From b4b0864da10c47d675c8da1343e5f78d050850d6 Mon Sep 17 00:00:00 2001 From: Ameya Chawla <88154798+ameyachawlaggsipu@users.noreply.github.com> Date: Tue, 2 Nov 2021 22:22:18 +0530 Subject: [PATCH] feat: Implemented Fast Fourier Transform algorithm (#1700) * feat ; Implemented Fast Fourier Transform * feat : Implemented Fast Fourier Transform * fix : added comments to header * Fixing Code Formatter errors * fix : updated the documentation * fix : removed bad practice of using namespace td * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * updating DIRECTORY.md * fix : fixed the integer values with their appropriate size * Fixed errors * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Fixed many errors * fix : fixed array errors * fix: fixing memory leak errors * fix: using delete instead of free as mentioned in errors * fix : using delete[ ] instead of delete * fix : fixing errors for memory leaks * Update fast_fourier_transform.cpp * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * clang-format and clang-tidy fixes for a6594c85 * fix : updated documentation * fix : added time complexity in documentation * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * fix : update code as per the namespace numerical_methods * fix : use of auto keyword to reduce complexity * fix : updated documentation * fix : fixed segmentation fault error * fix : fixing clang-tidy errors * fix : fixing clang-tidy errors * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * fix : updated documentation * fix : fixing warnings * Update fast_fourier_transform.cpp * Update fast_fourier_transform.cpp * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * clang-format and clang-tidy fixes for 198b4297 * Update fast_fourier_transform.cpp * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * Update numerical_methods/fast_fourier_transform.cpp Co-authored-by: David Leal * a * Apply suggestions from code review * fix : updating leak memeory * Update fast_fourier_transform.cpp Co-authored-by: David Leal Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> --- DIRECTORY.md | 1 + numerical_methods/fast_fourier_transform.cpp | 165 +++++++++++++++++++ 2 files changed, 166 insertions(+) create mode 100644 numerical_methods/fast_fourier_transform.cpp diff --git a/DIRECTORY.md b/DIRECTORY.md index 278a9ed69..781a4cbe0 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -225,6 +225,7 @@ * [Brent Method Extrema](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/brent_method_extrema.cpp) * [Durand Kerner Roots](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/durand_kerner_roots.cpp) * [False Position](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/false_position.cpp) + * [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) * [Lu Decompose](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/lu_decompose.cpp) diff --git a/numerical_methods/fast_fourier_transform.cpp b/numerical_methods/fast_fourier_transform.cpp new file mode 100644 index 000000000..86e9d6d79 --- /dev/null +++ b/numerical_methods/fast_fourier_transform.cpp @@ -0,0 +1,165 @@ +/** + * @file + * @brief [A fast Fourier transform + * (FFT)](https://medium.com/@aiswaryamathur/understanding-fast-fouriertransform-from-scratch-to-solve-polynomial-multiplication-8018d511162f) + * is an algorithm that computes the + * discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). + * @details + * This + * algorithm has application in use case scenario where a user wants to find points of a + * function + * in a short time by just using the coefficients of the polynomial + * function. + * It can be also used to find inverse fourier transform by just switching the + value of omega. + * Time complexity + * this algorithm computes the DFT in O(nlogn) time in comparison to traditional + O(n^2). + * @author [Ameya Chawla](https://github.com/ameyachawlaggsipu) + */ + +#include /// for assert +#include /// for mathematical-related functions +#include /// for storing points and coefficents +#include /// for IO operations +#include /// for std::vector + +/** + * @namespace numerical_methods + * @brief Numerical algorithms/methods + */ +namespace numerical_methods { +/** + * @brief FastFourierTransform is a recursive function which returns list of + * complex numbers + * @param p List of Coefficents in form of complex numbers + * @param n Count of elements in list p + * @returns p if n==1 + * @returns y if n!=1 + */ +std::complex *FastFourierTransform(std::complex *p, uint8_t n) { + if (n == 1) { + return p; /// Base Case To return + } + + double pi = 2 * asin(1.0); /// Declaring value of pi + + std::complex om = std::complex( + cos(2 * pi / n), sin(2 * pi / n)); /// Calculating value of omega + + auto *pe = new std::complex[n / 2]; /// Coefficients of even power + + auto *po = new std::complex[n / 2]; /// Coefficients of odd power + + int k1 = 0, k2 = 0; + for (int j = 0; j < n; j++) { + if (j % 2 == 0) { + pe[k1++] = p[j]; /// Assigning values of even Coefficients + + } else + po[k2++] = p[j]; /// Assigning value of odd Coefficients + } + + std::complex *ye = + FastFourierTransform(pe, n / 2); /// Recursive Call + + std::complex *yo = + FastFourierTransform(po, n / 2); /// Recursive Call + + auto *y = new std::complex[n]; /// Final value representation list + + k1 = 0, k2 = 0; + + for (int i = 0; i < n / 2; i++) { + y[i] = + ye[k1] + pow(om, i) * yo[k2]; /// Updating the first n/2 elements + y[i + n / 2] = + ye[k1] - pow(om, i) * yo[k2]; /// Updating the last n/2 elements + + k1++; + k2++; + } + + if(n!=2){ + + delete[] pe; + delete[] po; + + } + + delete[] ye; /// Deleting dynamic array ye + delete[] yo; /// Deleting dynamic array yo + return y; +} + +} // namespace numerical_methods + +/** + * @brief Self-test implementations + * @details + * Declaring two test cases and checking for the error + * in predicted and true value is less than 0.000000000001. + * @returns void + */ +static void test() { + /* descriptions of the following test */ + + auto *t1 = new std::complex[2]; /// Test case 1 + auto *t2 = new std::complex[4]; /// Test case 2 + + t1[0] = {1, 0}; + t1[1] = {2, 0}; + t2[0] = {1, 0}; + t2[1] = {2, 0}; + t2[2] = {3, 0}; + t2[3] = {4, 0}; + + uint8_t n1 = 2; + uint8_t n2 = 4; + std::vector> r1 = { + {3, 0}, {-1, 0}}; /// True Answer for test case 1 + + std::vector> r2 = { + {10, 0}, {-2, -2}, {-2, 0}, {-2, 2}}; /// True Answer for test case 2 + + std::complex *o1 = numerical_methods::FastFourierTransform(t1, n1); + std::complex *t3=o1; /// Temporary variable used to delete memory location of o1 + std::complex *o2 = numerical_methods::FastFourierTransform(t2, n2); + std::complex *t4=o2; /// Temporary variable used to delete memory location of o2 + for (uint8_t i = 0; i < n1; i++) { + assert((r1[i].real() - o1->real() < 0.000000000001) && + (r1[i].imag() - o1->imag() < + 0.000000000001)); /// Comparing for both real and imaginary + /// values for test case 1 + o1++; + } + + for (uint8_t i = 0; i < n2; i++) { + assert((r2[i].real() - o2->real() < 0.000000000001) && + (r2[i].imag() - o2->imag() < + 0.000000000001)); /// Comparing for both real and imaginary + /// values for test case 2 + o2++; + } + + + delete[] t1; + delete[] t2; + delete[] t3; + delete[] t4; + std::cout << "All tests have successfully passed!\n"; +} + +/** + * @brief Main function + * @param argc commandline argument count (ignored) + * @param argv commandline array of arguments (ignored) + * calls automated test function to test the working of fast fourier transform. + * @returns 0 on exit + */ + +int main(int argc, char const *argv[]) { + test(); // run self-test implementations + // with 2 defined test cases + return 0; +}