蓝桥集训之斐波那契数列

蓝桥集训之斐波那契数列

• 核心思想：矩阵乘法

  • 将原本O(n)的递推算法优化为O(log₂n)

  • 

  • 构造1x2矩阵f和2x2矩阵a

  • 发现f(n+1) = f(n) * a

    • 则f(n+1) = f(1) * aⁿ
    • 可以用快速幂优化

•   #include <iostream>#include <cstring>#include <algorithm>using namespace std;const int MOD = 10000;int f[2];int a[2][2];int n;void mul1(){int res[2];  //res = res*a 求1x2矩阵memset(res,0,sizeof res);for(int i=0;i<2;i++)for(int j=0;j<2;j++)res[i] = (res[i] + f[j] * a[j][i]) %MOD;  //计算f*amemcpy(f,res,sizeof f);}void mul2(){int res[2][2];  //a = a*a 求2x2矩阵memset(res,0,sizeof res);for(int i=0;i<2;i++)for(int j=0;j<2;j++)for(int k=0;k<2;k++)res[i][j] = (res[i][j] + a[i][k] * a[k][j])%MOD;  //计算a*amemcpy(a,res,sizeof a);}void qmi(int n){while (n)  //快速幂优化{ if(n&1) mul1();  //res = res*a%MODmul2();  //a = a*a%MODn>>=1;}}int main(){while(cin>>n , n!=-1){f[0] = 0,f[1] = 1;  //初始化第0 1项a[0][0] = 0,a[0][1] = 1,a[1][0] = 1,a[1][1] = 1;  //初始化a矩阵qmi(n); cout<<f[0]<<endl;}return 0;}