HDU 1588 Gauss Fibonacci (matrix embedding matrix)

Question:

Find the sum of K * I + B in Fibonacci.

Train of Thought Analysis:

Defines the matrix of the Fibonacci series

F (n) is the nth entry of Fibonacci

F (n) = f (n + 1)

F (N)

Then we can know the matrix.

A = 1 1

1 0

Make F (n) = A * F (n + 1)

Then we simplify the final answer.

Sum = F (B) + f (K + B) + F (2 * k + B )....

Sum = F (B) + A ^ k f (B) + A ^ 2 k f (B ).....

Sum = (E + A ^ K + A ^ 2 k...) * F (B)

Then we define matrix A ^ K as matrix K.

Then, calculate the sum formula above.

E * sum = sum + E

0 k e k

Therefore, construct a matrix of embedded matrices.

Then obtain and multiply by F (B.

#include <cstdio>#include <iostream>#include <cstring>#include <iostream>#define N 2using namespace std;typedef long long LL;LL mod;struct matrix{    LL data[N][N];    friend matrix operator * (const matrix A,const matrix B)    {        matrix res;        memset(res.data,0,sizeof res.data);        for(int i=0;i<N;i++)        for(int j=0;j<N;j++)        for(int k=0;k<N;k++)        res.data[i][j]+=(A.data[i][k]*B.data[k][j])%mod;        return res;    }    friend matrix operator + (const matrix A,const matrix B)    {        matrix res;        for(int i=0;i<N;i++)        for(int j=0;j<N;j++)        res.data[i][j]=(A.data[i][j]+B.data[i][j])%mod;        return res;    }    friend matrix operator - (const matrix A,const matrix B)    {        matrix res;        for(int i=0;i<N;i++)        for(int j=0;j<N;j++)        res.data[i][j]=((A.data[i][j]-B.data[i][j])+mod)%mod;        return res;    }    void print()    {        for(int i=0;i<N;i++)        {            for(int j=0;j<N;j++)            printf("%I64d ",data[i][j]);            puts("");        }    }}E,zero;struct supermax{    matrix ret[N][N];    friend supermax operator * (supermax A,supermax B)    {        supermax res;        for(int i=0;i<N;i++)        for(int j=0;j<N;j++)        res.ret[i][j]=zero;        for(int i=0;i<N;i++)        for(int j=0;j<N;j++)        for(int k=0;k<N;k++)        {            res.ret[i][j]=res.ret[i][j]+(A.ret[i][k]*B.ret[k][j]);            for(int p=0;p<N;p++)            for(int q=0;q<N;q++)            res.ret[i][j].data[p][q]%=mod;        }        return res;    }};matrix matmod(matrix origin,LL n){    matrix res=E;    while(n)    {        if(n&1)        res=res*origin;        n>>=1;        origin=origin*origin;    }    return res;}supermax Do(supermax origin,LL n){    supermax res;    for(int i=0;i<N;i++)    for(int j=0;j<N;j++)    res.ret[i][j]=zero;    for(int i=0;i<N;i++)    res.ret[i][i]=E;    while(n)    {        if(n&1)        res=res*origin;        n>>=1;        origin=origin*origin;    }    return res;}int main(){    memset(zero.data,0,sizeof zero.data);    memset(E.data,0,sizeof E.data);    for(int i=0;i<N;i++)    E.data[i][i]=1;    LL k,b,n;    while(scanf("%I64d%I64d%I64d%I64d",&k,&b,&n,&mod)!=EOF)    {        matrix fib;        fib.data[0][0]=1;        fib.data[0][1]=1;        fib.data[1][0]=1;        fib.data[1][1]=0;        matrix K=matmod(fib,k);        supermax o;        o.ret[0][0]=E;        o.ret[0][1]=E;        o.ret[1][0]=zero;        o.ret[1][1]=K;        supermax final=Do(o,n);        matrix tmp=(final.ret[0][0]*zero)+(final.ret[0][1]*E);        matrix B=matmod(fib,b);        matrix fibb,fib0;        fib0.data[0][0]=1;        fib0.data[1][0]=0;        fib0.data[0][1]=fib0.data[1][1]=0;        fibb=B*fib0;        matrix ans = tmp*fibb;        printf("%I64d\n",ans.data[1][0]%mod);    }    return 0;}