feat: add fibonacci_sum.cpp (#1343)

* Johnson's Algorithm added

* Update Johnson's algorithm.cpp

@Panquesito7 plzz review

* Update Johnson's algorithm.cpp

* remove johnson'algo

* add fibo_sum.cpp

* updating DIRECTORY.md

* Update math/fibonacci_sum.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: Krishna Vedala <7001608+kvedala@users.noreply.github.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: Krishna Vedala <7001608+kvedala@users.noreply.github.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: Krishna Vedala <7001608+kvedala@users.noreply.github.com>

* Update math/fibonacci_sum.cpp

Co-authored-by: Krishna Vedala <7001608+kvedala@users.noreply.github.com>

* clang-format and clang-tidy fixes for a054b759

Co-authored-by: SarthakSahu1009 <sarthaksashu1009@gmail.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: Krishna Vedala <7001608+kvedala@users.noreply.github.com>

DIRECTORY.md

@@ -137,6 +137,7 @@
   * [Fibonacci](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/fibonacci.cpp)
   * [Fibonacci Fast](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/fibonacci_fast.cpp)
   * [Fibonacci Large](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/fibonacci_large.cpp)
+  * [Fibonacci Sum](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/fibonacci_sum.cpp)
   * [Gcd Iterative Euclidean](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/gcd_iterative_euclidean.cpp)
   * [Gcd Of N Numbers](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/gcd_of_n_numbers.cpp)
   * [Gcd Recursive Euclidean](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/gcd_recursive_euclidean.cpp)

math/fibonacci_sum.cpp

@@ -0,0 +1,138 @@
+/**
+ * @file
+ * @brief An algorithm to calculate the sum of [Fibonacci
+ * Sequence](https://en.wikipedia.org/wiki/Fibonacci_number): \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.
+ * Reference source:
+ * https://stackoverflow.com/questions/4357223/finding-the-sum-of-fibonacci-numbers
+ * @author [Sarthak Sahu](https://github.com/SarthakSahu1009)
+ */
+
+#include <cassert>   /// for assert
+#include <iostream>  /// for std::cin and std::cout
+#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 {
+        using matrix = std::vector<std::vector<uint64_t> >;
+
+        /**
+         * Function to multiply two matrices
+         * @param T matrix 1
+         * @param A martix 2
+         * @returns resultant matrix
+         */
+        math::fibonacci_sum::matrix multiply(const math::fibonacci_sum::matrix &T, const math::fibonacci_sum::matrix &A) {
+            math::fibonacci_sum::matrix result(2, std::vector<uint64_t>(2, 0));
+
+            // multiplying matrices
+            result[0][0] = T[0][0]*A[0][0] + T[0][1]*A[1][0];
+            result[0][1] = T[0][0]*A[0][1] + T[0][1]*A[1][1];
+            result[1][0] = T[1][0]*A[0][0] + T[1][1]*A[1][0];
+            result[1][1] = T[1][0]*A[0][1] + T[1][1]*A[1][1];
+
+            return result;
+        }
+
+        /**
+         * Function to compute A^n where A is a matrix.
+         * @param T matrix
+         * @param ex power
+         * @returns resultant matrix
+         */
+        math::fibonacci_sum::matrix power(math::fibonacci_sum::matrix T, uint64_t ex) {
+            math::fibonacci_sum::matrix 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 uint64_t ans, the sum of sequence
+         */
+        uint64_t result(uint64_t n) {
+            math::fibonacci_sum::matrix T{{1, 1}, {1, 0}};
+            T = power(T, n);
+            uint64_t 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 uint64_t the sum of sequence
+         */
+        uint64_t fiboSum(uint64_t n, uint64_t 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() {
+    uint64_t n = 0, m = 3;
+    uint64_t test_1 = math::fibonacci_sum::fiboSum(n, m);
+    assert(test_1 == 4);
+    std::cout << "Passed Test 1!" << std::endl;
+
+    n = 3;
+    m = 5;
+    uint64_t test_2 = math::fibonacci_sum::fiboSum(n, m);
+    assert(test_2 == 10);
+    std::cout << "Passed Test 2!" << std::endl;
+
+    n = 5;
+    m = 7;
+    uint64_t test_3 = math::fibonacci_sum::fiboSum(n, m);
+    assert(test_3 == 26);
+    std::cout << "Passed Test 3!" << std::endl;
+
+    n = 7;
+    m = 10;
+    uint64_t test_4 = math::fibonacci_sum::fiboSum(n, m);
+    assert(test_4 == 123);
+    std::cout << "Passed Test 4!" << std::endl;
+
+    n = 9;
+    m = 12;
+    uint64_t 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;
+}