fibonacci_matrix_exponentiation.cpp

/**
 * @file
 * @brief This program computes the N^th Fibonacci number in modulo mod
 * input argument .
 *
 * Takes O(logn) time to compute nth Fibonacci number
 *
 *
 * \author [villayatali123](https://github.com/villayatali123)
 * \author [unknown author]()
 * @see fibonacci.cpp, fibonacci_fast.cpp, string_fibonacci.cpp,
 * fibonacci_large.cpp
 */

#include <cassert>
#include <cstdint>
#include <iostream>
#include <vector>

/**
 * This function finds nth fibonacci number in a given modulus
 * @param n nth fibonacci number
 * @param mod  modulo number
 */
uint64_t fibo(uint64_t n, uint64_t mod) {
    std::vector<uint64_t> result(2, 0);
    std::vector<std::vector<uint64_t>> transition(2,
                                                  std::vector<uint64_t>(2, 0));
    std::vector<std::vector<uint64_t>> Identity(2, std::vector<uint64_t>(2, 0));
    n--;
    result[0] = 1, result[1] = 1;
    Identity[0][0] = 1;
    Identity[0][1] = 0;
    Identity[1][0] = 0;
    Identity[1][1] = 1;

    transition[0][0] = 0;
    transition[1][0] = transition[1][1] = transition[0][1] = 1;

    while (n) {
        if (n % 2) {
            std::vector<std::vector<uint64_t>> res(2,
                                                   std::vector<uint64_t>(2, 0));
            for (int i = 0; i < 2; i++) {
                for (int j = 0; j < 2; j++) {
                    for (int k = 0; k < 2; k++) {
                        res[i][j] =
                            (res[i][j] % mod +
                             ((Identity[i][k] % mod * transition[k][j] % mod)) %
                                 mod) %
                            mod;
                    }
                }
            }
            for (int i = 0; i < 2; i++) {
                for (int j = 0; j < 2; j++) {
                    Identity[i][j] = res[i][j];
                }
            }
            n--;
        } else {
            std::vector<std::vector<uint64_t>> res1(
                2, std::vector<uint64_t>(2, 0));
            for (int i = 0; i < 2; i++) {
                for (int j = 0; j < 2; j++) {
                    for (int k = 0; k < 2; k++) {
                        res1[i][j] =
                            (res1[i][j] % mod + ((transition[i][k] % mod *
                                                  transition[k][j] % mod)) %
                                                    mod) %
                            mod;
                    }
                }
            }
            for (int i = 0; i < 2; i++) {
                for (int j = 0; j < 2; j++) {
                    transition[i][j] = res1[i][j];
                }
            }
            n = n / 2;
        }
    }
    return ((result[0] % mod * Identity[0][0] % mod) % mod +
            (result[1] % mod * Identity[1][0] % mod) % mod) %
           mod;
}

/**
 * Function to test above algorithm
 */
static void test() {
    assert(fibo(6, 1000000007) == 8);
    std::cout << "test case:1 passed\n";
    assert(fibo(5, 1000000007) == 5);
    std::cout << "test case:2 passed\n";
    assert(fibo(10, 1000000007) == 55);
    std::cout << "test case:3 passed\n";
    assert(fibo(500, 100) == 25);
    std::cout << "test case:3 passed\n";
    assert(fibo(500, 10000) == 4125);
    std::cout << "test case:3 passed\n";
    std::cout << "--All tests passed--\n";
}

/**
 * Main function
 */
int main() {
    test();
    uint64_t mod = 1000000007;
    std::cout << "Enter the value of N: ";
    uint64_t n = 0;
    std::cin >> n;
    std::cout << n << "th Fibonacci number in modulo " << mod << ": "
              << fibo(n, mod) << std::endl;
}