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