/**
 * @file
 * @brief An algorithm to calculate the sum of Fibonacci Sequence: \f$\mathrm{F}(n) + \mathrm{F}(n+1) + .. + \mathrm{F}(m)\f$
 * @details An algorithm to calculate the sum of Fibonacci Sequence: \f$\mathrm{F}(n) + \mathrm{F}(n+1) + .. + \mathrm{F}(m)\f$ where \f$\mathrm{F}(i)\f$
 * denotes the i-th Fibonacci Number . Note that F(0) = 0 and F(1) = 1.
 * The value of the sum is calculated using matrix exponentiation.
 * @source https://stackoverflow.com/questions/4357223/finding-the-sum-of-fibonacci-numbers
 * @author [Sarthak Sahu](https://github.com/SarthakSahu1009)
 */

#include <iostream> /// for std::cin and std::cout
#include <cassert>  /// for assert
#include <vector>   /// for std::vector

/**
 * @namespace math
 * @brief Mathematical algorithms
 */
namespace math {
    /**
     * @namespace fibonacci_sum
     * @brief Functions for the sum of the Fibonacci Sequence: \f$\mathrm{F}(n) + \mathrm{F}(n+1) + .. + \mathrm{F}(m)\f$
     */
    namespace fibonacci_sum {

        /**
         * Function to multiply two matrices
         * @param T matrix 1
         * @param A martix 2
         * @returns resultant matrix
         */
        std::vector<std::vector<int> > multiply(std::vector<std::vector<int> > T, std::vector<std::vector<int> > A) {

            std::vector<std::vector<int> > result(2,std::vector<int>(2));

            // multiplying matrices
            for(int i=0;i<2;i++) {
                for(int j=0;j<2;j++) {
                    result[i][j]=0;
                    for(int k=0;k<2;k++) {
                        result[i][j]=(result[i][j]+(T[i][k]*A[k][j]));
                    }
                }
            }

            return result;
        }

        /**
         * Function to compute A^n where A is a matrix.
         * @param T matrix
         * @param ex power
         * @returns resultant matrix
         */
        std::vector<std::vector<int> > power(std::vector<std::vector<int> > T, int ex) {

            std::vector<std::vector<int> > A{{1,1},{1,0}};
            if(ex == 0 || ex == 1) {
                return T;
            }

            T = power(T,ex/2);
            T = multiply(T,T);
            if(ex&1) {
                T = multiply(T,A);
            }
            return T;
        }

        /**
         * Function to compute sum of fibonacci sequence from 0 to n.
         * @param n number
         * @returns int ans, the sum of sequence
         */
        int result(int n) {
            std::vector<std::vector<int> > T{{1,1},{1,0}};
            T = power(T,n);
            int ans=T[0][1];
            ans = (ans - 1);
            return ans;
        }

        /**
         * Function to compute sum of fibonacci sequence from n to m.
         * @param n start of sequence
         * @param m end of sequence
         * @returns int the sum of sequence
         */
        int fiboSum(int n,int m){
            return (result(m+2) - result(n+1));
        }
    }  // namespace fibonacci_sum
}  // namespace math

/**
 * Function for testing fiboSum function.
 * test cases and assert statement.
 * @returns `void`
*/
static void test() {
    int n = 0, m = 3;
    int test_1 = math::fibonacci_sum::fiboSum(n,m);
    assert(test_1 == 4);
    std::cout << "Passed Test 1!" << std::endl;

    n = 3; m = 5;
    int test_2 = math::fibonacci_sum::fiboSum(n,m);
    assert(test_2 == 10);
    std::cout << "Passed Test 2!" << std::endl;

    n = 5; m = 7;
    int test_3 = math::fibonacci_sum::fiboSum(n,m);
    assert(test_3 == 26);
    std::cout << "Passed Test 3!" << std::endl;

    n = 7; m = 10;
    int test_4 = math::fibonacci_sum::fiboSum(n,m);
    assert(test_4 == 123);
    std::cout << "Passed Test 4!" << std::endl;

    n = 9; m = 12;
    int test_5 = math::fibonacci_sum::fiboSum(n,m);
    assert(test_5 == 322);
    std::cout << "Passed Test 5!" << std::endl;
}

/**
 * @brief Main function
 * @returns 0 on exit
 */
int main()
{
    test(); // execute the tests
    return 0;
}