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MathGroup Archive 2006

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Re: Program to calculate rational function with imbedded continued fraction

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69936] Re: [mg69875] Program to calculate rational function with imbedded continued fraction
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 28 Sep 2006 06:14:23 -0400 (EDT)

What do you mean by:

"do I plug (e(1) + 1)(T^2e(1)) in for k?." Are you using Mathematica  
syntax or something else? All your messages seem to contain weird  
mixture of Mathematica syntax, such as T^2^i and expressions which  
are syntactically impermissible in Mathematica, such as [1] or [i],  
etc. So what exactly did you "plug in" to get your error message?  
When I run this program I do not get any error messages, but I use  
correct Mathematica syntax. For example, let's define:

g[i_] := T^2^i - T

and take the following finite continued fraction:

e[1] = {0, g[1], g[2], g[1], g[3], g[1], g[2], g[1], g[4], g[1], g 
[2], g[1], g[
     3], g[1], g[2], g[1], g[5]};

Let's convert it and to a rational function and Simplify (very  
important!)

k = Simplify[FromContinuedFraction[e[1]]]

you will get a pretty long expression that I have decided to skip.  
Now do:

F[Numerator[k], Denominator[k]]

and you will get a continued fraction in terms of T; no error  
messages. This example works pretty fast but if your e[1] is  
extremely long I can't of course guarantee Mathematica will manage it.


Andrzej Kozlowski

On 27 Sep 2006, at 14:18, Diana Mecum wrote:

> I apologize if my example was not clear. I was trying to keep it  
> simple.
>
> I have an exponential function e(z), where z might equal 1, which  
> is a continued fraction defined as follows:
>
> e(1) = [0, [1], [2], [1], [3], [1], [2], [1], [4], [1], [2], [1],  
> [3], [1], [2], [1], [5], ...]
>
> [i] = T^2^i - T in characteristic 2.
>
> I am trying to learn how to evaluate
>
> (xe(1) + y)(ze(1) + w), where x, y, z, and w are polynomials in T,  
> and xw - yz != 0.
>
> So, for any given example, such as
>
> (e(1) + 1)(T^2e(1), I am trying to determine what the continued  
> fraction for it would be.
>
> I see your program. Thanks for forwarding it.
>
> To implement it, do I plug (e(1) + 1)(T^2e(1) in for k?
>
> When I run the routine, I get an error saying that the tags in In 
> [50] and In[51] are protected.
>
> Thanks for your time,
>
> Diana M.
>
>
>
>
> On 9/26/06, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: *This  
> On 26 Sep 2006, at 18:21, Diana wrote:
>
> > Hello all,
> >
> > Would someone be able to help me write a program to calculate the
> > rational function of (2x + 3)/(2x + 4), for example?
>
> I have once already posted such a program and so did Daniel
> Lichtblau. (Search the archive for the subject "How to do Continued
> fraction of polynomials" and compare our posts. (I can't remember any
> more what the difference between them was).
>
> Here is my version:
>
> F[f_, g_] :=
>       If[PolynomialReduce[f, g][[1, 1]] =!= 0, PolynomialReduce[f,
> g][[
>          1, 1]] + F[PolynomialReduce[f,
>       g][[2]], g], Module[{u = g, v = f, p, ls}, ls = Flatten[Last[
>        Reap[While[u =!= 0, p = PolynomialReduce[u, v]; u = v; v =
> p[[2]]; Sow[
>      p[[1]]]]]]]; 1/Fold[Function[{x, y}, y + 1/x], Infinity,
> Reverse[ls]]]]
>
> In the case of your example above you need to evaluate:
>
> In[50]:=
> k=(2x+3)/(2x+4);
>
> In[51]:=
> F[Numerator[k],Denominator[k]]
>
> Out[51]=
> 1 + 1/(-2*x - 4)
>
>
>
>
> >
> > Also, (tx + 2t)(3t^2x + t), for example.
>
>
> What exactly do you mean? This does not make sense to me; is that a
> quotient? If so why did't you cancel the t's? I think the case of
> rational functions in several variables will be much harder, and I
> would have to think more about it, for which I do not have the time
> at this moment, and in any case it might not be what you are asking  
> for.
>
>
> >
> > x is actually an infinite continued fraction, and I am trying to
> > figure
> > out how to divide one infinite continued fraction by another.
> >
> > (t [0, 1, 2, 1, 3, 1, 2, 1, 4, ...] + 2t)/(3t^2 [0, 1, 2, 1, 3,  
> 1, 2,
> > 1, 4, ...] +t)
> >
>
>
> I think this can be done combining the built in function
> FromContinuedFraction with the function F defined above.
>
> Andrzej Kozlowski
>
> Tokyo, Japan
>


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