Fibonacci–poj3070(矩阵快速幂)

程序员兰兰技术 2022年11月15日

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http://poj.org/problem?id=3070

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

Sample Output

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:

这道题就是快速幂http://blog.csdn.net/u013795055/article/details/38599321

还有矩阵相乘

#include <iostream>
#include <stdio.h>
#include <string.h>
#include <algorithm>
#include <math.h>
#include <stdlib.h>
#include <vector>
#include <queue>using namespace std;
#define memset(a,b) memset(a,b,sizeof(a))
#define N 4
#define INF 0xfffffff
struct node
{
    int a[N][N];
}e;node mm(node p,node q)
{
    node t;
    for(int i=;i<;i++)
    {
        for(int j=;j<;j++)
        {
            t.a[i][j]=;
            for(int k=;k<;k++)
            {
                t.a[i][j]=(t.a[i][j]+(p.a[i][k]*q.a[k][j]))%;
            }
        }
    }
    return t;
}node mul(node p,int n)
{    node q;
    for(int i=;i<;i++)
    {
        for(int j=;j<;j++)
        {
            q.a[i][j]=(i==j);
        }
    }///这个是为了当n是奇数是第一次跟q相乘是还是原来的p,为了下一次跟下一奇数相乘
    while(n)
    {
        if(n&)
           q=mm(q,p);
        n=n/;
        p=mm(p,p);
    }
    return q;
}
int main()
{
    int n;
    while(scanf("%d",&n),n!=-)
    {
        e.a[][]=;
        e.a[][]=;
        e.a[][]=;
        e.a[][]=;
        node b=mul(e,n);
        printf("%d\n",b.a[][]);
    }
    return ;
}