poj3070 Fibonacci 矩阵快速幂

Fibonacci

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 18084		Accepted: 12572

Description

In the Fibonacci integer sequence, F₀ = 0, F₁ = 1, and F_n = F_{n − 1} + F_{n − 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 F_n.

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 F_n. If the last four digits of F_n are all zeros, print ‘0’; otherwise, omit any leading zeros (i.e., print F_n 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

Stanford Local 2006

矩阵快速幂的知识

点击打开链接

#include<iostream>
using namespace std;
typedef long long  ll;
const int MOD=1e4;
#define mod(x) ((x)%MOD)
struct mat{
int m[2][2];
}unit;
mat operator *(mat a,mat b)
{
    mat ret;
    ll x;
    for(int i=0;i<2;i++)
        for(int j=0;j<2;j++)
    {
        x=0;
        for(int k=0;k<2;k++)
            x+=mod( (ll) (a.m[i][k]*b.m[k][j]) );
            ret.m[i][j]=mod(x);
    }
    return ret;
}
void init_unit()
{
    for(int i=0;i<2;i++)
        unit.m[i][i]=1;//单位矩阵，主对角线上元素为1，其余元素为0
    return;
}
mat pow_mat(mat a,ll n)
{
    mat ret=unit;
    while(n){
        if(n&1) ret=ret*a;
        a=a*a;
        n>>=1;
    }
    return ret;
}
int main()
{
	ios::sync_with_stdio(false);
    ll n;
    init_unit();
    while(cin>>n&&n!=-1){
        mat a;
        a.m[0][0]=a.m[0][1]=a.m[1][0]=1;
        a.m[1][1]=0;
        a=pow_mat(a,n);
        cout<<a.m[0][1]<<endl;
    }
	return 0;
}