快速傅里叶变换与多项式乘法优化

1. 快速傅里叶变换与多项式乘法优化 5120309037 赵卓越

2. 多项式乘法 • 有2n项，直接计算的时间复杂度为 • n-1次多项式也可以用n个不同的点表示

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

4. 快速傅里叶变换(FFT) • 二分思想 • 递归实现 • 迭代实现

5. 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; }

6. 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]); } }

7. 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(); • }