洛谷P1962 斐波那契数列 题解 矩阵快速幂

题目链接：https://www.luogu.com.cn/problem/P1962

解题思路：

矩阵快速幂模板题。

但是我用 递归写，交的时候总是莫名 RE 两组，而且 RE 的两组还是不固定的，有的时候这两组 RE，有的时候那两组 RE。

递归代码如下（可能会 RE）：

#include <bits/stdc++.h>
using namespace std;
const long long MOD = 1000000007;
long long n;
struct Matrix {
    int n, m;
    long long a[2][2];
    Matrix() {};
    Matrix(int _n, int _m) { n = _n; m = _m; }
    Matrix operator * (Matrix b) {
        Matrix c = Matrix(n, b.m);
        memset(c.a, 0, sizeof(c.a));
        for (int i = 0; i < n; i ++)
            for (int j = 0; j < b.m; j ++)
                for (int k = 0; k < m; k ++)
                    c.a[i][j] += a[i][k] * b.a[k][j] % MOD, c.a[i][j] %= MOD;
        return c;
    }
    void test() {
        cout << "n = " << n << ", m = " << m << endl;
        for (int i = 0; i < n; i ++) {
            for (int j = 0; j < m; j ++) cout << a[i][j] << " ";
            cout << endl;
        }
    }
};
Matrix solve(Matrix a, long long n) {
    if (n == 1) return a;
    Matrix b = solve(a, n/2);
    b = b * b;
    if (n % 2) b = b * a;
    return b;
}
int main() {
    cin >> n;
    if (n <= 2) cout << 1 << endl;
    else {
        Matrix a = Matrix(2, 2);
        a.a[0][0] = 0; a.a[0][1] = a.a[1][0] = a.a[1][1] = 1;
        Matrix b = Matrix(2, 1);
        b.a[0][0] = b.a[1][0] = 1;
        cout << (solve(a, n-1) * b).a[0][0] << endl;
    }
    return 0;
}

所以改成了非递归的写法，如下：

#include <bits/stdc++.h>
using namespace std;
const long long MOD = 1000000007;
long long n;
struct Matrix {
    int n, m;
    long long a[2][2];
    Matrix() {};
    Matrix(int _n, int _m) { n = _n; m = _m; }
    Matrix operator * (Matrix b) {
        Matrix c = Matrix(n, b.m);
        memset(c.a, 0, sizeof(c.a));
        for (int i = 0; i < n; i ++)
            for (int j = 0; j < b.m; j ++)
                for (int k = 0; k < m; k ++)
                    c.a[i][j] += a[i][k] * b.a[k][j] % MOD, c.a[i][j] %= MOD;
        return c;
    }
    void test() {
        cout << "n = " << n << ", m = " << m << endl;
        for (int i = 0; i < n; i ++) {
            for (int j = 0; j < m; j ++) cout << a[i][j] << " ";
            cout << endl;
        }
    }
};
Matrix solve(Matrix a, long long n) {
    Matrix c = Matrix(2, 2);
    memset(c.a, 0, sizeof(c.a));
    c.a[0][0] = c.a[1][1] = 1;
    while (n > 0) {
        if (n % 2) c = c * a;
        a = a * a;
        n /= 2;
    }
    return c;
}
int main() {
    cin >> n;
    if (n <= 2) cout << 1 << endl;
    else {
        Matrix a = Matrix(2, 2);
        a.a[0][0] = 0; a.a[0][1] = a.a[1][0] = a.a[1][1] = 1;
        Matrix b = Matrix(2, 1);
        b.a[0][0] = b.a[1][0] = 1;
        cout << (solve(a, n-1) * b).a[0][0] << endl;
    }
    return 0;
}

结果还是 RE。不知道怎么回事了囧、、（感觉是今天洛谷的测聘机出问题了（2020年11月7日 CSP-S复赛当天））

反复提交了多次，终于AC了。事实证明 (Rightarrow) 洛谷的测评机今天估计出问题了。