多项式全家桶

多项式全家桶

补一个多项式 ln ⁡ \ln ln
给定一个多项式 A ( x ) A(x) A(x),求一个多项式 F ( x ) F(x) F(x),满足 F ( x ) ≡ ln ⁡ A ( x )        ( m o d      x n ) F(x)\equiv \ln A(x)\;\;\;(mod\;\;x^n) F(x)lnA(x)(modxn)

考虑求导。
F ′ ( x ) = A ′ ( x ) A ( x ) F'(x)=\frac{A'(x)}{A(x)} F(x)=A(x)A(x)

因此只需要先求出 H ( x ) = A ′ ( x ) H(x)=A'(x) H(x)=A(x),再求出 G ( x ) = A − 1 ( x ) G(x)=A^{-1}(x) G(x)=A1(x),则 F = G ∗ H F=G*H F=GH,多项式乘法即能求出 F ′ ( x ) F'(x) F(x)

然后在积分回去即可。

A ( 0 ) = 1 A(0)=1 A(0)=1时, F ( 0 ) = 0 F(0)=0 F(0)=0

其他内容见牛顿迭代法

Code——多项式全家桶(多项式除法 咕咕咕)

#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <cctype>
#include <string>
#include <cstring>
#include <ctime>
#include <cassert>
#include <string.h>
//#include <unordered_set>
//#include <unordered_map>
//#include <bits/stdc++.h>

#define MP(A,B) make_pair(A,B)
#define PB(A) push_back(A)
#define SIZE(A) ((int)A.size())
#define LEN(A) ((int)A.length())
#define FOR(i,a,b) for(int i=(a);i<(b);++i)
#define fi first
#define se second

using namespace std;

template<typename T>inline bool upmin(T &x,T y) { return y<x?x=y,1:0; }
template<typename T>inline bool upmax(T &x,T y) { return x<y?x=y,1:0; }

typedef long long ll;
typedef unsigned long long ull;
typedef long double lod;
typedef pair<int,int> PR;
typedef vector<int> VI;

const lod eps=1e-11;
const lod pi=acos(-1);
const int oo=1<<30;
const ll loo=1ll<<62;
const int mods=998244353;
const int inv2=(mods+1)>>1;
const int G=3;
const int Gi=(mods+1)/G;
const int MAXN=600005;
const int INF=0x3f3f3f3f;//1061109567
/*--------------------------------------------------------------------*/
inline int read()
{
	int f=1,x=0; char c=getchar();
	while (c<'0'||c>'9') { if (c=='-') f=-1; c=getchar(); }
	while (c>='0'&&c<='9') { x=(x<<3)+(x<<1)+(c^48); c=getchar(); }
	return x*f;
}
int a[MAXN],F[MAXN];
namespace Poly
{
	int c[MAXN],f[MAXN],b[MAXN],d[MAXN],rev[MAXN],Limit,l;
	inline int upd(int x,int y){ return x+y>=mods?x+y-mods:x+y; }
	inline int quick_pow(int x,int y)
	{
		int ret=1;
		for (;y;y>>=1)
		{
			if (y&1) ret=1ll*ret*x%mods;
			x=1ll*x*x%mods;
		}
		return ret;
	}
	void Number_Theoretic_Transform(int *A,int n,int opt)
	{
		for (int i=0;i<Limit;i++) if (i<rev[i]) swap(A[i],A[rev[i]]);
		for (int mid=1;mid<Limit;mid<<=1)
		{
			int Wn=quick_pow((opt==1)?G:Gi,(mods-1)/(mid<<1));
			for (int j=0;j<Limit;j+=mid<<1)
				for (int k=j,w=1;k<j+mid;k++,w=1ll*w*Wn%mods)
				{
					int x=A[k],y=1ll*A[k+mid]*w%mods;
					A[k]=upd(x,y),A[k+mid]=upd(x,mods-y);
				}
		}
		
		if (opt==-1)
		{
			int invLimit=quick_pow(Limit,mods-2);
			for (int i=0;i<n;i++) A[i]=1ll*A[i]*invLimit%mods;
			for (int i=n;i<Limit;i++) A[i]=0;
		}
	}
	void getinv(int *F,int *G,int n) //F->invG 
	{
		if (n==1) { F[0]=quick_pow(G[0],mods-2); return; }
		getinv(F,G,(n+1)>>1);
		Limit=1,l=0;
		while (Limit<(n<<1)) Limit<<=1,l++;
		for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));
		for (int i=0;i<n;i++) c[i]=G[i];
		for (int i=n;i<Limit;i++) c[i]=0;
		Number_Theoretic_Transform(c,n,1);
		Number_Theoretic_Transform(F,n,1);
		for (int i=0;i<Limit;i++) F[i]=1ll*upd(2,mods-1ll*F[i]*c[i]%mods)*F[i]%mods;
		Number_Theoretic_Transform(F,n,-1);
		for (int i=n;i<Limit;i++) F[i]=0;
	}
	void getln(int *a,int n) //a->ln a (a[0]=1)
	{
		memset(f,0,sizeof f);
		getinv(f,a,n);
		for (int i=0;i<n;i++) a[i]=1ll*a[i+1]*(i+1)%mods;
		
		Limit=1,l=0;
		while (Limit<(n<<1)) Limit<<=1,l++;
		for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));
		Number_Theoretic_Transform(a,n,1);
		Number_Theoretic_Transform(f,n,1);
		for (int i=0;i<Limit;i++) f[i]=1ll*f[i]*a[i]%mods;
		Number_Theoretic_Transform(f,n,-1);
		
		for (int i=1;i<n;i++) a[i]=1ll*f[i-1]*quick_pow(i,mods-2)%mods; 
		for (int i=n;i<=Limit;i++) a[i]=0; 
		a[0]=0;
	}
	void getexp(int *F,int *a,int n) //F->exp a (a[0]=0)
	{
		memset(b,0,sizeof b);
		if (n==1) { F[0]=1; return; }
		getexp(F,a,(n+1)>>1);
		for (int i=0;i<n;i++) b[i]=F[i];
		getln(b,n);
		
		Limit=1,l=0;
		while (Limit<(n<<1)) Limit<<=1,l++;
		for (int i=0;i<Limit;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(l-1));
		
		for (int i=0;i<n;i++) b[i]=upd(a[i],mods-b[i]);
		for (int i=n;i<Limit;i++) b[i]=F[i]=0;
		b[0]++;
		
		Number_Theoretic_Transform(F,n,1);
		Number_Theoretic_Transform(b,n,1);
		for (int i=0;i<Limit;i++) F[i]=1ll*F[i]*b[i]%mods;
		Number_Theoretic_Transform(F,n,-1);
		for (int i=n;i<Limit;i++) F[i]=0; 
	}
	void getsqrt(int *F,int *a,int n,int k) //F->sqrt^k a(a[0]=1)
	{
		memset(d,0,sizeof d);
		for (int i=0;i<n;i++) d[i]=a[i];
		getln(d,n);
		int invk=quick_pow(k,mods-2);
		for (int i=0;i<n;i++) d[i]=1ll*d[i]*invk%mods;
		getexp(F,d,n);
	} 	
	void getpow(int *F,int *a,int n,int k) //F->pow^k a(a[0]=1)
	{
		memset(d,0,sizeof d);
		for (int i=0;i<n;i++) d[i]=a[i];
		getln(d,n);
		for (int i=0;i<n;i++) d[i]=1ll*d[i]*k%mods;
		getexp(F,d,n);
	}
}

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