From 0a5f5c61f73004cd90208ec5b998d22aec3ff246 Mon Sep 17 00:00:00 2001 From: dsmurrow <73198549+dsmurrow@users.noreply.github.com> Date: Thu, 13 Apr 2023 17:33:10 -0400 Subject: [PATCH] feat: added extended Euclidean algorithm (#1238) * chore: made it so math directory gets built * feat: added extended Euclidean algorithm * docs: added details qualifier Co-authored-by: David Leal * docs: added param qualifiers to functions that needed them * docs: added details qualifier Co-authored-by: David Leal * docs: small cleanup --------- Co-authored-by: David Leal --- math/euclidean_algorithm_extended.c | 154 ++++++++++++++++++++++++++++ 1 file changed, 154 insertions(+) create mode 100644 math/euclidean_algorithm_extended.c diff --git a/math/euclidean_algorithm_extended.c b/math/euclidean_algorithm_extended.c new file mode 100644 index 00000000..914eb98b --- /dev/null +++ b/math/euclidean_algorithm_extended.c @@ -0,0 +1,154 @@ +/** + * @{ + * @file + * @brief Program to perform the [extended Euclidean + * algorithm](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm) + * + * @details The extended Euclidean algorithm, on top of finding the GCD (greatest common + * divisor) of two integers a and b, also finds the values x and y such that + * ax+by = gcd(a, b) + */ + +#include /// for tests +#include /// for IO +#include /// for div function and corresponding div_t struct + +/** + * @brief a structure holding the values resulting from the extended Euclidean + * algorithm + */ +typedef struct euclidean_result +{ + int gcd; ///< the greatest common divisor calculated with the Euclidean + ///< algorithm + int x, y; ///< the values x and y such that ax + by = gcd(a, b) +} euclidean_result_t; + +/** + * @brief gives queue-like behavior to an array of two ints, pushing an element + * onto the end and pushing one off the front + * + * @param arr an array of ints acting as a queue + * @param newval the value being pushed into arr + * + * @returns void + */ +static inline void xy_push(int arr[2], int newval) +{ + arr[1] = arr[0]; + arr[0] = newval; +} + +/** + * @brief calculates the value of x or y and push those into the small 'queues' + * + * @details Both x and y are found by taking their value from 2 iterations ago minus the + * product of their value from 1 iteration ago and the most recent quotient. + * + * @param quotient the quotient from the latest iteration of the Euclidean + * algorithm + * @param prev the 'queue' holding the values of the two previous iterations + * + * @returns void + */ +static inline void calculate_next_xy(int quotient, int prev[2]) +{ + int next = prev[1] - (prev[0] * quotient); + xy_push(prev, next); +} + +/** + * @brief performs the extended Euclidean algorithm on integer inputs a and b + * + * @param a first integer input + * @param b second integer input + * + * @returns euclidean_result_t containing the gcd, and values x and y such that + * ax + by = gcd + */ +euclidean_result_t extended_euclidean_algorithm(int a, int b) +{ + int previous_remainder = 1; + int previous_x_values[2] = {0, 1}; + int previous_y_values[2] = {1, 0}; + div_t div_result; + euclidean_result_t result; + + /* swap values of a and b */ + if (abs(a) < abs(b)) + { + a ^= b; + b ^= a; + a ^= b; + } + + div_result.rem = b; + + while (div_result.rem > 0) + { + div_result = div(a, b); + + previous_remainder = b; + + a = b; + b = div_result.rem; + + calculate_next_xy(div_result.quot, previous_x_values); + calculate_next_xy(div_result.quot, previous_y_values); + } + + result.gcd = previous_remainder; + result.x = previous_x_values[1]; + result.y = previous_y_values[1]; + + return result; +} + +/** @} */ + +/** + * @brief perform one single check on the result of the algorithm with provided + * parameters and expected output + * + * @param a first paramater for Euclidean algorithm + * @param b second parameter for Euclidean algorithm + * @param gcd expected value of result.gcd + * @param x expected value of result.x + * @param y expected value of result.y + * + * @returns void + */ +static inline void single_test(int a, int b, int gcd, int x, int y) +{ + euclidean_result_t result; + + result = extended_euclidean_algorithm(a, b); + assert(result.gcd == gcd); + assert(result.x == x); + assert(result.y == y); +} + +/** + * @brief Perform tests on known results + * @returns void + */ +static void test() +{ + single_test(40, 27, 1, -2, 3); + single_test(71, 41, 1, -15, 26); + single_test(48, 18, 6, -1, 3); + single_test(99, 303, 3, -16, 49); + single_test(14005, 3507, 1, -305, 1218); + + printf("All tests have successfully passed!\n"); +} + +/** + * @brief Main Function + * @returns 0 upon successful program exit + */ +int main() +{ + test(); // run self-test implementations + return 0; +}