Luogu P5264 多项式三角函数

\(\mathtt{Description}\)

给定 \(\deg = n\) 的多项式 \(f\) 和 \(\text{type}\)，求一个多项式 \(g\) 使得

\[g\equiv \sin f \pmod {998244353}\quad(\text{type}=0) \]

\[g\equiv \cos f \pmod {998244353}\quad(\text{type}=1) \]

\(\mathtt{restrictions:}1\leq n \le10^5,a_i\in[0,998244353)\cap \mathbb{N}\)

\(\mathtt{Solution}\)

奇怪的多项式模板增加了！

\(\mathtt{Pre\;Knowledge}\)

• 多项式 \(\exp\)
• 欧拉公式

\(\mathtt{Main\;Body}\)

我们知道欧拉公式

\[e^{ix} = \cos x+ i\sin x \]

有

\[e^{-ix} = \cos x -i\sin x \]

故

\[\cos x=\dfrac {e^{ix}+e^{-ix}} {2} \]

\[\sin x = \dfrac{e^{ix}-e^{-ix}}{2i} \]

又有

\[i\equiv86583718\pmod{998244353} \]

所以只需要一个多项式 \(\exp\) 就可以了。

\(\mathtt{Code}\)

  inline auto Cos(vector <int> f) {
    const int len = f.size() ;
    for (int i = 0; i < len; ++i) f[i] = 1ll * f[i] * I % mod ;
    auto g = Exp(f),h = Inv(g) ;
    for (int i = 0; i < len; ++i) g[i] = 1ll * I2 * (g[i] + h[i]) % mod ;
    return g ;  
  }

  inline auto Sin(vector <int> f) {
    const int len = f.size() ;
    for (int i = 0; i < len; ++i) f[i] = 1ll * f[i] * I % mod ;
    auto g = Exp(f),h = Inv(g) ;
    const int x = ksm(I << 1) ;
    for (int i = 0; i < len; ++i) g[i] = 1ll * x * (g[i] - h[i] + mod) % mod ;
    return g ;
  }