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

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

FFT递归版

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

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

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

FFT迭代版