HDU 2256Problem of Precision(矩阵快速幂)

题意

求$(\sqrt{2} + \sqrt{3})^{2n} \pmod {1024}$

$n \leqslant 10^9$

sol

看到题解的第一感受：这玩意儿也能矩阵快速幂？？？

是的，它能qwq。。。。

首先我们把$2$的幂乘进去，变成了

$(5 + 2\sqrt{6})^n$

设$f(n) = a_n + \sqrt{6} b_n$

那么$f(n+1) = (a_n + \sqrt{6} b_n ) * (5 + 2\sqrt{6})$

乘出来得到

$a_{n + 1} = 5 a_n + 12 b_n$

$b_{n + 1} = 2a_n + b b_n$

那么不难得到转移矩阵

$$\begin{pmatrix} 5 & 12 \\ 2 & 5 \end{pmatrix}$$

这样好像就能做了。。

但是实际上后我们最终会得到一个类似于$a_n + \sqrt{6}b_n$的东西，这玩意儿还是不能取模

考虑如何把$\sqrt{6}$的影响消掉。

$(5 + 2 \sqrt{6})^n = a_n + \sqrt{6}b_n$

$(5 - 2 \sqrt{6})^n = a_n - \sqrt{6}b_n$

相加得

$(5 + 2 \sqrt{6})^n + (5 - 2 \sqrt{6})^n = 2a_n$

考虑到$0 < (5 - 2\sqrt{6})^n < 1$

那么

$$\lfloor (5 + 2\sqrt{6})^n \rfloor = 2a_n - 1$$

做完啦qwq

 

#include<cstdio>
#include<cstring>
#include<iostream>
#define pair pair<int, int> 
#define mp(x, y)
#define fi first
#define se second 
// #include<map>
using namespace std;
#define ll long long
const ll maxn = 101, mod = 1024;
inline ll read() {
    char c = getchar(); ll x = 0, f = 1;
    while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();}
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
int t, n;
struct matrix {
    ll m[5][5], n;
    matrix() {n = 2; memset(m, 0, sizeof(m));}
    matrix operator * (const matrix &rhs) const {
        matrix ans;
        for(int k = 1; k <= n; k++) 
            for(int i = 1; i <= n; i++)
                for(int j = 1; j <= n; j++)
                    (ans.m[i][j] += 1ll * m[i][k] * rhs.m[k][j] % mod) % mod;
        return ans;
    }
};

matrix fp(matrix a, int p) {
    matrix base; base.m[1][1] = 1; base.m[2][2] = 1;
    while(p) {
        if(p & 1) base = base * a; 
        a = a * a; p >>= 1;
    }
    return base;
}
int main() {
    t = read();
    while(t--) {
        n = read();
        matrix a; 
        a.m[1][1] = 5; a.m[1][2] = 12;
        a.m[2][1] = 2; a.m[2][2] = 5;
        a = fp(a, n - 1);
        ll ans = (5 * a.m[1][1] + 2 * a.m[1][2]) % mod;
        printf("%i64d\n", (2 * ans - 1) % mod);
    }
    return 0;
}

/**/