       Re: The most efficient Fibonacci algorithim?

• To: mathgroup at smc.vnet.net
• Subject: [mg23383] Re: [mg23357] The most efficient Fibonacci algorithim?
• From: Jack Goldberg <jackgold at math.lsa.umich.edu>
• Date: Fri, 5 May 2000 02:07:16 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```FYI, Roman Maeder's new book on Mathematica and Computer Programming discusses
this problam in a bit more detail than the book you quoted.  It appears
that Maeder's method is faster than the built-in one.  I too was started
by this...

Unfortunately, the method as it stands takes advantage of special features
or the companion matrix to the difference equation  f[n+1] = f[n]+f[n-1]

{{1,1},{1,0}]

namely, its symmetry.  Generally companion matrices are not symmetric so
the technique does not generalize in any obvious way.

On Thu, 4 May 2000 zeno at magicnet.net wrote:

> On page 128 of the book "The Mathematica Programmer" by Roman Maeder (the
> Mathematic 2.2 edition) is this Mathematica program to compute Fibonacci
> numbers...
>
> fibj[n_]:=
> Module[{r11=1,r12=0,r22=1,digits=IntegerDigits[n-1,2],i,t},
> Do[If[digits[[i]]==1,
> {r11,r22}={r11(r11+2r12),r12(r11+r22)};
> r12=r11-r22,
> t=r12(r11+r22);
> {r11,r12}={r11(r11+2r12)-t,t};
> r22=r11-r12],
> {i,Length[digits]-1}];
> If[digits[[-1]]==1,
> r11(r11+2r12),
> r11(r11+r22)-(-1)^((n-1)/2)]]
>
> The book says this is the most efficient one of a few mentioned in the book.
> Does anyone know of any other programs that are faster? This one really
> screams...I am curious if anyone has done anything even better.
>

```

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