# 多项式操作公式记录

## 快速傅里叶变换/数论变换

$A_(\omega_n^k)~=~A_0(\omega_m^k) + \omega_n^kA_1(\omega_m^k)$

$A(\omega_n^{k + m})~=~A_0(\omega_m^k) - \omega_n^kA_1(\omega_m^k)$

## 求乘法逆

$A(x) B(x) \equiv 1 \pmod {x^n}$

$B_{2k}(x) \equiv 2 B_{k}(x) - A_{2k}(x) \times B_{k}^{2}(x) \pmod {x^{2k}}$

## 求对数函数

$B(x) = \ln A(x)$

$B'(x)~\equiv~A'(x) A^{-1}(x) \pmod {x^n}$

$B(x)~=~\int B'(x) \text{d}x$

$f(x) = Cx^a~~\Rightarrow~~ f'(x) = C(a - 1)x^{a - 1}$

$f(x) = Cx^a~~ ~~\Rightarrow~~ \int f(x) \text{d}x = \frac{C}{a + 1} x^{a + 1}$

## 求指数函数

$B(x) = e^{A(x)}$

$B_{2k}(x)~=~B_k(x) \times(1 - \ln B_k(x) + A_{2k}(x))$

## 多项式 $k$ 次幂

$B(x) \equiv (A(x))^k \pmod {x^n}$

$B(x) = \exp (k \ln A(x))$

## 封装代码

namespace Poly {

const int G = 3;
const int MOD = 998244353;
const int PHI = 998244352;
const int DPHI = 998244351;
const int SzI = sizeof(int);
const int SzL = sizeof(long long int);

void Init(int *const A, const int N) {
for (int i = 0; i < N; ++i) {
qr(A[i]);
}
}

void Print(const int *const A, const int N) {
int DN = N - 1;
for (int i = 0; i < DN; ++i) {
qw(A[i], ' ', true);
}
qw(A[DN], '\n', true);
}

void GetN(int n, int &N) {
N = 1;
while (N < n) N <<= 1;
}

int mpow(int x, int y) {
int _ret = 1;
while (y) {
if (y & 1)  _ret = 1ll * x * _ret % MOD;
x = 1ll * x * x % MOD;
y >>= 1;
}
return _ret;
}

int tax[maxn], taxlen;

void GetRev(const int N) {
int p = 1, d = N >> 1;
for (int w = p; w < N; w = p) {
for (int i = 0; i < w; ++i) {
tax[p++] = tax[i] | d;
}
d >>= 1;
}
taxlen = N;
}

void MakeRev(int *const A, const int N) {
for (int i = 1; i < N; ++i) if (tax[i] > i) {
std::swap(A[i], A[tax[i]]);
}
}

void Modint(int &x) {
while (x >= MOD) x -= MOD;
while (x < 0) x += MOD;
}

void NTT(int *const A, const int N) {
if (taxlen != N) {
GetRev(N);
}
MakeRev(A, N);
for (int w = 2, M = 1; w <= N; w <<= 1) {
int Wn = mpow(G, PHI / w);
for (int L = 0; L < N; L += w) {
ll g = 1;
for (int i = L, lm = L + M, j = lm; i < lm; ++i, ++j) {
ll x = A[i], y = A[j];
A[i] = (x + g * y) % MOD; A[j] = (x - g * y) % MOD;
(g *= Wn) %= MOD;
}
}
M = w;
}
for (int i = 0; i < N; ++i) {
Modint(A[i]);
}
}

void GetInv(const int *const A, int *const B, const int N) {
static int C[maxn];
B[0] = mpow(A[0], DPHI);
for (int w = 2, M = 4; w <= N; M <<= 1) {
memcpy(C, A, w * SzI);
NTT(B, M); NTT(C, M);
for (int i = 0; i < M; ++i) {
B[i] = (B[i] << 1) % MOD - 1ll * C[i] * B[i] % MOD * B[i] % MOD;
Modint(B[i]);
}
NTT(B, M);
std::reverse(B + 1, B + M);
for (int i = 0, iv = mpow(M, DPHI); i < w; ++i) {
B[i] = 1ll * B[i] * iv % MOD;
}
memset(B + w, 0, w * SzI);
w = M;
}
memset(C, 0, (N << 1) * SzI);
}

void GetDer(const int *const A, int *const B, const int N) {
for (int i = 1; i < N; ++i) {
B[i - 1] = 1ll * A[i] * i % MOD;
}
B[N - 1] = 0;
}

void GetInte(const int *const A, int *const B, const int N) {
B[0] = 0;
for (int i = 1; i < N; ++i) {
B[i] = 1ll * A[i - 1] * mpow(i, DPHI) % MOD;
}
}

void GetLn(const int *const A, int *const B, const int N) {
static int C[maxn];
GetDer(A, B, N);
GetInv(A, C, N);
int M = N << 1;
NTT(B, M); NTT(C, M);
for (int i = 0; i < M; ++i) {
C[i] = 1ll * B[i] * C[i] % MOD;
}
NTT(C, M);
std::reverse(C + 1, C + M);
for (int i = 0, iv = mpow(M, DPHI); i < N; ++i) {
C[i] = 1ll * C[i] * iv % MOD;
}
GetInte(C, B, N);
memset(B + N, 0, N * SzI);
memset(C, 0, M * SzI);
}

void GetExp(const int *const A, int *const B, const int N) {
static int C[maxn];
if (N == 1) {
B[0] = 1;
return;
}
GetExp(A, B, N >> 1);
GetLn(B, C, N);
int M = N << 1;
for (int i = 0; i < N; ++i) {
C[i] = -C[i] + A[i];
Modint(C[i]);
}
C[0] += 1;
NTT(C, M); NTT(B, M);
for (int i = 0; i < M; ++i) {
B[i] = 1ll * B[i] * C[i] % MOD;
}
NTT(B, M);
std::reverse(B + 1, B + M);
for (int i = 0, iv = mpow(M, DPHI); i < N; ++i) {
B[i] = 1ll * B[i] * iv % MOD;
}
memset(B + N, 0, N * SzI);
memset(C, 0, M * SzI);
}

void GetPow(const int *const A, const int k, int *const B, const int N) {
static int C[maxn];
GetLn(A, C, N);
for (int i = 0; i < N; ++i) {
C[i] = 1ll * C[i] * k % MOD;
}
GetExp(C, B, N);
memset(C, 0, N * SzI);
}

}
posted @ 2020-01-06 17:31  一扶苏一  阅读(342)  评论(0编辑  收藏  举报