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Others/fibonacci.cpp

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+#include <iostream>
+using namespace std;
+
+
+/*
+	Efficient method for finding nth term in a fibonacci number sequence.
+	Uses Dvide and conquer approach
+	Enter Values from 1
+	Eg-
+	1st term = 0;
+	2nd term = 1;
+	3rd term = 1;
+	.
+	.
+	.
+	.
+*/
+int a[2][2] = {{1,1},{1,0}};//fibonacci matrix
+int ans[2][2] = {{1,1},{1,0}};//final ans matrix
+/*
+	Working Principal:
+
+	[F(k+1)	  F(k)]   [1  1]^k
+	|             | = |    |
+	[F(k)	F(k-1)]   [1  0]
+
+	where F(k) is the kth term of the Fibonacci Sequence. 
+
+
+*/
+
+void product(int b[][2],int k[][2])
+{
+	/*
+		Function for computing product of the two matrices b and k
+		and storing them into the variable ans.
+
+		Implementation :
+		Simple matrix multiplication of two (2X2) matrices.
+	*/
+	int c[2][2];//temporary stores the answer
+	c[0][0] = b[0][0]*k[0][0]+b[0][1]*k[1][0];
+	c[0][1] = b[0][0]*k[0][1]+b[0][1]*k[1][1];
+	c[1][0] = b[1][0]*k[0][0]+b[1][1]*k[1][0];
+	c[1][1] = b[1][0]*k[0][1]+b[1][1]*k[1][1];
+	ans[0][0] = c[0][0];
+	ans[0][1] = c[0][1];
+	ans[1][0] = c[1][0];
+	ans[1][1] = c[1][1];
+	
+} 
+
+void power_rec(int n)
+{	
+
+	/*
+		Function for calculating A^n(exponent) in a recursive fashion
+
+		Implementation:
+		A^n = { A^(n/2)*A^(n/2) 			if n is even
+			  { A^((n-1)/2)*A^((n-1)/2)*A   if n is odd
+	*/
+	if((n == 1)||(n==0))
+		return;
+	else
+	{
+		if((n%2) == 0)
+		{	
+			power_rec(n/2);
+			product(ans,ans);
+			
+		}	
+		else
+		{
+			power_rec((n-1)/2);	
+			product(ans,ans);
+			product(ans,a);
+			
+		}
+	}
+}
+
+int main()
+{
+	//Main Function
+	cout <<"Enter the value of n\n";
+	int n;
+	cin >>n;
+	if(n == 1)
+	{
+		cout<<"Ans: 0"<<endl;
+	}
+	else
+	{	
+		power_rec(n-1);
+		cout <<"Ans :"<<ans[0][1]<<endl;
+	}
+	return 0;
+}
+