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40 lines
865 B
Python
40 lines
865 B
Python
#!/usr/bin/env python3
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"""
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This program calculates the nth Fibonacci number in O(log(n)).
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It's possible to calculate F(1_000_000) in less than a second.
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"""
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from __future__ import annotations
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import sys
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def fibonacci(n: int) -> int:
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"""
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return F(n)
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>>> [fibonacci(i) for i in range(13)]
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[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
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"""
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if n < 0:
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raise ValueError("Negative arguments are not supported")
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return _fib(n)[0]
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# returns (F(n), F(n-1))
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def _fib(n: int) -> tuple[int, int]:
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if n == 0: # (F(0), F(1))
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return (0, 1)
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# F(2n) = F(n)[2F(n+1) − F(n)]
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# F(2n+1) = F(n+1)^2+F(n)^2
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a, b = _fib(n // 2)
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c = a * (b * 2 - a)
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d = a * a + b * b
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return (d, c + d) if n % 2 else (c, d)
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if __name__ == "__main__":
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n = int(sys.argv[1])
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print(f"fibonacci({n}) is {fibonacci(n)}")
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