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.