模板—FFT

卷积:$C[i]=sum limits_{j=0}^{i}A[j]*B[i-j]$可以画图理解一下其实就是交叉相乘的和。

卷积可以看作两个多项式乘积的形式，只不过求出的结果的项数不同。

FFT讲解

复数讲解

#include<iostream>
#include<cstring>
#include<complex>
#include<cstdio>
#define cp complex<double>
using namespace std;
const double pi=3.14159265358979;

void FFT(cp *a,int n,int inv)
{    
    if(n==1)return;
    int mid=n/2;static cp b[1000100];
    for(int i=0;i<=mid-1;i++)b[i]=a[i*2],b[i+mid]=a[i*2+1];
    for(int i=0;i<=n-1;i++)a[i]=b[i];
    FFT(a,mid,inv);FFT(a+mid,mid,inv);
    for(int i=0;i<=mid-1;i++)
    {
        cp x(cos(2*pi*i/n),inv*sin(2*pi*i/n));
        b[i]=a[i]+x*a[i+mid],b[i+mid]=a[i]-x*a[i+mid];
    }
    for(int i=0;i<=n-1;i++)a[i]=b[i];
}
int n,m;
cp a[1000010],b[1000010];int c[1000010];
signed main()
{
//    freopen("1.in","r",stdin);
//    freopen("out.out","w",stdout);

    cin>>n>>m;double tem;
    for(int i=0;i<=n;i++)scanf("%lf",&tem),a[i]=cp(tem,0);
    for(int i=0;i<=m;i++)scanf("%lf",&tem),b[i]=cp(tem,0);
    int len=n+m+1,now=1;
    for(;;now*=2)if(now>=len){len=now;break;}
    FFT(a,len,1);FFT(b,len,1);
    for(int i=0;i<len;i++)a[i]*=b[i];
    FFT(a,len,-1);
    for(int i=0;i<=n+m;i++)cout<<(int)(a[i].real()/len+0.5)<<" ";
}

#include<iostream>
#include<cstring>
#include<complex>
#include<cstdio>
#define cp complex<double>
using namespace std;
const double pi=3.14159265358979;

int rev[1000000];
void FFT(cp *a,int n,int inv)
{
    int bit=0;while((1<<bit)<n)bit++;
    for(int i=0;i<n;i++)rev[i]=(rev[i>>1]>>1)|((i&1)<<(bit-1));
    for(int i=0;i<n;i++)if(i<rev[i])swap(a[i],a[rev[i]]);
    for(int mid=1;mid<n;mid*=2)
    {    
        cp temp(cos(pi/mid),inv*sin(pi/mid));
        for(int i=0;i<n;i+=mid*2)
        {
            cp ome(1,0);
            for(int j=0;j<mid;j++,ome*=temp)
            {
                cp x=a[i+j],y=ome*a[i+j+mid];
                a[i+j]=x+y,a[i+j+mid]=x-y;
            }
        }
    }
}
int n,m;
cp a[1000010],b[1000010];int c[1000010];
signed main()
{
//    freopen("1.in","r",stdin);
//    freopen("out.out","w",stdout);

    cin>>n>>m;double tem;
    for(int i=0;i<=n;i++)scanf("%lf",&tem),a[i]=cp(tem,0);
    for(int i=0;i<=m;i++)scanf("%lf",&tem),b[i]=cp(tem,0);
    int len=n+m+1,now=1;
    for(;;now*=2)if(now>=len){len=now;break;}
    FFT(a,len,1);FFT(b,len,1);
    for(int i=0;i<len;i++)a[i]*=b[i];
    FFT(a,len,-1);
    for(int i=0;i<=n+m;i++)cout<<(int)(a[i].real()/len+0.5)<<" ";
}