From 405d21a54464f84dcc4cd79f96d17303946fdb56 Mon Sep 17 00:00:00 2001
From: David Leal <halfpacho@gmail.com>
Date: Sat, 6 Nov 2021 21:56:42 -0600
Subject: [PATCH] Create inverse_fast_fourier_transform.cpp

---
 .../inverse_fast_fourier_transform.cpp        | 161 ++++++++++++++++++
 1 file changed, 161 insertions(+)
 create mode 100644 numerical_methods/inverse_fast_fourier_transform.cpp

diff --git a/numerical_methods/inverse_fast_fourier_transform.cpp b/numerical_methods/inverse_fast_fourier_transform.cpp
new file mode 100644
index 000000000..28ae330a0
--- /dev/null
+++ b/numerical_methods/inverse_fast_fourier_transform.cpp
@@ -0,0 +1,161 @@
+/**
+ * @file
+ * @brief [An inverse fast Fourier transform
+ * (IFFT)](https://www.geeksforgeeks.org/python-inverse-fast-fourier-transformation/)
+ * is an algorithm that computes the inverse fourier transform.
+ * @details
+ * This algorithm has an application in use case scenario where a user wants find coefficients of
+ * a function in a short time by just using points generated by DFT.
+ * Time complexity
+ * this algorithm computes the IDFT in O(nlogn) time in comparison to traditional O(n^2).
+ * @author [Ameya Chawla](https://github.com/ameyachawlaggsipu)
+ */
+
+#include <cassert>   /// for assert
+#include <cmath>     /// for mathematical-related functions
+#include <complex>   /// for storing points and coefficents
+#include <iostream>  /// for IO operations
+#include <vector>    /// for std::vector
+
+/**
+ * @namespace numerical_methods
+ * @brief Numerical algorithms/methods
+ */
+namespace numerical_methods {
+/**
+ * @brief InverseFastFourierTransform 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<double> *InverseFastFourierTransform(std::complex<double> *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<double> om = std::complex<double>(
+        cos(2 * pi / n), sin(2 * pi / n));  /// Calculating value of omega
+        
+    om.real(om.real()/n); /// One change in comparison with DFT
+    om.imag(om.imag()/n); /// One change in comparison with DFT
+
+    auto *pe = new std::complex<double>[n / 2];  /// Coefficients of even power
+
+    auto *po = new std::complex<double>[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<double> *ye =
+        InverseFastFourierTransform(pe, n / 2);  /// Recursive Call
+
+    std::complex<double> *yo =
+        InverseFastFourierTransform(po, n / 2);  /// Recursive Call
+
+    auto *y = new std::complex<double>[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<double>[2];  /// Test case 1
+    auto *t2 = new std::complex<double>[4];  /// Test case 2
+
+    t1[0] = {3, 0};
+    t1[1] = {-1, 0};
+    t2[0] = {10, 0};
+    t2[1] = {-2, -2};
+    t2[2] = {-2, 0};
+    t2[3] = {-2, 2};
+
+    uint8_t n1 = 2;
+    uint8_t n2 = 4;
+    std::vector<std::complex<double>> r1 = {
+        {1, 0}, {2, 0}};  /// True Answer for test case 1
+
+    std::vector<std::complex<double>> r2 = {
+        {1, 0}, {2, 0}, {3, 0}, {4, 0}};  /// True Answer for test case 2
+
+    std::complex<double> *o1 = numerical_methods::InverseFastFourierTransform(t1, n1);
+    
+    std::complex<double> *o2 = numerical_methods::InverseFastFourierTransform(t2, n2);
+
+    for (uint8_t i = 0; i < n1; i++) {
+        assert((r1[i].real() - o1[i].real() < 0.000000000001) &&
+               (r1[i].imag() - o1[i].imag() <
+                0.000000000001));  /// Comparing for both real and imaginary
+                                   /// values for test case 1
+        
+    }
+
+    for (uint8_t i = 0; i < n2; i++) {
+        assert((r2[i].real() - o2[i].real() < 0.000000000001) &&
+               (r2[i].imag() - o2[i].imag() <
+                0.000000000001));  /// Comparing for both real and imaginary
+                                   /// values for test case 2
+        
+    }
+    
+    
+    delete[] t1;
+    delete[] t2;
+    delete[] o1;
+    delete[] o2;
+    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;
+}