P4725 模板多项式对数函数

Posted 2021-02-03 olinr

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给出 (n-1) 次多项式 (A(x))，求一个 (mod{:x^n})下的多项式 (B(x))，满足 (B(x) equiv ln A(x))

在 ( ext{mod } 998244353)下进行，且 (a_i in [0, 998244353] cap mathbb{Z})

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第一行一个整数 (n).

下一行有 (n) 个整数，依次表示多项式的系数 (a_0, a_1, cdots, a_{n-1})

保证 (a_0 = 1).

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输出 (n) 个整数，表示答案多项式中的系数 (a_0, a_1, cdots, a_{n-1}).

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6
1 927384623 878326372 3882 273455637 998233543

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0 927384623 817976920 427326948 149643566 610586717

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对于 (100\%)的数据，(n le 10^5).

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推一下式子

(B(x)=ln(A(x)))

(B'(x)=frac{A'(x)}{A(x)})

(B(x)=int frac{A'(x)}{A(x)})

积分相当于逆求导，由于是多项式，很好求

先求个逆，再求个导，再FFT一下，再积分一下就行了

#include<bits/stdc++.h>
#define LL long long
LL in() {
    char ch; LL x = 0, f = 1;
    while(!isdigit(ch = getchar()))(ch == '-') && (f = -f);
    for(x = ch ^ 48; isdigit(ch = getchar()); x = (x << 1) + (x << 3) + (ch ^ 48));
    return x * f;
}
using std::vector;
const int mod = 998244353;
const int maxn = 4e5 + 10;
int len, r[maxn];
LL ksm(LL x, LL y) {
    LL re = 1LL;
    while(y) {
        if(y & 1) re = re * x % mod;
        x = x * x % mod;
        y >>= 1;
    }
    return re;
}
void FNTT(vector<int> &A, int flag) {
    A.resize(len);
    for(int i = 0; i < len; i++) if(i < r[i]) std::swap(A[i], A[r[i]]);
    for(int l = 1; l < len; l <<= 1) {
        int w0 = ksm(3, (mod - 1) / (l << 1));
        for(int i = 0; i < len; i += (l << 1)) {
            int w = 1, a0 = i, a1 = i + l;
            for(int k = 0; k < l; k++, a0++, a1++, w = 1LL * w * w0 % mod) {
                int tmp = 1LL * A[a1] * w % mod;
                A[a1] = ((A[a0] - tmp) % mod + mod) % mod;
                A[a0] = (A[a0] + tmp) % mod;
            }
        }
    }
    if(flag == -1) {
        std::reverse(A.begin() + 1, A.end());
        int inv = ksm(len, mod - 2);
        for(int i = 0; i < len; i++) A[i] = 1LL * inv * A[i] % mod;
    }
}
vector<int> operator * (vector<int> A, vector<int> B) {
    int tot = A.size() + B.size() - 1;
    for(len = 1; len <= tot; len <<= 1);
    for(int i = 0; i < len; i++) r[i] = (r[i >> 1] >> 1) | ((i & 1) * (len >> 1));
    FNTT(A, 1), FNTT(B, 1);
    vector<int> ans;
    ans.resize(len);
    for(int i = 0; i < len; i++) ans[i] = 1LL * A[i] * B[i] % mod;
    FNTT(ans, -1);
    ans.resize(tot);
    return ans;
}
vector<int> operator - (const vector<int> &A, const vector<int> &B) {
    vector<int> ans;
    for(int i = 0; i < (int)std::min(A.size(), B.size()); i++) ans.push_back(A[i] - B[i]);
    if(A.size() > B.size()) for(int i = B.size(); i < (int)A.size(); i++) ans.push_back(A[i]);
    if(A.size() < B.size()) for(int i = A.size(); i < (int)B.size(); i++) ans.push_back(-B[i]);
    return ans;
}
vector<int> inv(const vector<int> &A) {
    if(A.size() == 1) {
        vector<int> ans;
        ans.push_back(ksm(A[0], mod - 2));
        return ans;
    }
    int n = A.size(), _ = (n + 1) >> 1;
    vector<int> ans, B = A;
    ans.push_back(2);
    B.resize(_);
    B = inv(B);
    ans = B * (ans - A * B);
    ans.resize(n);
    return ans;
}
vector<int> getd(const vector<int> &A) {
    vector<int> ans;
    ans.resize(A.size() - 1);
    for(int i = 1; i < (int)A.size(); i++) ans[i - 1] = 1LL * i * A[i] % mod;
    return ans;
}
vector<int> geti(const vector<int> &A) {
    vector<int> ans;
    ans.resize(A.size() + 1);
    for(int i = 1; i < (int)ans.size(); i++) ans[i] = 1LL * A[i - 1] * ksm(i, mod - 2) % mod;
    return ans;
}
int main() {
    int n = in();
    vector<int> a, b, c;
    for(int i = 0; i < n; i++) a.push_back(in());
    b = inv(a);
    a = getd(a);
    c = a * b;
    c = geti(c);
    for(int i = 0; i < n; i++) printf("%d%c", c[i], i == n - 1? '
' : ' ');
    return 0;
}