feat: add modular_inverse_simple (#1937)

* fix: power_of_two algorithm redundant conditional 'else'

* feat : add modular_inverse_simple

* updating DIRECTORY.md

* fix : according lint rules

* fix : removed macro and initialize variable aux

* fix: remove namespace std like default

* feat : add unsigned type for block negative number and improving description

* fix: defined in header <cstdint> for use uint64_t type

* fix: improving descriptive comments

* fix: remove redundant lib

* fix: improvinf "brief" and "details" acording suggestion in review

* fix: improving descrition of function imod acording suggestion in review

* fix: improving descrition acording suggestion in review

* Update modular_inverse_simple.cpp

* Apply suggestions from code review

* clang-format and clang-tidy fixes for 768a99c8

* Apply suggestions from code review

* Apply suggestions from code review

* updating DIRECTORY.md

Co-authored-by: John Law <johnlaw.po@gmail.com>
Co-authored-by: David Leal <halfpacho@gmail.com>

DIRECTORY.md

@@ -158,9 +158,6 @@
   * [Quadratic Probing Hash Table](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/hashing/quadratic_probing_hash_table.cpp)
   * [Sha1](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/hashing/sha1.cpp)
 
-## Linear Algebra
-  * [Gram Schmidt](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/linear_algebra/gram_schmidt.cpp)
-
 ## Machine Learning
   * [A Star Search](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/machine_learning/a_star_search.cpp)
   * [Adaline Learning](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/machine_learning/adaline_learning.cpp)
@@ -208,6 +205,7 @@
   * [Modular Division](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_division.cpp)
   * [Modular Exponentiation](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_exponentiation.cpp)
   * [Modular Inverse Fermat Little Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_fermat_little_theorem.cpp)
+  * [Modular Inverse Simple](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_simple.cpp)
   * [N Bonacci](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/n_bonacci.cpp)
   * [N Choose R](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/n_choose_r.cpp)
   * [Ncr Modulo P](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/ncr_modulo_p.cpp)
@@ -237,6 +235,7 @@
   * [Fast Fourier Transform](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/fast_fourier_transform.cpp)
   * [Gaussian Elimination](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/gaussian_elimination.cpp)
   * [Golden Search Extrema](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/golden_search_extrema.cpp)
+  * [Gram Schmidt](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/gram_schmidt.cpp)
   * [Inverse Fast Fourier Transform](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/inverse_fast_fourier_transform.cpp)
   * [Lu Decompose](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/lu_decompose.cpp)
   * [Lu Decomposition](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/lu_decomposition.h)

math/modular_inverse_simple.cpp

@@ -0,0 +1,60 @@
+/**
+ * @file
+ * @brief Simple implementation of [modular multiplicative
+ * inverse](https://en.wikipedia.org/wiki/Modular_multiplicative_inverse)
+ *
+ * @details
+ * this algorithm calculates the modular inverse x^{-1} \mod y iteratively
+ */
+
+#include <cassert>   /// for assert
+#include <iostream>  /// for IO operations
+
+/**
+ * @brief Function imod
+ * Calculates the modular inverse of x with respect to y, x^{-1} \mod y
+ * @param x number
+ * @param y number
+ * @returns the modular inverse
+ */
+uint64_t imod(uint64_t x, uint64_t y) {
+    uint64_t aux = 0;  // auxiliary variable
+    uint64_t itr = 0;  // iteration counter
+
+    do {  // run the algorithm while not find the inverse
+        aux = y * itr + 1;
+        itr++;
+    } while (aux % x);  // while module aux % x non-zero
+
+    return aux / x;
+}
+
+/**
+ * @brief  self-test implementations
+ * @returns void
+ */
+static void test() {
+    std::cout << "First case testing... \n";
+    // for a = 3 and b = 11 return 4
+    assert(imod(3, 11) == 4);
+    std::cout << "\nPassed!\n";
+
+    std::cout << "Second case testing... \n";
+    // for a = 3 and b = 26 return 9
+    assert(imod(3, 26) == 9);
+    std::cout << "\nPassed!\n";
+
+    std::cout << "Third case testing... \n";
+    // for a = 7 and b = 26 return 15
+    assert(imod(7, 26) == 15);
+    std::cout << "\nPassed!\n";
+
+    std::cout << "\nAll test cases have successfully passed!\n";
+}
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test();  // run self-test implementations
+};