#include <stdio.h>
#include <stdint.h>

/**
 * @brief Computes the n-th Fibonacci number using matrix exponentiation.
 *
 * This implementation uses the identity:
 *   [F(n+1) F(n)  ] = [1 1]^n
 *   [F(n)   F(n-1)]   [1 0]
 *
 * The matrix is exponentiated in O(log n) time using exponentiation by squaring.
 *
 * @param n The index of the Fibonacci number to compute.
 * @return The n-th Fibonacci number.
 */
uint64_t fibonacci_matrix(uint32_t n);

// 2x2 matrix structure for Fibonacci computation
typedef struct {
    uint64_t a, b;
    uint64_t c, d;
} FibMatrix;

/**
 * @brief Multiplies two 2x2 matrices.
 */
static FibMatrix matrix_multiply(FibMatrix x, FibMatrix y) {
    FibMatrix result;
    result.a = x.a * y.a + x.b * y.c;
    result.b = x.a * y.b + x.b * y.d;
    result.c = x.c * y.a + x.d * y.c;
    result.d = x.c * y.b + x.d * y.d;
    return result;
}

/**
 * @brief Raises a 2x2 matrix to the power of n using exponentiation by squaring.
 */
static FibMatrix matrix_power(FibMatrix base, uint32_t n) {
    FibMatrix result = {1, 0, 0, 1}; // Identity matrix
    while (n > 0) {
        if (n % 2 == 1)
            result = matrix_multiply(result, base);
        base = matrix_multiply(base, base);
        n /= 2;
    }
    return result;
}

uint64_t fibonacci_matrix(uint32_t n) {
    if (n == 0) return 0;
    FibMatrix base = {1, 1, 1, 0};
    FibMatrix result = matrix_power(base, n - 1);
    return result.a;
}

int main() {
    for (uint32_t i = 0; i <= 20; ++i) {
        printf("F(%u) = %lu\n", i, fibonacci_matrix(i));
    }
    return 0;
}