Re: Tail recursion and local functions

• To: mathgroup at smc.vnet.net
• Subject: [mg45720] Re: Tail recursion and local functions
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 21 Jan 2004 04:55:03 -0500 (EST)
• Organization: The University of Western Australia
• References: <buiv4p\$qtn\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <buiv4p\$qtn\$1 at smc.vnet.net>,
Matthew McDonnell <kebl1405 at herald.ox.ac.uk> wrote:

> I have recently been learning a bit of functional programming
> using OCaml and was trying to translate tail recursion using auxillary
> local functions into Mathematica.

Not sure why you really want or need these in Mathematica.

> My question is whether Mathematica allows local functions to be
> defined within functions, or just local variables?

Mathematica allows local functions to be defined within functions.

> More explanation follows:
>
> Consider the function to compute the factorial of n i.e. fact[n]=n!
> (not concerned with coping with negative n for the moment, define n! =1
> for negative n)
>
> Standard recursive solution:
> ===========================
> fact1[n_:Integer] := If[n <= 0, 1, n fact1[n - 1]];
>
> Works, but hits standard recursion limit of 256 for n>253.

You can get around this as follows:

fact1[n_:Integer] := Block[{\$RecursionLimit = Infinity},
If[n <= 0, 1, n fact1[n - 1]]]

Note that you can write

\$RecursionLimit = Infinity;

fac =  If[# == 1, 1, # #0[# - 1]] &;

> Tail recursive solution:
> =======================
> auxFact[n_:Integer, acc_:Integer] :=
> 		If[n <= 0, acc, auxFact[n - 1, n acc]];
> fact2[n_:Integer] := auxFact[n, 1];
>
> Works, hits the Iteration limit of 4096 for n>2046

Similarly, you can get around this by setting

\$IterationLimit = Infinity

> Tail recursive solution with local auxillary function:
> =====================================================
> fact3[n_:Integer] := Module[
>     {aux = Function[{i, acc}, If[i <= 0, 1, aux[i - 1, i acc]]]},
>     aux[n, 1]]
>
> Doesn't work eg fact3[100] gives aux[99,100]

Because the code is wrong. It should read

fact3[n_:Integer] := Module[
{aux = Function[{i, acc}, If[i <= 0, 1, i aux[i - 1, acc]]]},
aux[n, 1]]

> first class variables (eg able to be passed and returned from functions),
> is this correct?  Or am I just going the wrong way about it?

Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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