題意：多項式除法,A(x)=C(x)*B(x)+D(x),給定A(x),B(x)，求C(x),D(x)
題解：A(x)的度是n，B(x)的度是m
定義\(A'(x)=x^n*A(\frac{1}{x})\),可以發現\(A'(x)=A(n-x)\)
\(A(\frac{1}{x})=C(\frac{1}{x})*B(\frac{1}{x})+D(\frac{1}{x})\)
\(x^nA(\frac{1}{x})=x^{n-m}*C(\frac{1}{x})*x^{m}*B(\frac{1}{x})+x^{n-m+1}*x^{m}D(\frac{1}{x})\)
\(A'(x)=C'(x)*B'(x)+x^{n-m+1}*D'(x)\)
那么我們考慮在\(mod(x^{n-m+1})\)的等式，即可消去D'(x)
這樣我們便可求出\(C'(x)=\frac{A'(x)}{B'(x)}(mod(x^{n-m+1}))\)，用多項式求逆，再回退C，帶入可求出D

//#pragma GCC optimize(2) //#pragma GCC optimize(3) //#pragma GCC optimize(4) //#pragma GCC optimize("unroll-loops") //#pragma comment(linker, "/stack:200000000") //#pragma GCC optimize("Ofast,no-stack-protector") //#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") #include<bits/stdc++.h> #define fi first #define se second #define db double #define mp make_pair #define pb push_back #define pi acos(-1.0) #define ll long long #define vi vector<int> #define mod 998244353 #define ld long double #define C 0.5772156649 #define ls l,m,rt<<1 #define rs m+1,r,rt<<1|1 #define pll pair<ll,ll> #define pil pair<int,ll> #define pli pair<ll,int> #define pii pair<int,int> //#define cd complex<double> #define ull unsigned long long #define base 1000000000000000000 #define Max(a,b) ((a)>(b)?(a):(b)) #define Min(a,b) ((a)<(b)?(a):(b)) #define fin freopen("a.txt","r",stdin) #define fout freopen("a.txt","w",stdout) #define fio ios::sync_with_stdio(false);cin.tie(0) template<typename T> inline T const& MAX(T const &a,T const &b){return a>b?a:b;} template<typename T> inline T const& MIN(T const &a,T const &b){return a<b?a:b;} inline void add(ll &a,ll b){a+=b;if(a>=mod)a-=mod;} inline void sub(ll &a,ll b){a-=b;if(a<0)a+=mod;} inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;} inline ll qp(ll a,ll b){ll ans=1;while(b){if(b&1)ans=ans*a%mod;a=a*a%mod,b>>=1;}return ans;} inline ll qp(ll a,ll b,ll c){ll ans=1;while(b){if(b&1)ans=ans*a%c;a=a*a%c,b>>=1;}return ans;}using namespace std;const double eps=1e-8; const ll INF=0x3f3f3f3f3f3f3f3f; const int N=300000+10,maxn=400000+10,inf=0x3f3f3f3f;ll a[N<<3],b[N<<3],c[N<<3],d[N<<3],e[N<<3],tmp[N<<3]; int rev[N<<3]; void getrev(int bit) {for(int i=0;i<(1<<bit);i++)rev[i]=(rev[i>>1]>>1) | ((i&1)<<(bit-1)); } void ntt(ll *a,int n,int dft) {for(int i=0;i<n;i++)if(i<rev[i])swap(a[i],a[rev[i]]);for(int step=1;step<n;step<<=1){ll wn=qp(3,(mod-1)/(step*2));if(dft==-1)wn=qp(wn,mod-2);for(int j=0;j<n;j+=step<<1){ll wnk=1;for(int k=j;k<j+step;k++){ll x=a[k];ll y=wnk*a[k+step]%mod;a[k]=(x+y)%mod;a[k+step]=(x-y+mod)%mod;wnk=wnk*wn%mod;}}}if(dft==-1){ll inv=qp(n,mod-2);for(int i=0;i<n;i++)a[i]=a[i]*inv%mod;} } void pol_inv(int deg,ll *a,ll *b) {if(deg==1){b[0]=qp(a[0],mod-2);return ;}pol_inv((deg+1)>>1,a,b);int sz=0;while((1<<sz)<=deg)sz++;sz++;getrev(sz);int len=1<<sz;for(int i=0;i<deg;i++)tmp[i]=a[i];for(int i=deg;i<len;i++)tmp[i]=0;ntt(tmp,len,1),ntt(b,len,1);for(int i=0;i<len;i++)b[i]=(2ll-tmp[i]*b[i]%mod+mod)%mod*b[i]%mod;ntt(b,len,-1);for(int i=deg;i<len;i++)b[i]=0; } void pol_div(int n,int m,ll *a,ll *b,ll *c,ll *d) {for(int i=0;i<=n-m;i++)if(m>=i)c[i]=b[m-i];int sz=0;while((1<<sz)<=MAX(n-m+1,m))sz++;sz++;int len=1<<sz;pol_inv(len,c,d);for(int i=0;i<=n-m;i++)c[i]=a[n-i];for(int i=n-m+1;i<len;i++)c[i]=d[i]=0;getrev(sz);ntt(c,len,1);ntt(d,len,1);for(int i=0;i<len;i++)c[i]=c[i]*d[i]%mod;ntt(c,len,-1);reverse(c,c+n-m+1);for(int i=n-m+1;i<len;i++)c[i]=0;sz=0;while((1<<sz)<=n+1)sz++;sz++;len=1<<sz;getrev(sz);ntt(c,len,1);ntt(b,len,1);for(int i=0;i<len;i++)d[i]=c[i]*b[i]%mod;ntt(d,len,-1);ntt(c,len,-1);for(int i=0;i<len;i++)d[i]=(a[i]-d[i]+mod)%mod;for(int i=m;i<len;i++)d[i]=0; } int main() {int n,m;scanf("%d%d",&n,&m);for(int i=0;i<=n;i++)scanf("%lld",&a[i]);for(int i=0;i<=m;i++)scanf("%lld",&b[i]);pol_div(n,m,a,b,c,d);for(int i=0;i<=n-m;i++)printf("%lld ",c[i]);puts("");for(int i=0;i<m;i++)printf("%lld ",d[i]);puts("");return 0; } /******************** 5 1 1 9 2 6 0 8 1 7 ********************/

轉載于:https://www.cnblogs.com/acjiumeng/p/9530479.html

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