Re: Faster Recursive Fibonacci Numbers

References	<108ce447-10fa-4cf7-84ef-440ee18dbfd4@22g2000prx.googlegroups.com>
From	Ian Kelly <ian.g.kelly@gmail.com>
Date	2011-05-17 10:23 -0600
Subject	Re: Faster Recursive Fibonacci Numbers
Newsgroups	comp.lang.python
Message-ID	<mailman.1678.1305649459.9059.python-list@python.org> (permalink)

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On Tue, May 17, 2011 at 9:50 AM, RJB <rbotting@csusb.edu> wrote:
> I noticed some discussion of recursion..... the trick is to find a
> formula where the arguments are divided, not decremented.
> I've had a "divide-and-conquer" recursion for the Fibonacci numbers
> for a couple of years in C++ but just for fun rewrote it
> in Python.  It was easy.  Enjoy.  And tell me how I can improve it!
>
> def fibo(n):
>        """A Faster recursive Fibonaci function
> Use a formula from Knuth Vol 1 page 80, section 1.2.8:
>           If F[n] is the n'th Fibonaci number then
>                   F[n+m] = F[m]*F[n+1] + F[m-1]*F[n].
>  First set m = n+1
>   F[ 2*n+1 ] = F[n+1]**2 + F[n]*2.
>
>  Then put m = n in Knuth's formula,
>           F[ 2*n ] = F[n]*F[n+1] + F[n-1]* F[n],
>   and replace F[n+1] by F[n]+F[n-1],
>           F[ 2*n ] = F[n]*(F[n] + 2*F[n-1]).
> """
>        if n<=0:
>                return 0
>        elif n<=2:
>                return 1
>        elif n%2==0:
>                half=n//2
>                f1=fibo(half)
>                f2=fibo(half-1)
>                return f1*(f1+2*f2)
>        else:
>                nearhalf=(n-1)//2
>                f1=fibo(nearhalf+1)
>                f2=fibo(nearhalf)
>                return f1*f1 + f2*f2

Thanks for posting!  Actually, it looks like this is the same O(n)
algorithm that rusi posted.  There was also a O(log n) algorithm
discussed that is based on vector math.  You might want to take a
look.

Cheers,
Ian

Thread

Faster Recursive Fibonacci Numbers RJB <rbotting@csusb.edu> - 2011-05-17 08:50 -0700
  Re: Faster Recursive Fibonacci Numbers Ian Kelly <ian.g.kelly@gmail.com> - 2011-05-17 10:23 -0600
  Re: Faster Recursive Fibonacci Numbers rusi <rustompmody@gmail.com> - 2011-05-17 09:25 -0700
  Re: Faster Recursive Fibonacci Numbers rusi <rustompmody@gmail.com> - 2011-05-17 09:36 -0700
    Re: Faster Recursive Fibonacci Numbers geremy condra <debatem1@gmail.com> - 2011-05-17 10:02 -0700
      Re: Faster Recursive Fibonacci Numbers Jussi Piitulainen <jpiitula@ling.helsinki.fi> - 2011-05-17 20:19 +0300
        Re: Faster Recursive Fibonacci Numbers geremy condra <debatem1@gmail.com> - 2011-05-17 11:56 -0700
          Re: Faster Recursive Fibonacci Numbers Wolfram Hinderer <wolfram.hinderer@googlemail.com> - 2011-05-17 14:04 -0700
            Re: Faster Recursive Fibonacci Numbers geremy condra <debatem1@gmail.com> - 2011-05-17 14:59 -0700
            Re: Faster Recursive Fibonacci Numbers rusi <rustompmody@gmail.com> - 2011-05-17 21:43 -0700
      Re: Faster Recursive Fibonacci Numbers Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2011-05-19 22:47 +0000
        Re: Faster Recursive Fibonacci Numbers Chris Angelico <rosuav@gmail.com> - 2011-05-20 09:37 +1000
          Re: Faster Recursive Fibonacci Numbers Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2011-05-20 05:58 +0000
            Re: Faster Recursive Fibonacci Numbers Chris Angelico <rosuav@gmail.com> - 2011-05-20 16:06 +1000
              Re: Faster Recursive Fibonacci Numbers harrismh777 <harrismh777@charter.net> - 2011-05-20 01:26 -0500
                Re: Faster Recursive Fibonacci Numbers Chris Angelico <rosuav@gmail.com> - 2011-05-20 16:54 +1000
                Re: Faster Recursive Fibonacci Numbers Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2011-05-20 17:07 +0000
                Re: Faster Recursive Fibonacci Numbers geremy condra <debatem1@gmail.com> - 2011-05-20 11:43 -0700
                Re: Faster Recursive Fibonacci Numbers Chris Angelico <rosuav@gmail.com> - 2011-05-21 10:44 +1000
              Re: Faster Recursive Fibonacci Numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2011-05-21 20:49 +1200
            Re: Faster Recursive Fibonacci Numbers Ian Kelly <ian.g.kelly@gmail.com> - 2011-05-20 01:26 -0600
            Re: Faster Recursive Fibonacci Numbers Ian Kelly <ian.g.kelly@gmail.com> - 2011-05-20 01:32 -0600
              Re: Faster Recursive Fibonacci Numbers SigmundV <sigmundv@gmail.com> - 2011-05-20 03:53 -0700
        Re: Faster Recursive Fibonacci Numbers geremy condra <debatem1@gmail.com> - 2011-05-19 17:03 -0700
  Re: Faster Recursive Fibonacci Numbers Raymond Hettinger <python@rcn.com> - 2011-05-24 16:44 -0700

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