# Fibonacci Number

The **Fibonacci numbers**, commonly denoted `F(n)` form a sequence, called the **Fibonacci sequence**, such that each number is the sum of the two preceding ones, starting from `0` and `1`. That is,

```
F(0) = 0,   F(1) = 1
F(N) = F(N - 1) + F(N - 2), for N > 1.
```

Given `N`, calculate `F(N)`.

**Example 1:**

```
Input: 2
Output: 1
Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.
```

**Example 2:**

```
Input: 3
Output: 2
Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.
```

**Example 3:**

```
Input: 4
Output: 3
Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.
```

**Note:**

0 ≤ `N` ≤ 30.

```java
class Solution {
    public int fib(int N) {
        if (N == 0)
            return 0;
        int[][] x = { { 1, 1 }, { 1, 0 } };
        // Identity matrix
        int[][] ans = { { 1, 0 }, { 0, 1 } };
        while (N > 0) {
            if (N % 2 != 0)
                ans = multiply(ans, x);
            N = N / 2;
            x = multiply(x, x);
        }
        return ans[0][1];
    }

    public int[][] multiply(int[][] x, int[][] y) {
        int[][] res = new int[2][2];
        res[0][0] = x[0][0] * y[0][0] + x[0][1] * y[1][0];
        res[0][1] = x[0][0] * y[0][1] + x[0][1] * y[1][1];
        res[1][0] = x[1][0] * y[0][0] + x[1][1] * y[1][0];
        res[1][1] = x[1][0] * y[0][1] + x[1][1] * y[1][1];
        return res;
    }
}
```
