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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: [mg69958] Re: [mg69875] Program to calculate rational function with imbedded continued fraction
  • From: "Diana Mecum" <diana.mecum at gmail.com>
  • Date: Thu, 28 Sep 2006 06:15:39 -0400 (EDT)

Andrzej,

Yes, I was using two types of notation in my e-mail to you, and I apologize
for the sloppiness. For the purpose of Mathematica, I had set up a function
g[i_] := T^2^i - T
as you defined.

You have solved my problem, and greatly assisted me with my master's thesis
preparation. I now have the fodder I need to continue analyzing and drilling
down on the proofs of the articles I am studying.

Thank you very much for your time, I am very grateful!

Diana M.


On 9/27/06, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>
> 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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