Re: idiom for recurrence relations

• To: mathgroup at smc.vnet.net
• Subject: [mg27081] Re: [mg26714] idiom for recurrence relations
• From: spiralcenter314 at my-deja.com
• Date: Sat, 3 Feb 2001 04:59:20 -0500 (EST)
• References: <200101180557.AAA16636@smc.vnet.net> <948rtc\$kli@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <948rtc\$kli at smc.vnet.net>,
Daniel Lichtblau <danl at wolfram.com> wrote:
> maharrix at my-deja.com wrote:
> >
> > I have a problem which is mostly esthetic, but could have efficiency
> > repercussions. First, the general question: What is the best way to
> > implement recurrence relations (e.g. Fibonacci[n])?
> >
> > The obvious and simplest way is to define base cases and an
inductive
> > case (I'll use Fibonnaci as the generic example:
> >
> >   F[0]:=0; F[1]:=1; F[n_]:=F[n-1]+F[n-2]
> >
> > ). This can be sped up using memoization F[n_]:=F[n]=... Great, but
this
> > fills up internal memory (nothing can get garbage collected since
all
> > the computed values must hang around forever; this can be bad when
you
> > have lots of recurrence relations, graphics and a long session).
> >
> > So the next solution would be to implement it as an imperative
program
> > keeping only the necessary values to compute the next in the
sequence
> > (e.g.
> >
> >   While[i<=n,temp=f2; f2=f2+f1; f1=temp;i++]
> >
> > ). Again, great, but  the Mathematica docs repeatedly say that list
operations
> > are much faster than imperative ones.
> >
> > So my solution is to use Nest:
> >
> > F[n_] := First@Nest[{#[[2]], #[[2]]+#[[1]]} &, {0, 1}, n]];
> >
> > Is this the best (fastest, simplest, most appropriate, or most
memory
> > efficient, etc.) way to do it? It seems somehow ugly (too many [[,
#,
> > etc.) but it is shorter in number of characters to encode it (not
what I
> > would usually consider highest on my list of things to optimize (but
> > that's not to say terribly low either))
> >
> > The whole point is to be able to create recurrence relations easily
in a
> > single function efficiently. I'd also like to create them as lambdda
> > functions (as above, as Function[pair, {pair[[2]],pair[[2]]+pair
[[1]]}]
> > )
> >
> > So, how do the above rate and what are the reasonable alternatives?
> >
> > Thanks for any help,
> > Mitch
> >
> > Sent via Deja.com
> > http://www.deja.com/
>
> One good way might be via matrix powers. Below I show several
variations
> of the Fibonacci example, including one that uses such powers.
>
> fib1[0]:=0; fib1[1]:=1; fib1[n_]:=fib1[n-1]+fib1[n-2];
>
> fib2[0]=0; fib2[1]=1; fib2[n_]:=fib2[n]=fib2[n-1]+fib2[n-2];
>
> fib3[0]=0; fib3[1]=1;
> fib3[n_] := Module[{f1=1,f2=1,temp},
> 	Do[temp=f2; f2=f2+f1; f1=temp, {n-2}];
> 	f2]
>
> fib4[n_] := First@Nest[{#[[2]], #[[2]]+#[[1]]} &, {0, 1}, n];
>
> fib5[n_] := MatrixPower[{{1,1},{1,0}},n-1][[1,1]]
>
> fib6[n_] := Fibonacci[n]
>
> \$RecursionLimit=Infinity;
>
> The first two are basically useless for serious work. First is
> exponential in speed, second is O(n^2) (see below for reason) but
seems
> to have recursion stack problems at modest values.
>
> In[9]:= Timing[fib1[25];]
> Out[9]= {7.33 Second, Null}
>
> In[10]:= Timing[fib2[2000];]
> Out[10]= {0.41 Second, Null}
>
> Third and fourth are O(n^2) algorithms. One factor is for the #steps,
> the other is for adding numbers with that many bits (which is what we
do
> because fib[n]~GoldenRatio^n has O(n) bits).
>
> In[12]:= Timing[fib3[100000];]
> Out[12]= {20.04 Second, Null}
>
> In[13]:= Timing[fib4[100000];]
> Out[13]= {20.42 Second, Null}
>
> Fifth method uses a divide-and-conquer approach in MatrixPower. So it
> will be O(log(n)*M(n)) where M(n) is cost of multiplying a pair of n-
bit
> integers. The built-in Fibonacci code does something similar, I
believe.
>
> In[14]:= Timing[fib5[100000];]
> Out[14]= {0.32 Second, Null}
>
> In[15]:= Timing[fib6[100000];]
> Out[15]= {0.23 Second, Null}
>
> Daniel Lichtblau
> Wolfram Research

I have done some research on the Golden ratio, though not probably as
extensive as you.

But this may be pertinent,  1 / 89 = .011235955...
This has all the fibonacci in it. Which is:

0/(10^1)+
1/(10^2)+
1/(10^3)+
2/(10^4)+
3/(10^5)+
5/(10^6)+
etc...

Let me know if this is helpful !

>

Sent via Deja.com
http://www.deja.com/

```

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