多项式开方

描述

给定多项式 g\left(x\right),求 f\left(x\right),满足:

f^{2}\left(x\right)\equiv g\left(x\right) \pmod{x^{n}}

解法

倍增法

假设现在已经求出了 g\left(x\right) 在模 x^{\left\lceil\frac{n}{2}\right\rceil} 意义下的平方根 f_{0}\left(x\right),则有:

\begin{aligned} f_{0}^{2}\left(x\right)&\equiv g\left(x\right) &\pmod{x^{\left\lceil\frac{n}{2}\right\rceil}}\\ f_{0}^{2}\left(x\right)-g\left(x\right)&\equiv 0 &\pmod{x^{\left\lceil\frac{n}{2}\right\rceil}}\\ \left(f_{0}^{2}\left(x\right)-g\left(x\right)\right)^{2}&\equiv 0 &\pmod{x^{n}}\\ \left(f_{0}^{2}\left(x\right)+g\left(x\right)\right)^{2}&\equiv 4f_{0}^{2}\left(x\right)g\left(x\right) &\pmod{x^{n}}\\ \left(\frac{f_{0}^{2}\left(x\right)+g\left(x\right)}{2f_{0}\left(x\right)}\right)^{2}&\equiv g\left(x\right) &\pmod{x^{n}}\\ \frac{f_{0}^{2}\left(x\right)+g\left(x\right)}{2f_{0}\left(x\right)}&\equiv f\left(x\right) &\pmod{x^{n}}\\ 2^{-1}f_{0}\left(x\right)+2^{-1}f_{0}^{-1}\left(x\right)g\left(x\right)&\equiv f\left(x\right) &\pmod{x^{n}} \end{aligned}

倍增计算即可。

时间复杂度

T\left(n\right)=T\left(\frac{n}{2}\right)+O\left(n\log{n}\right)=O\left(n\log{n}\right)

还有一种常数较小的写法就是在倍增维护 f\left(x\right) 的时候同时维护 f^{-1}\left(x\right) 而不是每次都求逆。

\left[x^{0}\right]g\left(x\right)\neq 1 时,可能需要使用二次剩余来计算 \left[x^{0}\right]f\left(x\right)

洛谷模板题 P5205 【模板】多项式开根 参考代码
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#include <bits/stdc++.h>
using namespace std;

const int maxn = 1 << 20, mod = 998244353;

int a[maxn], b[maxn], g[maxn], gg[maxn];

int qpow(int x, int y) {  // 快速幂
  int ans = 1;

  while (y) {
    if (y & 1) {
      ans = (long long)1 * ans * x % mod;
    }
    x = (long long)1 * x * x % mod;
    y >>= 1;
  }
  return ans;
}

int inv2 = qpow(2, mod - 2);  // 逆元

inline void change(int *f, int len) {
  for (int i = 1, j = len / 2; i < len - 1; i++) {
    if (i < j) {
      swap(f[i], f[j]);
    }

    int k = len / 2;
    while (j >= k) {
      j -= k;
      k /= 2;
    }
    if (j < k) {
      j += k;
    }
  }
}

inline void NTT(int *f, int len, int type) {  // NTT
  change(f, len);

  for (int q = 2; q <= len; q <<= 1) {
    int nxt = qpow(3, (mod - 1) / q);
    for (int i = 0; i < len; i += q) {
      int w = 1;

      for (int k = i; k < i + (q >> 1); k++) {
        int x = f[k];
        int y = (long long)1 * w * f[k + (q / 2)] % mod;

        f[k] = (x + y) % mod;
        f[k + (q / 2)] = (x - y + mod) % mod;
        w = (long long)1 * w * nxt % mod;
      }
    }
  }

  if (type == -1) {
    reverse(f + 1, f + len);
    int iv = qpow(len, mod - 2);

    for (int i = 0; i < len; i++) {
      f[i] = (long long)1 * f[i] * iv % mod;
    }
  }
}

inline void inv(int deg, int *f, int *h) {  // 求逆元
  if (deg == 1) {
    h[0] = qpow(f[0], mod - 2);
    return;
  }

  inv(deg + 1 >> 1, f, h);

  int len = 1;
  while (len < deg * 2) {  // 倍增
    len *= 2;
  }

  copy(f, f + deg, gg);
  fill(gg + deg, gg + len, 0);

  NTT(gg, len, 1);
  NTT(h, len, 1);
  for (int i = 0; i < len; i++) {
    h[i] = (long long)1 * (2 - (long long)1 * gg[i] * h[i] % mod + mod) % mod *
           h[i] % mod;
  }

  NTT(h, len, -1);
  fill(h + deg, h + len, 0);
}

int n, t[maxn];

// deg:次数
// f:被开根数组
// h:答案数组
inline void sqrt(int deg, int *f, int *h) {
  if (deg == 1) {
    h[0] = 1;
    return;
  }

  sqrt(deg + 1 >> 1, f, h);

  int len = 1;
  while (len < deg * 2) {  // 倍增
    len *= 2;
  }
  fill(g, g + len, 0);
  inv(deg, h, g);
  copy(f, f + deg, t);
  fill(t + deg, t + len, 0);
  NTT(t, len, 1);
  NTT(g, len, 1);
  NTT(h, len, 1);

  for (int i = 0; i < len; i++) {
    h[i] = (long long)1 * inv2 *
           ((long long)1 * h[i] % mod + (long long)1 * g[i] * t[i] % mod) % mod;
  }
  NTT(h, len, -1);
  fill(h + deg, h + len, 0);
}

int main() {
  cin >> n;

  for (int i = 0; i < n; i++) {
    scanf("%d", &a[i]);
  }
  sqrt(n, a, b);

  for (int i = 0; i < n; i++) {
    printf("%d ", b[i]);
  }

  return 0;
}

Newton's Method

参见 Newton's Method.

例题

  1. 「Codeforces Round #250」E. The Child and Binary Tree

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