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Re: Root Finding Methods Gaurenteed to Find All Root Between (xmin, xmax)

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  • Subject: [mg112769] Re: Root Finding Methods Gaurenteed to Find All Root Between (xmin, xmax)
  • From: "Ted Ersek" <ersekt at md.metrocast.net>
  • Date: Thu, 30 Sep 2010 04:49:41 -0400 (EDT)

It was mentioned that some of the methods to find all roots in a certain
region require interval analysis. 
However, Interval arithmetic in Mathematica is limited. First of all
Mathematica only knows how to evaluate 
elementary function and simple functions such as (Abs, Min) when given an
Interval as an argument.  
So for example Mathematica does nothing with   Gamma[ Interval[ {2,3} ] ]
even though Gamma[x] is continuous and monotonic over this interval with
Gamma[2] == 1; Gamma[3] ==2  

Besides that Mathematica performs calculations as if each instance of
Interval[{xmin_, xmax_}] 
is a value between (xmin, xmax) that is unrelated to any other instance of
Interval[{xmin_, xmax_}]. 
So I present the following as a worst case example of what this can give.

In[1]:= poly=Function[x, Evaluate[ChebyshevT[15,x]]]
Out[1]= Function[x,-15 x+560 x^3-6048 x^5+28800 x^7-70400 x^9+92160
x^11-61440 x^13+16384 x^15]

In[2]:= poly[Interval[{-1,1}]]
Out[2]= Interval[{-275807,275807}]

The above interval isn't very useful when we can see from  Plot[ poly[x],
{x,-1,1} ] 
that  poly[x] oscillates between (-1,1) over this interval.
For a general function we can approximate the interval we really want using

NMinimize[ { f[x], xmin<x<xmax}, x ],  NMaximize[ { f[x], xmin<x<xmax}, x ].

However, that's about as difficult as finding a root of f[x].
Actually that may be more difficult than finding a root of f[x].

Ted Ersek




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