Polynomial Notes
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\(\text{Inv}\)
\[\because FG\equiv1(\bmod\ x^n)
\]
\[\therefore FG\equiv1(\bmod\ x^{\frac{n}{2}})
\]
令
\[FG_0\equiv1(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore F(G-G_0)\equiv0(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore G-G_0\equiv 0(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore (G-G_0)^2\equiv 0(\bmod \ x^n)
\]
\[\therefore G^2-2GG_0+G_0^2\equiv 0(\bmod\ x^n)
\]
\[\therefore F(G^2-2GG_0+G_0^2)\equiv 0(\bmod\ x^n)
\]
\[\therefore G-2G_0+FG_0^2\equiv0(\bmod\ x^n)
\]
\[\therefore G\equiv2G_0-FG_0^2(\bmod\ x^n)
\]
\(T(n) = T(\frac{n}{2})+O(n\log n)=O(n\log n)\)
\(\text{Sqrt}\)
\[\because F\equiv G^2(\bmod\ x^n)
\]
\[\therefore F\equiv G^2(\bmod\ x^{\frac{n}{2}})
\]
令
\[F\equiv G_0^2(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore F(G^2-G_0^2)\equiv0(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore G^2-G_0^2\equiv0(\bmod\ x^{\frac{n}{2}})
\]
\[\therefore (G^2-G_0^2)^2\equiv0(\bmod\ x^n)
\]
\[\therefore G^4-2G^2G_0^2+G_0^4\equiv0(\bmod\ x^n)
\]
\[\therefore G^4+2G^2G_0^2+G_0^4\equiv 4G^2G_0^2(\bmod\ x^n)
\]
\[\therefore (G^2+G_0^2)^2\equiv 4G^2G_0^2(\bmod\ x^n)
\]
\[\therefore G^2+G_0^2\equiv 2GG_0(\bmod\ x^n)
\]
\[\therefore F+G_0^2\equiv 2GG_0(\bmod\ x^n)
\]
\[\therefore G\equiv\dfrac{F+G_0^2}{2G_0}(\bmod x^n)
\]
求逆即可。
\(T(n)=T(\frac{n}{2})+O(n\log n)=O(n\log n)\)
\(\ln\)
\[G\equiv \ln(F)(\bmod\ x^n)
\]
令
\[f(x)=\text{ln}(x)
\]
\[G\equiv f(F)(\bmod\ x^n)
\]
\[\therefore G'\equiv f'(F)F'(\bmod\ x^n)
\]
由 \(\text{ln}(x)=\frac{1}{x}\) 得
\[G'\equiv \dfrac{F'}{F}(\bmod\ x^n)
\]
\[\therefore \int G'\equiv\int\dfrac{F'}{F}(\bmod\ x^n)
\]
\[\therefore G\equiv\int\dfrac{F'}{F}(\bmod\ x^n)
\]
求逆即可。
//n 表示项数
const int N = 2.7e5 + 10;
namespace Polynomial {
const int mod = 998244353, G = 3, Gi = 332748118, inv2 = 499122177, img = 86583718;
int lim, rev[N], a[N], b[N], c[N];
int qpow(int a, int k) {
int res = 1;
for(; k; a = 1ll * a * a % mod, k >>= 1)
if(k & 1) res = 1ll * res * a % mod;
return res;
}
void NTT(int *f, int T) {
for(int i = 0; i < lim; i++)
if(i < rev[i])
swap(f[i], f[rev[i]]);
for(int mid = 1; mid < lim; mid <<= 1) {
int wn = qpow(T == 1 ? G : Gi, (mod - 1) / (mid << 1));
int len = mid << 1;
for(int i = 0; i < lim; i += mid << 1) {
int w = 1;
for(int j = 0; j < mid; j++, w = 1ll * w * wn % mod) {
int x = f[i + j], y = 1ll * w * f[i + j + mid] % mod;
f[i + j] = (x + y) % mod;
f[i + j + mid] = (x - y + mod) % mod;
}
}
}
if(T == -1) {
int inv = qpow(lim, mod - 2);
for(int i = 0; i < lim; i++)
f[i] = 1ll * f[i] * inv % mod;
}
}
void init(int n) {
for(lim = 1; lim < n; lim <<= 1);
for(int i = 0; i < lim; i++)
rev[i] = (rev[i >> 1] >> 1) | ((i & 1) * (lim >> 1));
}
void mul(int *f, int *g, int *h, int n, int m) {
static int a[N], b[N];
init(n + m - 1);
memset(a, 0, lim << 2);
memcpy(a, f, n << 2);
memset(b, 0, lim << 2);
memcpy(b, g, m << 2);
NTT(a, 1), NTT(b, 1);
for(int i = 0; i < lim; i++)
h[i] = 1ll * a[i] * b[i] % mod;
NTT(h, -1);
}
void inv(int *f, int *g, int n) {
if(n == 1) { g[0] = qpow(f[0], mod - 2); return; }
inv(f, g, n + 1 >> 1);
init(n << 1);
copy(f, f + n, a);
fill(a + n, a + lim, 0);
NTT(a, 1), NTT(g, 1);
for(int i = 0; i < lim; i++)
g[i] = (2 - 1ll * a[i] * g[i] % mod + mod) % mod * g[i] % mod;
NTT(g, -1);
fill(g + n, g + lim, 0);
}
void div(int *f, int *g, int *q, int *r, int n, int m) {
static int a[N], b[N], c[N];
reverse_copy(f, f + n, a);
reverse_copy(g, g + m, b);
int len = n - m + 1;
inv(b, c, len);
memset(b, 0, m << 2);
mul(a, c, a, len, len);
reverse_copy(a, a + len, q);
fill(q + len, q + n, 0);
copy(g, g + m, b);
mul(b, q, b, n, n);
for(int i = 0; i < m - 1; i++)
r[i] = (f[i] - b[i] + mod) % mod;
memset(a, 0, lim << 2);
memset(b, 0, lim << 2);
memset(c, 0, lim << 2);
}
void sqrt(int *f, int *g, int n) {
if(n == 1) { g[0] = 1; return; }
sqrt(f, g, n + 1 >> 1);
memset(b, 0, n << 2);
inv(g, b, n);
mul(f, b, b, n, n);
for(int i = 0; i < n; i++)
g[i] = 1ll * (g[i] + b[i]) * inv2 % mod;
}
void dev(int *f, int *g, int n) {
for(int i = 1; i < n; i++)
g[i - 1] = 1ll * i * f[i] % mod;
g[n - 1] = 0;
}
void idev(int *f, int *g, int n) {
for(int i = n - 1; i; i--)
g[i] = 1ll * f[i - 1] * qpow(i, mod - 2) % mod;
g[0] = 0;
}
void ln(int *f, int *g, int n) {
static int a[N];
init(n << 1);
memset(a, 0, lim << 2);
inv(f, a, n);
dev(f, g, n);
mul(g, a, g, n, n);
idev(g, g, n);
}
void exp(int *f, int *g, int n) {
if(n == 1) { g[0] = 1; return; }
exp(f, g, n + 1 >> 1);
ln(g, c, n);
for(int i = 0; i < n; i++)
c[i] = (!i + f[i] - c[i] + mod) % mod;
fill(c + n, c + lim, 0);
mul(g, c, g, n, n);
fill(g + n, g + lim, 0);
}
void pow(int *f, int *g, int n, int k) {
static int a[N];
memset(a, 0, n << 2);
ln(f, a, n);
for(int i = 0; i < n; i++)
a[i] = 1ll * a[i] * k % mod;
exp(a, g, n);
}
void sin(int *f, int *g, int n) {
static int a[N], b[N], c[N];
for(int i = 0; i < n; i++)
a[i] = 1ll * f[i] * img % mod;
exp(a, b, n);
inv(b, c, n);
for(int i = 0; i < n; i++)
a[i] = (b[i] - c[i] + mod) % mod;
int v = qpow(img * 2, mod - 2);
for(int i = 0; i < n; i++)
g[i] = 1ll * a[i] * v % mod;
}
void cos(int *f, int *g, int n) {
static int a[N], b[N], c[N];
for(int i = 0; i < n; i++)
a[i] = 1ll * f[i] * img % mod;
exp(a, b, n);
inv(b, c, n);
for(int i = 0; i < n; i++)
a[i] = (b[i] + c[i]) % mod;
int v = qpow(2, mod - 2);
for(int i = 0; i < n; i++)
g[i] = 1ll * a[i] * v % mod;
}
void arcsin(int *f, int *g, int n) {
static int a[N], b[N], c[N];
dev(f, a, n);
mul(f, f, b, n, n);
for(int i = 0; i < n; i++)
b[i] = (mod - b[i]) % mod;
b[0] = (b[0] + 1) % mod;
sqrt(b, c, n);
memset(b, 0, lim << 2);
inv(c, b, n);
memset(c, 0, lim << 2);
mul(a, b, c, n, n);
idev(c, g, n);
}
void arctan(int *f, int *g, int n) {
static int a[N], b[N], c[N];
dev(f, a, n);
mul(f, f, b, n, n);
b[0] = (b[0] + 1) % mod;
inv(b, c, n);
memset(b, 0, lim << 2);
mul(a, c, b, n, n);
idev(b, g, n);
}
} using namespace Polynomial;
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