setup = setup,
+						  stmt = code,
+						  number = 100)
+
+	print ("With matrix exponentiation the average execution time is ", exec_time/100)
+
+
+def simple_fibonacci_time():
+	setup = '''
+from random import randint
+from __main__ import simple_fibonacci
+'''
+
+	code = '''
+n = randint(1,70000)
+
+f1 = 1
+f2 = 1
+
+
+simple_fibonacci(n, f1, f2)
+'''
+	exec_time = timeit.timeit(setup = setup,
+						  stmt = code,
+						  number = 100)
+
+	print ("Without matrix exponentiation the average execution time is ", exec_time/100)
+
+
+def main():
+	matrix_exponentiation_time()
+	simple_fibonacci_time()
+
+
+if __name__ == "__main__":
+	main()