矩阵快速幂

P3390 【模板】矩阵快速幂

早期代码忘记搬了

#include<bits/stdc++.h>
#define ll long long
#define il inline
#define ri register int
using namespace std;
const int MOD=1e9+7;
ll n,k;
struct Matrix
{
	ll a[105][105];
	Matrix() {memset(a,0,sizeof(a));}
	il void build(){for(ri i=1;i<=n;++i)a[i][i]=1;}//单位矩阵 
};
Matrix operator *(const Matrix &x,const Matrix &y)
{
	Matrix z;
	for(ri k=1;k<=n;++k)
	for(ri i=1;i<=n;++i)
	for(ri j=1;j<=n;++j)
	z.a[i][j]+=x.a[i][k]*y.a[k][j]%MOD,z.a[i][j]%=MOD;	
	return z;
}
il ll read()
{
	ll as=0,f=1;char ch=getchar();
	while(ch<'0'||ch>'9') f=ch=='-'?-1:1,ch=getchar();
	while(ch>='0'&&ch<='9') as=(as<<3)+(as<<1)+(ch^48),ch=getchar();
	return as*f;
}
il Matrix FPower(Matrix bs,ll pw)
{
	Matrix ans; ans.build();
	while(pw>0)
	{
		if(pw&1) ans=ans*bs;
		pw>>=1,bs=bs*bs;
	}
	return ans;
}
int main()
{
	n=read(),k=read();
	Matrix a;
	for(ri i=1;i<=n;++i)
		for(ri j=1;j<=n;++j)
			a.a[i][j]=read();
	a=FPower(a,k);
	for(ri i=1;i<=n;++i){
		for(ri j=1;j<=n;++j) 
			printf("%d ",a.a[i][j]);
		putchar('\n');	
	}
	return 0;
}

T1 P1962 斐波那契数列

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#include<bits/stdc++.h>
#define int long long
#define il inline
#define ri register int
using namespace std;
const int MOD=1e9+7;
int n;
struct Matrix{
	int a[3][3];
	Matrix(){memset(a,0,sizeof(a));}
	il void build(){a[1][1]=a[2][2]=1;}
	il void abuild(){a[1][1]=a[1][2]=1;}
}s;
il int gcd(int a,int b) {return a%b==0?b:gcd(b,a%b);}
il Matrix mul(Matrix a,Matrix b,bool f)
{
	Matrix as;
	for(ri k=1;k<=2;++k)
	for(ri i=1;i<=(f?1:2);++i)
	for(ri j=1;j<=2;++j)
	as.a[i][j]+=(a.a[i][k]*b.a[k][j])%MOD,as.a[i][j]%=MOD;
	return as;
} 
il Matrix FastPower(Matrix base,int power)
{
	Matrix as;as.build();
	while(power>0)
	{
		if(power&1) as=mul(as,base,0);
		power>>=1,base=mul(base,base,0);
	}
	return as;
}
signed main()
{
	scanf("%lld",&n);
	if(n<=2) printf("1");
	else
	{
		Matrix a,b;//a构造 b答案 
		b.abuild(),a.a[1][2]=a.a[2][2]=a.a[2][1]=1;
		a=FastPower(a,n-1),b=mul(b,a,1);
		printf("%lld",b.a[1][1]%MOD);
	}
	return 0;
 } 

T2 P1349 广义斐波那契数列

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#include<bits/stdc++.h>
#define int long long  //不开long long____
#define il inline
#define ri register int
using namespace std;
const int MOD=1e8;
int n,m;
struct Matrix{
	int a[3][3];
	Matrix(){memset(a,0,sizeof(a));}
	il void build(){a[1][1]=a[2][2]=1;}
	il void abuild(){a[1][1]=a[1][2]=1;}
}s;
il int gcd(int a,int b) {return a%b==0?b:gcd(b,a%b);}
il Matrix mul(Matrix a,Matrix b,bool f)
{
	Matrix as;
	for(ri k=1;k<=2;++k)
	for(ri i=1;i<=(f?1:2);++i)
	for(ri j=1;j<=2;++j)
	as.a[i][j]+=(a.a[i][k]*b.a[k][j])%MOD,as.a[i][j]%=MOD;
	return as;
} 
il Matrix FastPower(Matrix base,int power)
{
	Matrix as;as.build();
	while(power>0)
	{
		if(power&1) as=mul(as,base,0);
		power>>=1,base=mul(base,base,0);
	}
	return as;
}
signed main()
{
	scanf("%d%d",&n,&m);
	int id=gcd(n,m);
	if(id<=2) printf("1");
	else
	{
		Matrix a,b;//a构造 b答案 
		b.abuild(),a.a[1][2]=a.a[2][2]=a.a[2][1]=1;
		a=FastPower(a,id-1),b=mul(b,a,1);
		printf("%d",b.a[1][1]%MOD);
	}
	return 0;
 } 

T3 P1306 斐波那契公约数

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#include<bits/stdc++.h>
#define int long long  //不开long long____
#define il inline
#define ri register int
using namespace std;
const int MOD=1e8;
int n,m;
struct Matrix{
	int a[3][3];
	Matrix(){memset(a,0,sizeof(a));}
	il void build(){a[1][1]=a[2][2]=1;}
	il void abuild(){a[1][1]=a[1][2]=1;}
}s;
il int gcd(int a,int b) {return a%b==0?b:gcd(b,a%b);}
il Matrix mul(Matrix a,Matrix b,bool f)
{
	Matrix as;
	for(ri k=1;k<=2;++k)
	for(ri i=1;i<=(f?1:2);++i)
	for(ri j=1;j<=2;++j)
	as.a[i][j]+=(a.a[i][k]*b.a[k][j])%MOD,as.a[i][j]%=MOD;
	return as;
} 
il Matrix FastPower(Matrix base,int power)
{
	Matrix as;as.build();
	while(power>0)
	{
		if(power&1) as=mul(as,base,0);
		power>>=1,base=mul(base,base,0);
	}
	return as;
}
signed main()
{
	scanf("%d%d",&n,&m);
	int id=gcd(n,m);
	if(id<=2) printf("1");
	else
	{
		Matrix a,b;//a构造 b答案 
		b.abuild(),a.a[1][2]=a.a[2][2]=a.a[2][1]=1;
		a=FastPower(a,id-1),b=mul(b,a,1);
		printf("%d",b.a[1][1]%MOD);
	}
	return 0;
 } 

T4 P1939 【模板】矩阵加速（数列）

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#include<bits/stdc++.h>
#define il inline
#define int long long
#define ri register int
#define ru register unsigned int
using namespace std;
const int MOD=1e9+7;
il int read()
{
	int as=0,f=1;char ch=getchar();
	while(ch<'0'||ch>'9') f=ch=='-'?-1:1,ch=getchar();
	while(ch>='0'&&ch<='9') as=(as<<3)+(as<<1)+(ch^48),ch=getchar();
	return as*f;
}
struct Matrix{
	int a[5][5];
	Matrix(){memset(a,0,sizeof(a));}
	il void build() {a[1][1]=a[2][2]=a[3][3]=1;}
	il void abuild() {a[1][1]=a[1][2]=a[1][3]=1;}
};
il Matrix mul(Matrix a,Matrix b,bool f)
{
	Matrix as;
	for(ri k=1;k<=3;++k)
		for(ri i=1;i<=(f?1:3);++i)
			for(ri j=1;j<=3;++j)
	as.a[i][j]+=(a.a[i][k]*b.a[k][j])%MOD,as.a[i][j]%=MOD;
	return as;
}
il Matrix FastPower(Matrix base,int power)
{
	Matrix rt; rt.build();
	while(power>0)
	{
		if(power&1) rt=mul(rt,base,0);
		power>>=1,base=mul(base,base,0);		
	}
	return rt;
}
signed main()
{
	int T=read();
	for(ri t=1,n;t<=T;++t)
	{
		Matrix b,ans; n=read(),ans.abuild();//b构造 ans答案 
		if(n<=3) {printf("1"),putchar('\n');continue;}
		b.a[1][1]=b.a[1][2]=b.a[2][3]=b.a[3][1]=1;
		b=FastPower(b,n-3);
		ans=mul(ans,b,1),
		printf("%d",ans.a[1][1]),putchar('\n');
	}
	return 0;
}

T5 P2044 [NOI2012] 随机数生成器

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#include<bits/stdc++.h>
#define int __int128
#define il inline
#define ri register int
#define ru register unsigned int
using namespace std;
int m,a,c,x0,n,g;
struct Matrix{
	int a[3][3];
	Matrix(){memset(a,0,sizeof(a));}
	il void build(){a[1][1]=a[2][2]=1;}
};
il int read()
{
	int as=0,f=1;char ch=getchar();
	while(ch<'0'||ch>'9') f=ch=='-'?-1:1,ch=getchar();
	while(ch>='0'&&ch<='9') as=(as<<3)+(as<<1)+(ch^48),ch=getchar();
	return as*f;
}
il void write(int x)
{
	if(x==0) return;
	char ch=x%10+'0';
	write(x/10),putchar(ch);
}
il Matrix mul(Matrix x,Matrix y,bool f)
{
	Matrix rt;
	for(ru k=1;k<=2;++k)
	for(ru i=1;i<=(f?1:2);++i)
	for(ru j=1;j<=2;++j)
	rt.a[i][j]+=(x.a[i][k]*y.a[k][j])%m,rt.a[i][j]%=m;
	return rt;
}
il Matrix FastPower(Matrix x,int y)
{
	Matrix o;o.build();
	while(y>0)
	{
		if(y&1) o=mul(o,x,0);
		y>>=1,x=mul(x,x,0);
	}
	return o;
}
signed main()
{
	Matrix bs,ans;
	m=read(),a=read(),c=read(),x0=read(),n=read(),g=read();
	ans.a[1][1]=1,ans.a[1][2]=x0,
	bs.a[1][1]=1,bs.a[1][2]=c,bs.a[2][2]=a;
	bs=FastPower(bs,n),ans=mul(ans,bs,1);
	write(ans.a[1][2]%g);
	return 0;
}