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Re: Root again

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109130] Re: Root again
  • From: Bill Rowe <readnews at sbcglobal.net>
  • Date: Mon, 19 Apr 2010 04:08:31 -0400 (EDT)

On 4/12/10 at 11:01 PM, hemphill at hemphills.net (Scott Hemphill)
wrote:

>Bill Rowe <readnews at sbcglobal.net> writes:
>
>>In[18]:= r = Table[Root[1 + t*#1 + #1^6 &, k], {k, 6}];

>>In[19]:= Plot[r, {t, 0, 10}]

>>The resulting plot indicates half the roots are increasing with
>>increasing t and the other half are decreasing with increasing t.
>>This alone means there will be problems with using NMinimize to get
>>the answer.

>I don't understand why this is the case.

After reading your post, I realize I mis-interpreted what I saw
in the plot. The plot shows two curves one increasing with
increasing t and one decreasing for increasing t with two plot
colors. If all of the roots had real values for t > 1.569, there
should have been 6 curves with 6 colors unless some number of
the curves plotted exactly the same. In that case, only the last
curve color would show. For some rather inane reason, I
interpreted the plot as multiple curves plotting in the same position.

What I should have realized is that any root object that had no
real values for t > 0 would not be plotted by plot.
Consequently, the correct interpretation would have been only
two of the root objects had real values for t > 0. And had I
paid closer attention to the colors I saw I would have realized
only the first two root objects have real values for some values
of t > 0. A better way to have created the plot would have been

r = Table[Root[1 + t*#1 + #1^6 &, k], {k, 6}];
GraphicsGrid[Partition[Plot[#, {t, 0, 10}] & /@ r, 2],
  ImageSize -> Large]

The result clearly shows only the first two root objects have
real values for t > 1.569. The remaining 4 root objects have no
real values for 0 < t <10 and presumably no real values for t > 0.

In any case, it seems to me the first step for exploring these
root objects is to plot them. And given root objects are exact
expressions, it seems to me Maximize/Minimize are more
appropriate than NMaximize/NMinimize. Finally, since the first
root object is decreasing with increasing t and the second root
object is increasing with increasing t, using you cannot get a
good solution by always looking for either minimum or a maximum
in all cases for t > 0.



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