[karamazov]

Or use the closed-form solution.

[jacobolus]

The "closed-form solution" is slower than standard method. It just uses arbitrary-precision fractional arithmetic (square root, exponentiation, division) instead of arbitrary-precision integer arithmetic (exponentiation of a 2x2 integer matrix).

[singhrac]

The best solution is to use the O(log n) time exponentiation of a matrix, which is fast enough to be constant.