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Re: Recursive algorithm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97852] Re: [mg97835] Recursive algorithm
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Tue, 24 Mar 2009 05:28:33 -0500 (EST)
  • References: <200903230903.EAA27090@smc.vnet.net>

athanase wrote:
> hello all,
> 
>        i am having headaches trying to produce this recursive
> algorithm in mathematica:
> 
> the algorithm devides a reduced fraction r where r > 1  into n steps
> in the form (k+1)/k
> 
> 1)	try the largest step (k+1)/k (say s) that will fit in r;
> 2)	find out how to divide r/s into n-1 steps;
> 3)	try the next biggest step, etc;
> 4)	until the first step is small enough that n of them are smaller
> than r, then you are done.
> 
> 
> so for r=8 and n=3
> 
> the result is
> 
> (2/1,2/1,2/1)
> 
> and for r=7/5, n=2
> 
> the result is
> 
> (4/3,21/20),(6/5,7/6)
> 
> 
> i have found this algorithm written in another system but attempt to
> translate it fails for n>2
> 
> below is the code;
> 
> sincere thanks for considering this problem,
> 
>        athanase
> 
> 
> spsubdiv := proc(r:rational,n:integer)
> local i,j,l,s;
> if n=1
> then
>    if numer(r)=denom(r)+1
>    then [r]
>    else ( NULL )
>    fi;
> else
>    s := NULL;
>    for i from floor(1/(r-1))+1 while (1+1/i)^n >= r do
>       l := [spsubdiv( r/(1+1/i), n-1 )];
>       for j to nops(l) do
>          if op(1,op(j,l)) <= (1+1/i)
>          then s := s, [(1+1/i),op(op(j,l))];
>          fi
>       od;
>    od;
>    s;
> fi;
> end:

One way (probably among many possibilities):

spsubdiv[2,1] = {fractionSequence[2]};

spsubdiv[r_Rational,1]  /; Numerator[r]==Denominator[r]+1 :=
   {fractionSequence[r]}

spsubdiv[_,1] = {};

spsubdiv[r_Rational|r_Integer, n_Integer /; n>1] :=
   Union[Map[Sort,Flatten[Table[With[{j=1+1/i},
     Map[Join[fractionSequence[j],#]&,spsubdiv[r/j,n-1]]]
     ,{i,Floor[1/(r-1)]+1,Floor[1/(r^(1/n)-1)]}]]]]

Examples:

In[64]:= InputForm[spsubdiv[8,3]]
Out[64]//InputForm= {fractionSequence[2, 2, 2]}

In[65]:= InputForm[spsubdiv[7/5,2]]
Out[65]//InputForm= {fractionSequence[4/3, 21/20],
   fractionSequence[6/5, 7/6]}

In[66]:= InputForm[spsubdiv[7/5,3]]
Out[66]//InputForm=
{fractionSequence[441/440, 22/21, 4/3], fractionSequence[231/230, 23/22,
   4/3], fractionSequence[161/160, 24/23, 4/3],
  fractionSequence[126/125, 25/24, 4/3], fractionSequence[126/125, 10/9, 
5/4],
  fractionSequence[105/104, 26/25, 4/3], fractionSequence[91/90, 27/26, 
4/3],
  fractionSequence[81/80, 28/27, 4/3], fractionSequence[63/62, 31/30, 4/3],
  fractionSequence[56/55, 33/32, 4/3], fractionSequence[56/55, 11/10, 5/4],
  fractionSequence[51/50, 35/34, 4/3], fractionSequence[49/48, 36/35, 4/3],
  fractionSequence[49/48, 8/7, 6/5], fractionSequence[42/41, 41/40, 4/3],
  fractionSequence[36/35, 7/6, 7/6], fractionSequence[28/27, 9/8, 6/5],
  fractionSequence[26/25, 14/13, 5/4], fractionSequence[21/20, 16/15, 5/4],
  fractionSequence[21/20, 10/9, 6/5], fractionSequence[21/20, 8/7, 7/6],
  fractionSequence[16/15, 9/8, 7/6], fractionSequence[14/13, 13/12, 6/5],
  fractionSequence[12/11, 11/10, 7/6]}

As per comments at URL below, if you do large examples and speed becomes 
an issue, you might want to memoize values using the construct

spsubdiv[r_Rational|r_Integer, n_Integer /; n>1] :=
   spsubdiv[r,n] = ...

https://home.comcast.net/~dcanright/super/app.htm

This makes around a factor of three speed difference for

In[77]:= Timing[Length[spsubdiv[2,6]]]
Out[77]= {54.0718, 49513}

That is, with the alteration

spsubdiv[r_Rational|r_Integer, n_Integer /; n>1] :=
   spsubdiv[r,n] =
   Union[Map[Sort,Flatten[Table[With[{j=1+1/i},
     Map[Join[fractionSequence[j],#]&,spsubdiv[r/j,n-1]]]
     ,{i,Floor[1/(r-1)]+1,Floor[1/(r^(1/n)-1)]}]]]]

this instead is around 18 seconds.

I will point out that this version of the code will not handle 
spsubdiv[2,7] because iterators become too large.

Daniel Lichtblau
Wolfram Research


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