多项式板子
FFT
给定一个\(n\)次多项式\(F(x)\)，和一个\(m\)次多项式\(G(x)\)，请求出\(F(x)\)和\(G(x)\)的卷积。
#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cmath>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=3e6+1000;
const double pi=acos(-1);
int n,m,r[maxn],len;
struct complex{double x,y;}a[maxn],b[maxn];
complex operator-(complex a,complex b){return (complex){a.x-b.x,a.y-b.y};}
complex operator+(complex a,complex b){return (complex){a.x+b.x,a.y+b.y};}
complex operator*(complex a,complex b){return (complex){a.x*b.x-a.y*b.y,a.x*b.y+b.x*a.y};}
void fft(complex *a,int f)
{
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[r[i]],a[i]);
for(rg int i=1;i<n;i<<=1)
{
complex wn=(complex){cos(pi/i),f*sin(pi/i)};
for(rg int j=0;j<n;j+=(i<<1))
{
complex w=(complex){1,0};
for(rg int k=0;k<i;k++)
{
complex x=a[j+k],y=w*a[j+k+i];
a[j+k]=x+y,a[j+k+i]=x-y,w=w*wn;
}
}
}
if(f==-1)for(rg int i=0;i<=m;i++)a[i].x=a[i].x/n+0.1;
}
int main()
{
read(n),read(m);
for(rg int i=0,x;i<=n;i++)read(x),a[i].x=x;
for(rg int i=0,x;i<=m;i++)read(x),b[i].x=x;
m+=n;for(n=1;n<=m;n<<=1)len++;
for(rg int i=0;i<n;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
fft(a,1),fft(b,1);
for(rg int i=0;i<=n;i++)a[i]=a[i]*b[i];
fft(a,-1);
for(rg int i=0;i<=m;i++)printf("%d ",(int)a[i].x);
}
NTT
给定一个\(n\)次多项式\(F(x)\)，和一个\(m\)次多项式\(G(x)\)，请求出\(F(x)\)和\(G(x)\)的卷积，并对\(998244353\)取模。
#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cmath>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=3e6+10,mod=998244353,g=3,ginv=332748118;
int n,m,a[maxn],b[maxn],len,r[maxn];
int mi(int a,int b)
{
int ans=1;
while(b)
{
if(b&1)ans=1ll*ans*a%mod;
b>>=1,a=1ll*a*a%mod;
}
return ans;
}
void ntt(int *a,int f)
{
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[r[i]],a[i]);
for(rg int i=1;i<n;i<<=1)
{
int wn=mi(f?g:ginv,(mod-1)/(i<<1));
for(rg int j=0;j<n;j+=(i<<1))
{
int w=1;
for(rg int k=0;k<i;k++)
{
int x=a[j+k],y=1ll*a[j+k+i]*w%mod;
a[j+k]=(x+y)%mod,a[j+k+i]=(x-y+mod)%mod,w=1ll*w*wn%mod;
}
}
}
}
int main()
{
read(n),read(m);
for(rg int i=0;i<=n;i++)read(a[i]);
for(rg int i=0;i<=m;i++)read(b[i]);
m+=n;for(n=1;n<=m;n<<=1)len++;
for(rg int i=0;i<n;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(a,1),ntt(b,1);
for(rg int i=0;i<n;i++)a[i]=1ll*a[i]*b[i]%mod;
ntt(a,0);int inv=mi(n,mod-2);
for(rg int i=0;i<=m;i++)printf("%d ",1ll*inv*a[i]%mod);
}
多项式求逆
给定一个多项式\(F(x)\) ，请求出一个多项式\(G(x)\)， 满足\(F(x)∗G(x)≡1(mod\ x^n)\)。系数对 \(998244353\)取模。
#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cmath>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=3e6+10,mod=998244353,g=3,ginv=332748118;
int n,m,a[maxn],b[maxn],c[maxn],r[maxn];
int mi(int a,int b)
{
int ans=1;
while(b)
{
if(b&1)ans=1ll*ans*a%mod;
b>>=1,a=1ll*a*a%mod;
}
return ans;
}
void ntt(int *a,int n,int f)
{
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[r[i]],a[i]);
for(rg int i=1;i<n;i<<=1)
{
int wn=mi(f?g:ginv,(mod-1)/(i<<1));
for(rg int j=0;j<n;j+=(i<<1))
{
int w=1;
for(rg int k=0;k<i;k++)
{
int x=a[j+k],y=1ll*a[j+k+i]*w%mod;
a[j+k]=(x+y)%mod,a[j+k+i]=(x-y+mod)%mod,w=1ll*w*wn%mod;
}
}
}
if(f)return ;
int inv=mi(n,mod-2);
for(rg int i=0;i<n;i++)a[i]=1ll*a[i]*inv%mod;
}
void solve(int n,int *a,int *b)
{
if(n==1){b[0]=mi(a[0],mod-2);return ;}
solve((n+1)>>1,a,b);int m,len=0;
for(m=1;m<=n*2;m<<=1)len++;
for(rg int i=0;i<n;i++)c[i]=a[i];
for(rg int i=n;i<m;i++)c[i]=0;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(b,m,1),ntt(c,m,1);
for(rg int i=0;i<m;i++)b[i]=((2ll*b[i]-1ll*c[i]*b[i]%mod*b[i]%mod)+mod)%mod;
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
int main()
{
read(n);
for(rg int i=0;i<n;i++)read(a[i]);
solve(n,a,b);
for(rg int i=0;i<n;i++)printf("%d ",b[i]);
}
多项式求导
其实就是幂函数的求导
\[f(x)=\sum_{i=0}a_ix_i\\
f'(x)=\sum_{i=1}a_{i}x^{i-1}
\]
多项式求ln
给出\(n-1\)次多项式\(A(x)\)，求一个 \(\bmod{\:x^n}\)下的多项式\(B(x)\)，满足\(B(x)\equiv \ln A(x)\).
在\(\text{mod } 998244353\)下进行，且\(a_i\in [0, 998244353] \cap \mathbb{Z}\)
代码：
#include<cstdio>
#include<iostream>
#include<algorithm>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=8e5+10,mod=998244353,g=3,gi=332748118;
int n,m,a[maxn],b[maxn],c[maxn],len,r[maxn],d[maxn],e[maxn];
int add(int x,int y){return x+y>=mod?x+y-mod:x+y;}
int del(int x,int y){return x-y<0?x-y+mod:x-y;}
int mul(int x,int y){return 1ll*x*y-1ll*x*y/mod*mod;}
int mi(int a,int b)
{
int ans=1;
while(b)
{
if(b&1)ans=mul(a,ans);
b>>=1,a=mul(a,a);
}
return ans;
}
void ntt(int *a,int n,int f)
{
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[i],a[r[i]]);
for(rg int i=1;i<n;i<<=1)
{
int wn=mi((f?g:gi),(mod-1)/(i<<1));
for(rg int j=0;j<n;j+=(i<<1))
{
int w=1;
for(rg int k=0;k<i;k++)
{
int x=a[j+k],y=mul(a[j+k+i],w);
a[j+k]=add(x,y),a[j+k+i]=del(x,y),w=mul(w,wn);
}
}
}
if(f)return ;int inv=mi(n,mod-2);
for(rg int i=0;i<n;i++)a[i]=mul(a[i],inv);
}
void solve(int n,int *a,int *b)
{
if(n==1){b[0]=mi(a[0],mod-2);return ;}
solve((n+1)>>1,a,b);len=0;
for(m=1;m<=n*2;m<<=1)len++;
for(rg int i=0;i<n;i++)c[i]=a[i];
for(rg int i=n;i<m;i++)c[i]=0;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(c,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)b[i]=del(mul(2,b[i]),mul(mul(b[i],b[i]),c[i]));
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
void getln(int *a,int n)
{
solve(n,a,b);len=0;
for(rg int i=1;i<n;i++)d[i-1]=mul(a[i],i);
for(m=1;m<=n;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(d,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)d[i]=mul(d[i],b[i]);
ntt(d,m,0);e[0]=0;
for(rg int i=1;i<m;i++)e[i]=mul(d[i-1],mi(i,mod-2));
}
int main()
{
read(n);
for(rg int i=0;i<n;i++)read(a[i]);
for(m=1;m<=n;m<<=1);getln(a,m);
for(rg int i=0;i<n;i++)printf("%d ",e[i]);
}
多项式求exp
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\(B(x)≡e^A(x)\).
代码：
#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cstring>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=8e5+10,mod=998244353,g=3,gi=332748118;
int n,a[maxn],r[maxn],c[maxn],f[maxn],b[maxn],s[maxn],w[maxn],h[maxn];
int mul(int x,int y){return 1ll*x*y-1ll*x*y/mod*mod;}
int add(int x,int y){return x+y>=mod?x+y-mod:x+y;}
int del(int x,int y){return x-y<0?x-y+mod:x-y;}
int mi(int a,int b){
int ans=1;
while(b){
if(b&1)ans=mul(ans,a);
b>>=1,a=mul(a,a);
}
return ans;
}
void ntt(int *a,int n,int f){
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[i],a[r[i]]);
for(rg int i=1;i<n;i<<=1){
int wn=mi(f?g:gi,(mod-1)/(i<<1));
for(rg int j=0;j<n;j+=(i<<1)){
int w=1;
for(rg int k=0;k<i;k++){
int x=a[j+k],y=mul(w,a[j+k+i]);
a[j+k]=add(x,y),a[j+k+i]=del(x,y),w=mul(w,wn);
}
}
}
if(f)return ;int inv=mi(n,mod-2);
for(rg int i=0;i<n;i++)a[i]=mul(a[i],inv);
}
void get_inv(int *a,int *b,int n){
if(n==1)return b[0]=mi(a[0],mod-2),void();
get_inv(a,b,(n+1)>>1);
int m,len=0;
for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
for(rg int i=0;i<n;i++)c[i]=a[i];
for(rg int i=n;i<m;i++)c[i]=0;
ntt(c,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)
b[i]=del(mul(2,b[i]),mul(c[i],mul(b[i],b[i])));
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
void get_ln(int *a,int *b,int n){
for(rg int i=0;i<n;i++)w[i]=mul(a[i+1],i+1);
get_inv(a,h,n);int m,len=0;
for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(w,m,1),ntt(h,m,1);
for(rg int i=0;i<m;i++)w[i]=mul(w[i],h[i]);
ntt(w,m,0);b[0]=0;
for(rg int i=0;i<m;i++)b[i+1]=mul(w[i],mi(i+1,mod-2)),h[i]=0;
for(rg int i=n;i<m;i++)b[i]=0;
}
void get_exp(int *a,int *b,int n){
if(n==1)return b[0]=1,void();
get_exp(a,b,(n+1)>>1);
get_ln(b,s,n);
int m,len=0;
for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
for(rg int i=n;i<m;i++)s[i]=0;
for(rg int i=0;i<n;i++)s[i]=del(a[i],s[i]);s[0]=add(s[0],1);
ntt(s,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)b[i]=mul(b[i],s[i]);
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
int main()
{
read(n);
for(rg int i=0;i<n;i++)read(a[i]);
get_exp(a,f,n);
for(rg int i=0;i<n;i++)printf("%d ",f[i]);
}
多项式快速幂
给定一个\(n-1\)次多项式\(A(x)\)，求一个在\(mod\ x^n\)意义下的多项式\(B(x)\),使\(B(x)≡A^k(x) (mod \ x^n)\)
代码：
#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cstring>
using namespace std;
void read(int &x) {
char ch; bool ok;
for(ok=0,ch=getchar(); !isdigit(ch); ch=getchar()) if(ch=='-') ok=1;
for(x=0; isdigit(ch); x=x*10+ch-'0',ch=getchar()); if(ok) x=-x;
}
#define rg register
const int maxn=8e5+10,mod=998244353,g=3,gi=332748118;
char ch[maxn];
int n,m,tot,k,now=1,r[maxn],a[maxn],b[maxn],w[maxn],h[maxn],d[maxn],c[maxn],f[maxn],s[maxn];
int mul(int x,int y){return 1ll*x*y-1ll*x*y/mod*mod;}
int add(int x,int y){return x+y>=mod?x+y-mod:x+y;}
int del(int x,int y){return x-y<0?x-y+mod:x-y;}
int mi(int a,int b){
int ans=1;
while(b){
if(b&1)ans=mul(ans,a);
b>>=1,a=mul(a,a);
}
return ans;
}
void ntt(int *a,int n,int f){
for(rg int i=0;i<n;i++)if(r[i]>i)swap(a[i],a[r[i]]);
for(rg int i=1;i<n;i<<=1){
int wn=mi(f?g:gi,(mod-1)/(i<<1));
for(rg int j=0;j<n;j+=i<<1){
int w=1;
for(rg int k=0;k<i;k++){
int x=a[j+k],y=mul(w,a[j+k+i]);
a[j+k]=add(x,y),a[j+k+i]=del(x,y),w=mul(w,wn);
}
}
}
if(f)return ;int inv=mi(n,mod-2);
for(rg int i=0;i<n;i++)a[i]=mul(a[i],inv);
}
void get_inv(int *a,int *b,int n){
if(n==1)return b[0]=mi(a[0],mod-2),void();
get_inv(a,b,(n+1)>>1);int m,len=0;for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
for(rg int i=0;i<n;i++)c[i]=a[i];
for(rg int i=n;i<m;i++)c[i]=0;
ntt(c,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)b[i]=del(mul(2,b[i]),mul(c[i],mul(b[i],b[i])));
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
void get_ln(int *a,int *b,int n){
for(rg int i=0;i<n;i++)w[i]=mul(a[i+1],i+1);
get_inv(a,h,n);int m,len=0;for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
ntt(h,m,1),ntt(w,m,1);
for(rg int i=0;i<m;i++)w[i]=mul(w[i],h[i]);
ntt(w,m,0);b[0]=0;
for(rg int i=0;i<m;i++)b[i+1]=mul(w[i],mi(i+1,mod-2)),h[i]=w[i]=0;
for(rg int i=n;i<m;i++)b[i]=0;
}
void get_exp(int *a,int *b,int n){
if(n==1)return b[0]=1,void();
get_exp(a,b,(n+1)>>1),get_ln(b,s,n);
int m,len=0;for(m=1;m<=n<<1;m<<=1)len++;
for(rg int i=0;i<m;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(len-1));
for(rg int i=0;i<n;i++)s[i]=del(a[i],s[i]);s[0]=add(s[0],1);
ntt(s,m,1),ntt(b,m,1);
for(rg int i=0;i<m;i++)b[i]=mul(b[i],s[i]);
ntt(b,m,0);
for(rg int i=n;i<m;i++)b[i]=0;
}
int main()
{
read(n),read(k);
for(rg int i=0;i<n;i++)read(a[i]);
get_ln(a,d,n);for(rg int i=0;i<n;i++)d[i]=mul(k,d[i]);
get_exp(d,f,n);for(rg int i=0;i<n;i++)printf("%d ",f[i]);
}
