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Re: How to do Continued fraction of polynomials

• To: mathgroup at smc.vnet.net
• Subject: [mg53206] Re: [mg53052] How to do Continued fraction of polynomials
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 28 Dec 2004 23:12:43 -0500 (EST)
• References: <200412220952.EAA04405@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Milind Gupta wrote:
> How can we do a continued fraction expansion in Mathematica.
> 
>    Suppose we have
>              x^3 + 2x
>              ------------
>               x^2 + 1
> 
> and we want:
> 
>      x  +       1
>              -------------
>               x   +    1
>                         ----
>                           x
> 
>      Can we get this by using some function or some technique in Mathematica??
> 
> Regards,

One method is to iterate taking a quotient and remainder, keeping the 
quotients and dividing previous remainders by new ones.

polynomialContinuedFraction[p1_,p2_,x_] := Module[{quot,rem,res},
   Reap[NestWhileList[
     ({quot,rem} = Developer`PolynomialDivision[#[[1]],#[[2]],x];
     Sow[quot,res];
     {#[[2]],rem})&,
     {p1,p2},
     !Developer`ZeroQ[#[[2]]]&]][[2,1]]
   ]

For your example:

In[12]:= convergents = polynomialContinuedFraction[x^3+2*x, x^2+1, x]
Out[12]= {x, x, x}

To change to the customary format you can use

pcfForm[{l1_}] := l1
pcfForm[ll_List] := First[ll] + 1/pcfForm[Rest[ll]]

In[16]:= InputForm[pcfForm[convergents]]
Out[16]//InputForm= x + (x^(-1) + x)^(-1)


Daniel Lichtblau
Wolfram Research

• References:
  • How to do Continued fraction of polynomials
    • From: Milind Gupta <milind.gupta@gmail.com>