快速傅里叶变换与多项式乘法优化 赵卓越

多项式乘法

离散傅里叶变换 (DFT)  定义  逆变换（ IDFT ）  时间复杂度

快速傅里叶变换 (FFT)  二分思想  递归实现  迭代实现

递归实现 Vector Recursive_FFT(Vector x){ if (n == 1) return x; Vector x0, x1; for ( auto i = x.begin(); i!=x.end();){ x0.push_back(*i++); x1.push_back(*i++); } x0 = Recursive_FFT(x0); x1 = Recursive_FFT(x1); Complex w0(1.0); Complex w1(exp(Complex(0.0, - 2 * pi / x.size()))); for (int i = 0; i < x0.size(); ++i, w0 *= w1){ x[i] = x0[i] + w0 * x1[i]; x[i + x0.size()] = x0[i] - w0 * x1[i]; } return x; }

迭代实现  调整向量中分量的顺序 rev(x) inline unsigned rev(unsigned x, unsigned l){ int ret = 0; for (; l--; x >>= 1) ret = (ret << 1) | (x & 1); return ret; } inline void bit_reverse(Vector& a){ unsigned s = a.size(); unsigned n = 0; // length of a's indexes while (s >>= 1) { n+=1; } for (unsigned i = 0; i < a.size(); ++i){ s = rev(i, n); if (s > i) swap(a[i], a[s]); }

 void iterative_fft(Vector& a, int sign = -1){  bit_reverse(a);  for (unsigned s = 1; s < a.size(); s <<= 1){  Complex w0(exp(Complex(0.0, sign * pi_mul2 / (s << 1))));  for (unsigned i = 0; i < a.size(); i += (s << 1)){  Complex w(1.0);  for (unsigned k = 0; k < s; ++k, w*=w0){  Complex t1 = w * a[i + k + s];  Complex t2 = a[i + k];  a[i + k] = t2 + t1;  a[i + k + s] = t2 - t1;  }  inline void inverse_fft(Vector& v){  iterative_fft(v, 1);  for (Vector::iterator i = v.begin(); i != v.end(); ++i) *i /= v.size();  }