LuoguP4893 GodFly的求导工具 题解
Description
题目链接
给定一个 n 次整系数函数 f(x)，问 f(x) 的 k 阶导在 x_i 处的导数。
1leq n leq 100,kleq n,mleq 10,a_ileq 10^5,x_ileq 10^5
Solution
很好奇这题是怎么评紫的（？
咳咳，对于这道题，你先要有亿点简单高等数学知识。
- f(x)=x^p 的 k 阶导函数为 f^{(k)}(x)=prod_{i=1}^k(p-i 1)x^{p-k}
- f(x)=g(x) h(x) 的 k 阶导函数为 f^{(k)}(x)=g^{(k)}(x) h^{(k)}(x)
证明略（
假设你已经熟知了以上结论，那么就可以愉快的把这道题切掉了。
直接暴力拆出每一项的系数，暴力进行求导，找到其 k 阶导函数，对于每个询问，直接暴力带入即可。
因为 a_ileq 10^5,x_i leq 10^5 可能会导致答案爆 int，所以你还要耐心地写一个高精套一个高精模板。
Code
代码语言：javascript复制#include<bits/stdc .h>
#define Tp template<typename Ty>
#define Ts template<typename Ty,typename... Ar>
#define W while
#define I inline
#define RI register int
#define LL long long
#define Cn const
#define gc getchar
#define DD isdigit(c=gc())
#define pc(c) putchar((c))
#define min(x,y) ((x)<(y)?(x):(y))
#define max(x,y) ((x)>(y)?(x):(y))
using namespace std;
namespace Debug{
Tp I void _debug(Cn char* f,Ty t){cerr<<f<<'='<<t<<endl;}
Ts I void _debug(Cn char* f,Ty x,Ar... y){W(*f!=',') cerr<<*f ;cerr<<'='<<x<<",";_debug(f 1,y...);}
Tp ostream& operator<<(ostream& os,Cn vector<Ty>& V){os<<"[";for(Cn auto& vv:V) os<<vv<<",";os<<"]";return os;}
#define gdb(...) _debug(#__VA_ARGS__,__VA_ARGS__)
}using namespace Debug;
namespace FastIO{
Tp I void read(Ty& x){char c;int f=1;x=0;W(!DD) f=c^'-'?1:-1;W(x=(x<<3) (x<<1) (c&15),DD);x*=f;}
Ts I void read(Ty& x,Ar&... y){read(x),read(y...);}
Tp I void write(Ty x){x<0&&(pc('-'),x=-x,0),x<10?(pc(x '0'),0):(write(x/10),pc(x '0'),0);}
Tp I void writeln(Cn Ty& x){write(x),pc('n');}
}using namespace FastIO;
Cn int N=110;
class Int{//Manchery的高精模板
#define BASE 1000000000
public:
typedef long long value;
void New(size_t l){
if (a!=NULL)delete[] a;a=new value[l];
len=1;a[0]=0;sign=1;
}
Int():a(NULL),base(BASE){New(1);}
Int(value x):a(NULL),base(BASE){New(1);*this=x;}
Int(value x,value _base):a(NULL),base(_base){New(1);*this=x;}
Int(const Int &B):a(NULL),base(BASE){New(1);*this=B;}
~Int(){delete[] a;}
Int& operator =(value x){
size_t l=1;for (value x1=max(x,-x);x1>=base; l,x1/=base);New(l);
if (x<0)x=-x,sign=0;else sign=1;
len=0;while (x)a[len ]=x�se,x/=base;
if (!len)a[len ]=0;
return *this;
}
Int& operator =(const Int &A){
New(A.len);len=A.len;memcpy(a,A.a,sizeof(value)*len);
base=A.base;sign=A.sign;return *this;
}
friend Int operator -(Int A){A.sign=1-A.sign;return A;}
bool operator !(){if (len==1&&a[0]==0)return 1;else return 0;}
friend Int operator (Int A,Int B){
if (A.sign!=B.sign){B.sign=1-B.sign;return A-B;}
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
Int res;res.set_base(A.base); int len=A.len>B.len?A.len:B.len;
res.New(len 1);res.sign=A.sign;
memset(res.a,0,(len 1)*sizeof(value));
for (int i=0;i<len; i){
if (i<A.len)res.a[i] =A.a[i];
if (i<B.len)res.a[i] =B.a[i];
}
for (int i=0;i<len; i)
if (res.a[i]>=res.base) res.a[i 1],res.a[i]-=res.base;
if (res.a[len])res.len=len 1;else res.len=len;
if (!res)res.sign=1;return res;
}
friend Int operator -(Int A,Int B){
if (A.sign!=B.sign){B.sign=1-B.sign;return A B;}
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
if (small(A,B))swap(A,B),A.sign=1-A.sign;
Int res;res.set_base(A.base); int len=A.len>B.len?A.len:B.len;
res.New(len);res.sign=A.sign;
memset(res.a,0,len*sizeof(value));
for (int i=0;i<len; i){
if (i>=B.len)res.a[i] =A.a[i];
else res.a[i] =A.a[i]-B.a[i];
if (res.a[i]<0)res.a[i] =res.base,--res.a[i 1];
}
while (len>1&&!res.a[len-1])--len;res.len=len;
if (!res)res.sign=1;return res;
}
friend Int operator *(Int A,Int B){
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
Int res;res.set_base(A.base); int len=A.len B.len;
res.New(len);res.sign=(A.sign==B.sign);
memset(res.a,0,len*sizeof(value));
for (int i=0;i<A.len; i)
for (int j=0;j<B.len; j){
res.a[i j] =A.a[i]*B.a[j];
res.a[i j 1] =res.a[i j]/res.base;
res.a[i j]%=res.base;
}
/*
for (int i=0;i<A.len; i)
for (int j=0;j<B.len; j)res.a[i j] =A.a[i]*B.a[j];
for (int i=0;i<len-1; i)res.a[i 1] =res.a[i]/res.base,res.a[i]%=res.base;
*/
while (len>1&&!res.a[len-1])--len;res.len=len;
return res;
}
friend pair<Int,Int> divide(Int A,Int B){
if (!B){puts("error:div zero!");for (;;);}
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
if (small(A,B))return make_pair(Int(0),A);
Int C,D;C.set_base(A.base);D.set_base(A.base);C.New(A.len);C.len=A.len;
bool Csign=(A.sign==B.sign),Dsign=A.sign;A.sign=B.sign=1;
for (int i=A.len-1;i>=0;--i){
C.a[i]=0;D=D*D.base;D.a[0]=A.a[i];
int l=0,r=A.base-1,mid;
while (l<r){
mid=(l r 1)>>1;
if (small(B*mid,D 1))l=mid;
else r=mid-1;
}
C.a[i]=l;D=D-B*l;
}
C.sign=Csign;D.sign=Dsign;if (!D)D.sign=1;
while (C.len>1&&!C.a[C.len-1])--C.len;
return make_pair(C,D);
}
Int operator /(value x){
if (!x){puts("error:div zero!");for (;;);}
value d=0;Int res;res.set_base(base);res.New(len);res.len=len;
if (x<0)x=-x,res.sign=(sign==0);
else res.sign=(sign==1);
for (int i=len-1;i>=0;--i)
d=d*base a[i],res.a[i]=d/x,d%=x;
while (res.len>1&&!res.a[res.len-1])--res.len;
return res;
}
Int operator %(value x){
value d=0;if (x<0)x=-x;
for (int i=len-1;i>=0;--i)d=(d*base a[i])%x;
return d;
}
friend Int abs(Int A){A.sign=1;return A;}
friend bool small(Int A,Int B){
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
if (A.len!=B.len)return A.len<B.len;
for (int i=A.len-1;i>=0;--i)
if (A.a[i]!=B.a[i])return A.a[i]<B.a[i];
return 0;
}
friend bool operator <(Int A,Int B){
if (A.sign!=B.sign)return A.sign<B.sign;
return A.sign==1?small(A,B):small(B,A);
}
friend bool operator ==(Int A,Int B){
if (A.base!=B.base)
if (A.base>B.base)B.set_base(A.base);
else A.set_base(B.base);
if (A.sign!=B.signA.len!=B.len)return 0;
for (int i=0;i<A.len; i)if (A.a[i]!=B.a[i])return 0;
return 1;
}
friend bool operator !=(Int A,Int B){return !(A==B);}
friend bool operator >(Int A,Int B){return !(A<BA==B);}
friend bool operator <=(Int A,Int B){return A<BA==B;}
friend bool operator >=(Int A,Int B){return A>BA==B;}
Int operator /(Int B){return divide(*this,B).first;}
Int operator %(Int B){return divide(*this,B).second;}
Int& operator =(Int B){*this=*this B;return *this;}
Int& operator -=(Int B){*this=*this-B;return *this;}
Int& operator *=(Int B){*this=*this*B;return *this;}
Int& operator /=(Int B){*this=*this/B;return *this;}
Int& operator %=(Int B){*this=*this%B;return *this;}
Int& operator (){*this=*this 1;return *this;}
Int& operator --(){*this=*this-1;return *this;}
Int operator (int){Int res(*this);*this=*this 1;return res;}
Int operator --(int){Int res(*this);*this=*this-1;return res;}
Int operator (value x){return *this Int(x,this->base);}
Int operator -(value x){return *this-Int(x,this->base);}
Int operator *(value x){return *this*Int(x,this->base);}
//Int operator /(value x){Int T;T=x;return *this/T;}
//Int operator %(value x){Int T;T=x;return *this%T;}
Int& operator *=(value x){*this=*this*x;return *this;}
Int& operator =(value x){*this=*this x;return *this;}
Int& operator -=(value x){*this=*this-x;return *this;}
Int& operator /=(value x){*this=*this/x;return *this;}
Int& operator %=(value x){*this=*this%x;return *this;}
bool operator ==(value x){return *this==Int(x,this->base);}
bool operator !=(value x){return *this!=Int(x,this->base);}
bool operator <=(value x){return *this<=Int(x,this->base);}
bool operator >=(value x){return *this>=Int(x,this->base);}
bool operator <(value x){return *this<Int(x,this->base);}
bool operator >(value x){return *this>Int(x,this->base);}
friend Int gcd(Int x,Int y){
Int t;int cnt=0;
while (1){
if (x<y)t=x,x=y,y=t;
if (y==0){
while (cnt--)x*=2;
return x;
}
if (x%2==0&&y%2==0)x/=2,y/=2, cnt;
else if (x%2==0)x/=2;
else if (y%2==0)y/=2;
else {t=x;x=y;y=t-y;}
}
}
void to_arr(char *c){
char *c1=c;
for (int i=0;i<len-1; i)
for (value x=a[i],b=base/10;b>=1;b/=10)*c1 ='0' x,x/=10;
for (value x=a[len-1];x>0;x/=10)*c1 ='0' x;
if (len==1&&a[len]==0)*c1 ='0';
if (sign==0)*c1 ='-';*c1=0;reverse(c,c1);
}
void from_arr(char *c){
size_t base_l=get_basel(),b=1;int cl=strlen(c);value x=0;
New((cl base_l-1)/base_l);len=0;
if (*c=='-')sign=0, c,--cl;else sign=1;
while (--cl>=0){
x =(c[cl]-'0')*b;b*=10;if (b==base)a[len ]=x,x=0,b=1;
}
if (!lenx)a[len ]=x;
while (len>1&&!a[len-1])--len;
}
void set_base(int _base){
if (base==_base)return;
char *c=new char[len*get_basel() 1];
to_arr(c);base=_base;from_arr(c);
delete[] c;
}
void set_basel(int _l){
size_t _base=1;while (_l--)_base*=10;set_base(_base);
}
void read(){
vector<char> s;char ch;
scanf(" %c",&ch);if (ch=='-')s.push_back('-'),ch=getchar();
for (;ch>='0'&&ch<='9';ch=getchar())s.push_back(ch);
char *c=new char[s.size() 1];
for (int i=0;i<s.size(); i)c[i]=s[i];c[s.size()]=0;
from_arr(c);delete[] c;
if (!*this)this->sign=1;
}
void print(){
if (!sign)putchar('-');
printf("%d",int(a[len-1]));
for (int i=len-2;i>=0;--i){
for (int j=base/10;j>=10;j/=10)
if (a[i]<j)putchar('0');
else break;
printf("%d",(int)a[i]);
}
}
void println(){print();putchar('n');}
private:
value *a,base;int len;bool sign; //0="-"
size_t get_basel()const{
size_t res=0;for (int b=base/10;b>=1;b/=10, res);
return res;
}
#undef BASE
};
Int num[N],Ans[N],top=1,ans;
int n,m,k,len;
string str;
I Int fpow(int a,int b){Int A=a,s=1;W(b!=0){if(b%2==1) s*=A;A*=A;b=b/2;}return s;}//高精快速幂
int main(){
RI i,j,s;for(read(n,k),cin>>str,len=str.length(),i=5;i<len;)
if(isdigit(str[i])){j=i 1;s=str[i]-'0';W(j<len&&isdigit(str[j])) s=s*10 str[j]-'0',j ;top=s,i=j;}//处理系数
else if(str[i]=='x'){j=i 1;s=0;W(j<len&&str[j]!='^') j;j ;W(j<len&&isdigit(str[j])) s=s*10 str[j]-'0', j;num[s] =top;top=1;i=j;}//处理次数
else i ;
for(i=0;i<=n;i )
if(i-k>=0) for(Ans[i-k]=num[i],j=1;j<=k;j ) Ans[i-k]*=i-j 1;//直接求导套公式
for(read(m);m--;){for(read(s),ans=j=0;j<=n;j ) ans=ans Ans[j]*fpow(s,j);ans.println();}//对于每个询问直接暴力计算
return 0;
}
