Time Limit: 1000MS | Memory Limit: 65536K | |
Total Submissions: 2134 | Accepted: 1471 |
Description
In the Fibonacci integer sequence, F0 = 0, F1 = 1, and Fn = Fn − 1 + Fn − 2 for n ≥ 2. For example, the first ten terms of the Fibonacci sequence are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
An alternative formula for the Fibonacci sequence is
.
Given an integer n, your goal is to compute the last 4 digits of Fn.
Input
The input test file will contain multiple test cases. Each test case consists of a single line containing n (where 0 ≤ n ≤ 1,000,000,000). The end-of-file is denoted by a single line containing the number −1.
Output
For each test case, print the last four digits of Fn. If the last four digits of Fn are all zeros, print ‘0’; otherwise, omit any leading zeros (i.e., print Fn mod 10000).
Sample Input
0 9 999999999 1000000000 -1
Sample Output
0 34 626 6875
Hint
As a reminder, matrix multiplication is associative, and the product of two 2 × 2 matrices is given by
.
Also, note that raising any 2 × 2 matrix to the 0th power gives the identity matrix:
.
Source
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{{F(n+1),F(n)},
{F(n),F(n-1)}}={{1,1} ^n
{1,0}}
令A(n)表示一个矩阵序列
A(n)={{F(n),F(n-1)},
{F(n-1),F(n-2)}那么A(2)={{1,1},{1,0}},由那个表达式得到:A(n)=A(2)^(n-1),A(2)是己知的2*2矩阵,现在的问题就是如何求
A(2)^n因为方阵的乘法有结合律,所以A(2)^n=A(2)^(n/2)*A(2)^(n/2),不妨设n是偶数
所以求A(n)就可以化成求A(n/2)并作一次乘法,所以递归方程是:T(n)=T(n/2)+O(1)
*/
//5144566 11410 3070 Accepted 204K 0MS C++ 1049B 2009-05-13 10:13:17
//用矩阵做Fibonacci数
#include <iostream>
#define MAX 2
using namespace std;
typedef struct node
{
int matrix[MAX][MAX];
}Matrix;
Matrix unit,init,result;
int n;
void Init() //初始化
{
int i,j;
for(i=0;i<2;i++)
for(j=0;j<2;j++)
{
init.matrix[i][j]=1;
unit.matrix[i][j]=(i==j);
}
init.matrix[1][1]=0;
}
Matrix Mul(Matrix a,Matrix b) //矩阵乘法
{
Matrix c;
int i,j,k;
for(i=0;i<2;i++)
for(j=0;j<2;j++)
{
c.matrix[i][j]=0;
for(k=0;k<2;k++)
c.matrix[i][j]+=a.matrix[i][k]*b.matrix[k][j];
if(c.matrix[i][j]>=10000) //取余
c.matrix[i][j]%=10000;
}
return c;
}
Matrix Cal(int exp) //求幂
{
Matrix p,q;
p=unit;
q=init;
while(exp!=1)
{
if(exp&1)
{
exp--;
p=Mul(p,q);
}
else
{
exp>>=1;
q=Mul(q,q);
}
}
p=Mul(p,q);
return p;
}
int main()
{
while(scanf("%d",&n)!=EOF&&n!=-1)
{
if(n==0)
{
printf("0\n");
continue;
}
if(n==2||n==1)
{
printf("1\n");
continue;
}
Init();
result=Cal(n-1); //求n-1次幂就好
printf("%d\n",result.matrix[0][0]);
}
return 0;
}