What difficulties can arise when calculating very large Fibonacci numbers using recursion, for example, a million?

Calculating very large Fibonacci numbers recursively, for example, for a million, encounters several serious problems:

• Exponential time complexity: the classic recursive formula causes repeated calculations of the same values, leading to exponential growth in computation time.
• Stack overflow: the recursion depth for a million is too large, which will cause a stack overflow and crash the program.
• Memory: storing intermediate results in recursive calls requires a lot of memory.

To solve these problems, iterative methods or memoization are used. For example, an iterative approach in C++:

#include <iostream>
#include <vector>

// Calculate the n-th Fibonacci number iteratively
unsigned long long fib(unsigned int n) {
    if (n <= 1) return n;
    unsigned long long a = 0, b = 1, c;
    for (unsigned int i = 2; i <= n; ++i) {
        c = a + b;
        a = b;
        b = c;
    }
    return b;
}

int main() {
    unsigned int n = 50; // For a million, a different approach is needed due to number size
    std::cout << "Fib(" << n << ") = " << fib(n) << std::endl;
    return 0;
}

For very large n, algorithms with fast matrix exponentiation or Binet's formula with arbitrary-precision arithmetic are used.