多项式三角函数

描述

给定多项式 f\left(x\right) ，求模 x^{n} 意义下的 \sin{f\left(x\right)}, \cos{f\left(x\right)} 与 \tan{f\left(x\right)} 。

解法

首先由 Euler's formula \left(e^{ix} = \cos{x} + i\sin{x}\right) 可以得到 三角函数的另一个表达式：

\begin{aligned} \sin{x} &= \frac{e^{ix} - e^{-ix}}{2i} \\ \cos{x} &= \frac{e^{ix} + e^{-ix}}{2} \end{aligned}

那么代入 f\left(x\right) 就有：

\begin{aligned} \sin{f\left(x\right)} &= \frac{\exp{\left(if\left(x\right)\right)} - \exp{\left(-if\left(x\right)\right)}}{2i} \\ \cos{f\left(x\right)} &= \frac{\exp{\left(if\left(x\right)\right)} + \exp{\left(-if\left(x\right)\right)}}{2} \end{aligned}

直接按上述表达式编写程序即可得到模 x^{n} 意义下的 \sin{f\left(x\right)} 与 \cos{f\left(x\right)} 。再由 \tan{f\left(x\right)} = \frac{\sin{f\left(x\right)}}{\cos{f\left(x\right)}} 可求得 \tan{f\left(x\right)} 。

代码

多项式三角函数

注意到我们是在 \mathbb{Z}_{998244353} 上做 NTT，那么相应地，虚数单位 i 应该被换成 86583718 或 911660635 ： i = \sqrt{-1} \equiv \sqrt{998244352} \equiv 86583718 \equiv 911660635 \pmod{998244353} 。

constexpr int maxn = 262144;
constexpr int mod = 998244353;
constexpr int imgunit = 86583718; /* sqrt(-1) = sqrt(998233452) */

using i64 = long long;
using poly_t = int[maxn];
using poly = int *const;

void polytri(const poly &h, const int n, poly &sin_t, poly &cos_t) {
  /* sin(f) = (exp(i * f) - exp(- i * f)) / 2i */
  /* cos(f) = (exp(i * f) + exp(- i * f)) / 2 */
  /* tan(f) = sin(f) / cos(f) */
  assert(h[0] == 0);
  static poly_t tri1_t, tri2_t;

  for (int i = 0; i != n; ++i) tri2_t[i] = (i64)h[i] * imgunit % mod;
  polyexp(tri2_t, n, tri1_t);
  polyinv(tri1_t, n, tri2_t);

  if (sin_t != nullptr) {
    const int invi = fpow(pls(imgunit, imgunit), mod - 2);
    for (int i = 0; i != n; ++i)
      sin_t[i] = (i64)(tri1_t[i] - tri2_t[i] + mod) * invi % mod;
  }
  if (cos_t != nullptr) {
    for (int i = 0; i != n; ++i) cos_t[i] = div2(pls(tri1_t[i], tri2_t[i]));
  }
}
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