library

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interpolate (one point)
(FPS/interpolate.hpp)

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• Last update: 2025-05-11 13:37:16+09:00
• Include: #include "FPS/interpolate.hpp"

Depends on

• Combination (Math/comb.hpp)

Required by

• $\sum_{k} r^k\cdot poly(k)$ (FPS/sumofpolyexp.hpp)

Verified with

• Verify/LC_sum_of_exponential_times_polynomial.test.cpp
• Verify/LC_sum_of_exponential_times_polynomial_limit.test.cpp

Code

#pragma once
#include "Math/comb.hpp"

// f(0),..,f(d) -> f(t)
template <typename T> T Interpolate(vector<T> &ys, ll t) {
    int d = ys.size() - 1;
    if (t <= d)
        return ys[t];
    vector<T> L(d + 1, 1), R(d + 1, 1);
    rep(i, 0, d) L[i + 1] = L[i] * (t - i);
    for (int i = d; i; i--)
        R[i - 1] = R[i] * (t - i);
    T ret;
    rep(i, 0, d + 1) {
        T add = ys[i] * L[i] * R[i] * Fact<T>(i, 1) * Fact<T>(d - i, 1);
        if ((d - i) & 1)
            ret -= add;
        else
            ret += add;
    }
    return ret;
}

/**
 * @brief interpolate (one point)
 */

#line 2 "Math/comb.hpp"

template <typename T> T Inv(ll n) {
    static int md;
    static vector<T> buf({0, 1});
    if (md != T::get_mod()) {
        md = T::get_mod();
        buf = vector<T>({0, 1});
    }
    assert(n > 0);
    n %= md;
    while (SZ(buf) <= n) {
        int k = SZ(buf), q = (md + k - 1) / k;
        buf.push_back(buf[k * q - md] * q);
    }
    return buf[n];
}

template <typename T> T Fact(ll n, bool inv = 0) {
    static int md;
    static vector<T> buf({1, 1}), ibuf({1, 1});
    if (md != T::get_mod()) {
        md = T::get_mod();
        buf = ibuf = vector<T>({1, 1});
    }
    assert(n >= 0 and n < md);
    while (SZ(buf) <= n) {
        buf.push_back(buf.back() * SZ(buf));
        ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
    }
    return inv ? ibuf[n] : buf[n];
}

template <typename T> T nPr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
// sum = n, r tuples
template <typename T> T nHr(int n, int r, bool inv = 0) {
    return nCr<T>(n + r - 1, r - 1, inv);
}
// sum = n, a nonzero tuples and b tuples
template <typename T> T choose(int n, int a, int b) {
    if (n == 0)
        return !a;
    return nCr<T>(n + b - 1, a + b - 1);
}

/**
 * @brief Combination
 */
#line 3 "FPS/interpolate.hpp"

// f(0),..,f(d) -> f(t)
template <typename T> T Interpolate(vector<T> &ys, ll t) {
    int d = ys.size() - 1;
    if (t <= d)
        return ys[t];
    vector<T> L(d + 1, 1), R(d + 1, 1);
    rep(i, 0, d) L[i + 1] = L[i] * (t - i);
    for (int i = d; i; i--)
        R[i - 1] = R[i] * (t - i);
    T ret;
    rep(i, 0, d + 1) {
        T add = ys[i] * L[i] * R[i] * Fact<T>(i, 1) * Fact<T>(d - i, 1);
        if ((d - i) & 1)
            ret -= add;
        else
            ret += add;
    }
    return ret;
}

/**
 * @brief interpolate (one point)
 */

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