Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
1998
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 1998

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Series of a Root object

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14296] Re: Series of a Root object
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 13 Oct 1998 01:21:14 -0400
  • Organization: University of Western Australia
  • References: <6vf5ol$dht@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

RENZONI_FERRUCCIO wrote:

> I would like to know why the following program, meant to expand the Root
> object r in serie of g and s up to the second order, fails to give an
> answer.

I'm not completely sure but I think the problem is that the there does
not exist a Taylor expansion for this particular root.  To give you a
trivial example (which is actually closely related to your specific
example), consider the following:

In[1]:= (roots = Solve[16 g^2 - 8 g + 16 x^2 + 8 x + 1 == 0, x]) Out[1]=
                                  2
       -1 - 2 Sqrt[2] Sqrt[g - 2 g ] {{x ->
-----------------------------}, 
                     4
 
                                   2
        -1 + 2 Sqrt[2] Sqrt[g - 2 g ]
  {x -> -----------------------------}}
                      4

If you now expand these roots into a series you obtain

In[2]:= (x/.roots) + O[g]^2
Out[2]=
                    3/2
   1    Sqrt[g]    g            2
{-(-) - ------- + ------- + O[g] , 
   4    Sqrt[2]   Sqrt[2]
 
                     3/2
    1    Sqrt[g]    g            2
  -(-) + ------- - ------- + O[g] }
    4    Sqrt[2]   Sqrt[2]

It is clear that this series is NOT analytic (since it involves Sqrt[g])
and no Taylor series exists.

Note that you can perhaps see more clearly what is happening using Plot:

In[3]:= Plot[Evaluate[x /. roots], {g, -1, 1}, 
   PlotStyle -> Table[Hue[i], {i, 0, 1, 1/Length[roots]}]];

You will see why no Taylor series exists at g=0.

For your root, 

> r = Root[-8*b*g^4*s^2 + 8*b^2*g^4*s^2 - 4*g^4*#1 + 8*b*g^4*#1 +
>     4*b^3*g^2*s^2*#1 - 16*g^4*s^2*#1 + 24*b*g^4*s^2*#1 + 16*b*g^4*s^4*#1 -
>     2*b*g^2*#1^2 + 6*b^2*g^2*#1^2 + 16*g^4*#1^2 + 20*b^2*g^2*s^2*#1^2 +
>     32*g^4*s^2*#1^2 + 16*g^4*s^4*#1^2 + b^3*#1^3 - 4*g^2*#1^3 +
>     22*b*g^2*#1^3 + 36*b*g^2*s^2*#1^3 + 5*b^2*#1^4 + 20*g^2*#1^4 +
>     20*g^2*s^2*#1^4 + 8*b*#1^5 + 4*#1^6 & , 3];

if you extract the original polynomial, 

poly = (-8*b*g^4*s^2 + 8*b^2*g^4*s^2 - 4*g^4*#1 + 8*b*g^4*#1 +
     4*b^3*g^2*s^2*#1 - 16*g^4*s^2*#1 + 24*b*g^4*s^2*#1 +
16*b*g^4*s^4*#1 -
     2*b*g^2*#1^2 + 6*b^2*g^2*#1^2 + 16*g^4*#1^2 + 20*b^2*g^2*s^2*#1^2 +
     32*g^4*s^2*#1^2 + 16*g^4*s^4*#1^2 + b^3*#1^3 - 4*g^2*#1^3 +
     22*b*g^2*#1^3 + 36*b*g^2*s^2*#1^3 + 5*b^2*#1^4 + 20*g^2*#1^4 +
     20*g^2*s^2*#1^4 + 8*b*#1^5 + 4*#1^6 &)[x];

and then visualize the roots for specific b, s, and g, you may be able
to more clearly see what is going on.

Cheers,
	Paul

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at physics.uwa.edu.au  AUSTRALIA                       
http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________


  • Prev by Date: Re: Images in Mathematica
  • Next by Date: I Want More Digits in TraditionalForm
  • Previous by thread: Series of a Root object
  • Next by thread: PROBLEM ENTERING 2D USING CTRL