洛谷题单指南-基础线性代数-P1939 矩阵加速（数列）

原题链接：https://www.luogu.com.cn/problem/P1939

题意解读：求数列第n项。

解题思路：矩阵快速幂加速递推。

100分代码：

#include <bits/stdc++.h>
using namespace std;

typedef long long LL;
const int MOD = 1e9 + 7;
struct Matrix
{
    LL a[5][5];

    Matrix()
    {
        memset(a, 0, sizeof(a));
    }

    Matrix operator * (const Matrix &to) const
    {
        Matrix res;
        for(int i = 1; i <= 3; i++)
            for(int j = 1; j <= 3; j++)
                for(int k = 1; k <= 3; k++)
                    res.a[i][j] = (res.a[i][j] + a[i][k] * to.a[k][j]) % MOD;
        return res;
    }
} b3, A, ans;
LL t, n;

Matrix ksm(Matrix &a, LL b)
{
    Matrix res;
    res.a[1][1] = res.a[2][2] = res.a[3][3] = 1;
    while(b)
    {
        if(b & 1) res = res * a;
        b >>= 1;
        a = a * a;  
    }
    return res;
}

int main()
{
    cin >> t;
    while(t--)
    {
        cin >> n;
        if(n <= 3)
        {
            cout << 1 << endl;
            continue;
        }
        b3.a[1][1] = 1, b3.a[1][2] = 1, b3.a[1][3] = 1; 
        A.a[1][1] = 1, A.a[1][2] = 1, A.a[1][3] = 0,
        A.a[2][1] = 0, A.a[2][2] = 0, A.a[2][3] = 1,
        A.a[3][1] = 1, A.a[3][2] = 0, A.a[3][3] = 0; 
        ans = b3 * ksm(A, n - 3);
        cout << ans.a[1][1] << endl;
    }
    return 0;
}