Lagrange Polynomial(多項式補間, 係数) (math/combinatorics/lagrange-polynomial-2.hpp)

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• Last update: 2024-04-27 01:27:28+09:00
• Include: #include "math/combinatorics/lagrange-polynomial-2.hpp"

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/**
 * @brief Lagrange Polynomial(多項式補間, 係数)
 */
template <typename Mint>
vector<Mint> lagrange_polynomial(const vector<Mint> &x, const vector<Mint> &y) {
  int k = (int)x.size() - 1;

  vector<Mint> f(k + 1), dp(k + 2);
  dp[0] = 1;
  for (int j = 0; j <= k; j++) {
    for (int l = k + 1; l > 0; l--) {
      dp[l] = dp[l] * -x[j] + dp[l - 1];
    }
    dp[0] *= -x[j];
  }

  for (int i = 0; i <= k; i++) {
    Mint d = 1;
    for (int j = 0; j <= k; j++) {
      if (i != j) {
        d *= x[i] - x[j];
      }
    }
    Mint mul = y[i] / d;
    if (x[i] == 0) {
      for (int j = 0; j <= k; j++) {
        f[j] += dp[j + 1] * mul;
      }
    } else {
      Mint inv = Mint(1) / (-x[i]), pre = 0;
      for (int j = 0; j <= k; j++) {
        Mint cur = (dp[j] - pre) * inv;
        f[j] += cur * mul;
        pre = cur;
      }
    }
  }
  return f;
}

#line 1 "math/combinatorics/lagrange-polynomial-2.hpp"
/**
 * @brief Lagrange Polynomial(多項式補間, 係数)
 */
template <typename Mint>
vector<Mint> lagrange_polynomial(const vector<Mint> &x, const vector<Mint> &y) {
  int k = (int)x.size() - 1;

  vector<Mint> f(k + 1), dp(k + 2);
  dp[0] = 1;
  for (int j = 0; j <= k; j++) {
    for (int l = k + 1; l > 0; l--) {
      dp[l] = dp[l] * -x[j] + dp[l - 1];
    }
    dp[0] *= -x[j];
  }

  for (int i = 0; i <= k; i++) {
    Mint d = 1;
    for (int j = 0; j <= k; j++) {
      if (i != j) {
        d *= x[i] - x[j];
      }
    }
    Mint mul = y[i] / d;
    if (x[i] == 0) {
      for (int j = 0; j <= k; j++) {
        f[j] += dp[j + 1] * mul;
      }
    } else {
      Mint inv = Mint(1) / (-x[i]), pre = 0;
      for (int j = 0; j <= k; j++) {
        Mint cur = (dp[j] - pre) * inv;
        f[j] += cur * mul;
        pre = cur;
      }
    }
  }
  return f;
}

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