# [Newbies] Re: Beginners Digest, Vol 18, Issue 2

Bert Freudenberg bert at freudenbergs.de
Thu Oct 4 07:23:01 UTC 2007

```On Oct 4, 2007, at 6:41 , Gregory Hinton wrote:

>
> On Oct 3, 2007, at 7:31 AM, Michael Davies wrote:
>
>> If you're
>> interested in the numbers rather than the technique, there's a couple
>> of alternative approaches to speed it up, iteration and matrix
>> manipulation.
>
> Actually, in the case of the Fibonacci sequence, the best way to
> speed it up is to use the closed-form formula:
>
> nthFibonacci
> 	|phi1 phi2 term1 term2|
> 	phi1 := (1 + 5 sqrt) / 2.
> 	phi2 := (1 - 5 sqrt) / 2.
> 	term1 := phi1 raisedTo: self.
> 	term2 := phi2 raisedTo: self.
> 	^((term1 - term2) / (5 sqrt)) rounded.
>
> Sorry, I realize the OP requested a recursive solution, but I
> couldn't resist. :)

This is inaccurate for many numbers and fails for most:

"100 nthFibonacci" gives 354224848179262259200 (actually
354224848179261915075)

"1500 nthFibonacci" error

whereas the iterative version gives exact results and works as long
as we can store the result in memory. Iterative fib of 1500 takes 1.3
ms on my machine.

- Bert -

```