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。
1 #include<iostream> 2 #include<cstring> 3 #include<complex> 4 #include<cstdio> 5 #define cp complex<double> 6 using namespace std; 7 const double pi=3.14159265358979; 8 9 void FFT(cp *a,int n,int inv) 10 { 11 if(n==1)return; 12 int mid=n/2;static cp b[1000100]; 13 for(int i=0;i<=mid-1;i++)b[i]=a[i*2],b[i+mid]=a[i*2+1]; 14 for(int i=0;i<=n-1;i++)a[i]=b[i]; 15 FFT(a,mid,inv);FFT(a+mid,mid,inv); 16 for(int i=0;i<=mid-1;i++) 17 { 18 cp x(cos(2*pi*i/n),inv*sin(2*pi*i/n)); 19 b[i]=a[i]+x*a[i+mid],b[i+mid]=a[i]-x*a[i+mid]; 20 } 21 for(int i=0;i<=n-1;i++)a[i]=b[i]; 22 } 23 int n,m; 24 cp a[1000010],b[1000010];int c[1000010]; 25 signed main() 26 { 27 // freopen("1.in","r",stdin); 28 // freopen("out.out","w",stdout); 29 30 cin>>n>>m;double tem; 31 for(int i=0;i<=n;i++)scanf("%lf",&tem),a[i]=cp(tem,0); 32 for(int i=0;i<=m;i++)scanf("%lf",&tem),b[i]=cp(tem,0); 33 int len=n+m+1,now=1; 34 for(;;now*=2)if(now>=len){len=now;break;} 35 FFT(a,len,1);FFT(b,len,1); 36 for(int i=0;i<len;i++)a[i]*=b[i]; 37 FFT(a,len,-1); 38 for(int i=0;i<=n+m;i++)cout<<(int)(a[i].real()/len+0.5)<<" "; 39 }
1 #include<iostream> 2 #include<cstring> 3 #include<complex> 4 #include<cstdio> 5 #define cp complex<double> 6 using namespace std; 7 const double pi=3.14159265358979; 8 9 int rev[1000000]; 10 void FFT(cp *a,int n,int inv) 11 { 12 int bit=0;while((1<<bit)<n)bit++; 13 for(int i=0;i<n;i++)rev[i]=(rev[i>>1]>>1)|((i&1)<<(bit-1)); 14 for(int i=0;i<n;i++)if(i<rev[i])swap(a[i],a[rev[i]]); 15 for(int mid=1;mid<n;mid*=2) 16 { 17 cp temp(cos(pi/mid),inv*sin(pi/mid)); 18 for(int i=0;i<n;i+=mid*2) 19 { 20 cp ome(1,0); 21 for(int j=0;j<mid;j++,ome*=temp) 22 { 23 cp x=a[i+j],y=ome*a[i+j+mid]; 24 a[i+j]=x+y,a[i+j+mid]=x-y; 25 } 26 } 27 } 28 } 29 int n,m; 30 cp a[1000010],b[1000010];int c[1000010]; 31 signed main() 32 { 33 // freopen("1.in","r",stdin); 34 // freopen("out.out","w",stdout); 35 36 cin>>n>>m;double tem; 37 for(int i=0;i<=n;i++)scanf("%lf",&tem),a[i]=cp(tem,0); 38 for(int i=0;i<=m;i++)scanf("%lf",&tem),b[i]=cp(tem,0); 39 int len=n+m+1,now=1; 40 for(;;now*=2)if(now>=len){len=now;break;} 41 FFT(a,len,1);FFT(b,len,1); 42 for(int i=0;i<len;i++)a[i]*=b[i]; 43 FFT(a,len,-1); 44 for(int i=0;i<=n+m;i++)cout<<(int)(a[i].real()/len+0.5)<<" "; 45 }