POJ 3070 Fibonacci (矩阵快速幂)
作者:互联网
题目链接:POJ 3070
Problem Description
In the Fibonacci integer sequence, \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n − 1} + F_{n − 2}\) for \(n \ge 2\). 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 \(0 \le n \le 1,000,000,000\)). 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
Solution
题意
给定正整数 \(n\),求斐波那契数列第 \(n\) 项。
题解
矩阵快速幂
矩阵快速幂模板题。
\[ \left[ \begin{matrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{matrix} \right] = \left[ \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix} \right] ^ n \]
Code
#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstring>
using namespace std;
typedef long long ll;
const int maxn = 1e2 + 5;
const ll mod = 1e4;
struct Matrix {
int n, m;
ll a[maxn][maxn];
Matrix(int n = 0, int m = 0) : n(n), m(m) {}
void input() {
for(int i = 1; i <= n; ++i) {
for(int j = 1; j <= m; ++j) {
scanf("%lld", &a[i][j]);
}
}
}
void output() {
for(int i = 1; i <= n; ++i) {
for(int j = 1; j <= m; ++j) {
printf("%lld", a[i][j]);
printf("%s", j == m? "\n": " ");
}
}
}
void init() {
memset(a, 0, sizeof(a));
}
void unit() {
init();
for(int i = 1; i <= n; ++i) {
a[i][i] = 1;
}
}
Matrix operator *(const Matrix b) {
Matrix c(n, b.m);
c.init();
for(int i = 1; i <= c.n; ++i) {
for(int k = 1; k <= m; ++k) {
for(int j = 1; j <= c.m; ++j) {
c.a[i][j] = (c.a[i][j] + a[i][k] * b.a[k][j]) % mod;
}
}
}
return c;
}
Matrix qmod(ll b) {
Matrix ans = Matrix(n, n);
ans.unit();
if(!b) return ans;
while(b) {
if(b & 1) ans = ans * (*this);
*this = (*this) * (*this);
b >>= 1;
}
return ans;
}
};
int main() {
int n;
while(~scanf("%d", &n)) {
if(n == -1) break;
ll ans = 0;
if(n) {
Matrix m(2, 2);
m.a[1][1] = 1;
m.a[1][2] = 1;
m.a[2][1] = 1;
m.a[2][2] = 0;
m = m.qmod(n - 1);
ans = m.a[1][1] % mod;
}
printf("%lld\n", ans);
}
return 0;
}
标签:matrix,int,ll,3070,POJ,Fibonacci,ans,include 来源: https://www.cnblogs.com/wulitaotao/p/11420478.html