# C++ Program to Find Fibonacci Numbers using Matrix Exponentiation

The Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, i.e; each number is the sum of the two preceding ones, starting from 0 and 1. That is −

F0 = 0 and F1 = 1
And
Fn = Fn-1 + Fn-2
for n > 1.

## Algorithm

Begin
Take two 2 dimensional array
Create a function and Perform matrix multiplication
Create another function to find out power of matrix
Create a function then to find out the Fibonacci number
Multiply(arr1[2][2], arr2[2][2])
Take 4 variables a, b, c, d
a = arr1[0][0] * arr2[0][0] + arr1[0][1] * arr2[1][0]
b= arr1[0][0] * arr2[0][1] + arr1[0][1] * arr2[1][1]
c = arr1[1][0] * arr2[0][0] + arr1[1][1] * arr2[1][0]
d = arr1[1][0] * arr2[0][1] + arr1[1][1] * arr2[1][1]
arr1[0][0] = a
arr1[0][1] = b
arr1[1][0] = c
arr1[1][1] = d
Power(arr1[2][2], take integer n as input)
if (n == 0 or n == 1)
return;
arr1 [2][2] = {{1,1}, {1,0}}
power(arr1, n / 2)
multiply(arr1, arr1)
if (n mod 2 not equal to 0)
multiply(arr1, arr2)
fibonacci_matrix(n)
arr1[2][2] = {{1,1}, {1,0}}
if n ==0
return 0
power(arr1 n - 1)
return arr1[0][0]
End

## Example Code

#include <iostream>
using namespace std;
void multiply(int F[2][2], int M[2][2]) {
int a = F[0][0] * M[0][0] + F[0][1] * M[1][0];
int b= F[0][0] * M[0][1] + F[0][1] * M[1][1];
int c = F[1][0] * M[0][0] + F[1][1] * M[1][0];
int d = F[1][0] * M[0][1] + F[1][1] * M[1][1];
F[0][0] = a;
F[0][1] = b;
F[1][0] = c;
F[1][1] = d;
}
void power(int F[2][2], int n) {
if (n == 0 || n == 1)
return;
int M[2][2] = {{1,1},{1,0}};
power(F, n / 2);
multiply(F, F);
if (n % 2 != 0)
multiply(F, M);
}
int fibonacci_matrix(int n) {
int F[2][2] = {{1,1},{1,0}};
if (n == 0)
return 0;
power(F, n - 1);
return F[0][0];
}
int main() {
int n;
while (1) {
cout<<"Enter the integer n to find nth fibonacci no. (enter 0 to exit):";
cin>>n;
if (n == 0)
break;
cout<<fibonacci_matrix(n)<<endl;
}
return 0;
}

## Output

Enter the integer n to find nth fibonacci no. (enter 0 to exit): 2
1
Enter the integer n to find nth fibonacci no. (enter 0 to exit): 6
8
Enter the integer n to find nth fibonacci no. (enter 0 to exit): 7
13
Enter the integer n to find nth fibonacci no. (enter 0 to exit): 0