HDU 5950 Recursive sequence(矩阵快速幂)
2017-08-30 12:10
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Recursive sequence
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others)Total Submission(s): 29 Accepted Submission(s): 19
Problem Description
Farmer John likes to play mathematics games with his N cows. Recently, they are attracted by recursive sequences. In each turn, the cows would stand in a line, while John writes two positive numbers a and b on a blackboard. And then, the cows would say their
identity number one by one. The first cow says the first number a and the second says the second number b. After that, the i-th cow says the sum of twice the (i-2)-th number, the (i-1)-th number, and i4.
Now, you need to write a program to calculate the number of the N-th cow in order to check if John’s cows can make it right.
Input
The first line of input contains an integer t, the number of test cases. t test cases follow.
Each case contains only one line with three numbers N, a and b where N,a,b < 231 as
described above.
Output
For each test case, output the number of the N-th cow. This number might be very large, so you need to output it modulo 2147493647.
Sample Input
2
3 1 2
4 1 10
Sample Output
85
369
HintIn the first case, the third number is 85 = 2*1十2十3^4.
In the second case, the third number is 93 = 2*1十1*10十3^4 and the fourth number is 369 = 2 * 10 十 93 十 4^4.
题意:
已知递推公式为F(n)=F(n-1)+F(n-2)+n^4,F(1)=a,F(2)=b,给出n,求出F(n)。
思路:
一道矩阵快速幂的裸题,难点在于由已知的递推式构造矩阵,大家应该大一都学了线性代数,构造矩阵也就不难了,具体的步骤就不说了。
代码:
#include<iostream> #include<cstring> using namespace std; #define mod 2147493647 #define LL long long struct matrix{ LL m[7][7]; }mat; int t,n,f1,f2; void init(){ memset(mat.m,0,sizeof(mat));///此处也可以用一个初始化的二维数组通过来赋值,避免一个一个麻烦的赋值。 mat.m[0][1]=1,mat.m[1][0]=2,mat.m[1][1]=1,mat.m[1][2]=1,mat.m[1][3]=4,mat.m[1][4]=6,mat.m[1][5]=4,mat.m[1][6]=1; mat.m[2][2]=1,mat.m[2][3]=4,mat.m[2][4]=6,mat.m[2][5]=4,mat.m[2][6]=1; mat.m[3][3]=1,mat.m[3][4]=3,mat.m[3][5]=3,mat.m[3][6]=1; mat.m[4][4]=1,mat.m[4][5]=2,mat.m[4][6]=1; mat.m[5][5]=mat.m[5][6]=mat.m[6][6]=1; } matrix mul(matrix a,matrix b){ matrix c; memset(c.m,0,sizeof(c.m)); for(int i=0; i<7; i++) for(int j=0; j<7; j++) for(int k=0; k<7; k++) c.m[i][j]=(c.m[i][j]+a.m[i][k]*b.m[k][j])%mod;///注意要使用long long因为此处可能会越界。 return c; } int slove(){ matrix ans; memset(ans.m,0,sizeof(ans.m)); ans.m[0][0]=f1,ans.m[1][0]=f2,ans.m[2][0]=16,ans.m[3][0]=8,ans.m[4][0]=4,ans.m[5][0]=2,ans.m[6][0]=1; n--; while(n){ if(n&1) ans=mul(mat,ans);///顺序不能改变,矩阵乘法不满足交换侓。 n>>=1; mat=mul(mat,mat); } return ans.m[0][0]; } int main(){ cin>>t; while(t--){ cin>>n>>f1>>f2; init(); cout<<slove()<<endl; } }
参考博客:
http://blog.csdn.net/spring371327/article/details/52973534
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