RecursiveFFT.cpp

Author: lxylxy123456

README.md

@@ -192,6 +192,9 @@
 | 29 | Simplex.cpp					| Pivot									|  869 |
 | 29 | Simplex.cpp					| Simplex								|  871 |
 | 29 | Simplex.cpp					| Initialize Simplex					|  887 |
+| 30 | RecursiveFFT.cpp				| Recursive FFT							|  911 |
+| 30 | RecursiveFFT.cpp				| Inverse FFT							|  913 |
+| 30 | RecursiveFFT.cpp				| Polynomial Multiply					|  914 |
 
 # Supplementary Files
 * `utils.h`: Utils

RecursiveFFT.cpp

@@ -0,0 +1,144 @@
+//  
+//  algorithm - some algorithms in "Introduction to Algorithms", third edition
+//  Copyright (C) 2018  lxylxy123456
+//  
+//  This program is free software: you can redistribute it and/or modify
+//  it under the terms of the GNU Affero General Public License as
+//  published by the Free Software Foundation, either version 3 of the
+//  License, or (at your option) any later version.
+//  
+//  This program is distributed in the hope that it will be useful,
+//  but WITHOUT ANY WARRANTY; without even the implied warranty of
+//  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+//  GNU Affero General Public License for more details.
+//  
+//  You should have received a copy of the GNU Affero General Public License
+//  along with this program.  If not, see <https://www.gnu.org/licenses/>.
+//  
+
+#ifndef MAIN
+#define MAIN
+#define MAIN_RecursiveFFT
+#endif
+
+#ifndef FUNC_RecursiveFFT
+#define FUNC_RecursiveFFT
+
+#include <cmath>
+#include "utils.h"
+
+#include "SquareMatrixMultiply.cpp"
+
+template <typename T>
+class Complex {
+	public:
+		Complex(): real(0), imag(0) {}
+		Complex(T x): real(x), imag(0) {}
+		Complex(T x, T y): real(x), imag(y) {}
+		Complex<T> operator+(const Complex<T>& rhs) const {
+			return Complex<T>(real + rhs.real, imag + rhs.imag); 
+		}
+		Complex<T> operator-(const Complex<T>& rhs) const {
+			return Complex<T>(real - rhs.real, imag - rhs.imag); 
+		}
+		Complex<T> operator*(const Complex<T>& rhs) const {
+			return Complex<T>(real * rhs.real - imag * rhs.imag, 
+								real * rhs.imag + imag * rhs.real); 
+		}
+		Complex<T>& operator*=(const Complex<T>& rhs) {
+			T r = real * rhs.real - imag * rhs.imag; 
+			T i = real * rhs.imag + imag * rhs.real; 
+			real = r; 
+			imag = i; 
+			return *this; 
+		}
+		Complex<T>& operator/=(const T& rhs) {
+			real /= rhs; 
+			imag /= rhs; 
+			return *this; 
+		}
+		friend std::ostream& operator<<(std::ostream& os, const Complex<T>& r) {
+			if (abs(r.imag) > 0.0000000001)
+				return os << r.real << " + " << r.imag << " i"; 
+			else
+				return os << r.real; 
+		}
+		T real, imag; 
+}; 
+
+template <typename T>
+Complex<T> expi(T x) {
+	return Complex<T>(cos(x), sin(x)); 
+}
+
+template <typename T>
+Matrix<T> RecursiveFFT(Matrix<T>& a, bool neg = false) {
+	const size_t n = a.rows; 
+	if (n == 1)
+		return a; 
+	assert(n % 2 == 0); 
+	T wn = expi((neg ? -1 : 1) * 2 * M_PI / n); 
+	T w = 1; 
+	Matrix<T> a0(n / 2, 1), a1(n / 2, 1); 
+	for (size_t i = 0; i < n; i += 2) {
+		a0.data.push_back(a[i]); 
+		a1.data.push_back(a[i + 1]); 
+	}
+	Matrix<T> y0 = RecursiveFFT(a0, neg); 
+	Matrix<T> y1 = RecursiveFFT(a1, neg); 
+	Matrix<T> y(n, 1, 0); 
+	for (size_t k = 0; k < n / 2; k++) {
+		y[k][0] = y0[k][0] + w * y1[k][0]; 
+		y[k + n/2][0] = y0[k][0] - w * y1[k][0]; 
+		w *= wn; 
+	}
+	return y; 
+}
+
+template <typename T>
+Matrix<T> InverseFFT(Matrix<T>& a) {
+	const size_t n = a.rows; 
+	Matrix<T> ans = RecursiveFFT(a, true); 
+	for (size_t i = 0; i < n; i++)
+		ans[i][0] /= n; 
+	return ans; 
+}
+
+template <typename T>
+Matrix<T> PolynomialMultiply(Matrix<T>& a, Matrix<T>& b) {
+	const size_t n = a.rows; 
+	assert(n == b.rows); 
+	Matrix<T> n0(n, 1, 0); 
+	Matrix<T> aa = a.concat_v(n0); 
+	Matrix<T> bb = b.concat_v(n0); 
+	Matrix<T> fa = RecursiveFFT(aa); 
+	Matrix<T> fb = RecursiveFFT(bb); 
+	Matrix<T> fc(2 * n, 1, 0); 
+	for (size_t i = 0; i < 2 * n; i++)
+		fc[i][0] = fa[i][0] * fb[i][0]; 
+	return InverseFFT(fc); 
+}
+#endif
+
+#ifdef MAIN_RecursiveFFT
+int main(int argc, char *argv[]) {
+	const size_t n = get_argv(argc, argv, 1, 16); 
+	std::vector<int> int_a, int_b; 
+	random_integers(int_a, -n, n, n); 
+	random_integers(int_b, -n, n, n); 
+	using T = Complex<double>; 
+	std::vector<T> buf_a(n), buf_b(n); 
+	for (size_t i = 0; i < int_a.size(); i++)
+		buf_a[i] = int_a[i]; 
+	for (size_t i = 0; i < int_a.size(); i++)
+		buf_b[i] = int_b[i]; 
+	Matrix<T> a(n, 1, buf_a); 
+	std::cout << a << std::endl; 
+	Matrix<T> b(n, 1, buf_b); 
+	std::cout << b << std::endl; 
+	Matrix<T> c = PolynomialMultiply(a, b); 
+	std::cout << c << std::endl; 
+	return 0; 
+}
+#endif
+