Q about Interval arithmetic
- To: mathgroup@smc.vnet.net
- Subject: [mg10465] Q about Interval arithmetic
- From: Ersek_Ted%PAX1A@mr.nawcad.navy.mil
- Date: Fri, 16 Jan 1998 04:34:50 -0500
In the following lines Interval arithmetic gives an Interval that
contains the range of poly[x] for the domain = Interval[{ -1.0, 0.5
}].
In[1]:=
poly[x_] := 15*x^7 + 15*x^2 - 13*x + 15;
In[2]:=
poly[ Interval [{ -1.0, 0.5 }] ]
Out[2]=
Interval[{-6.5, 43.1172}]
Roots[eqn, vars] can not find the roots of poly'[x] in closed form.
This is probably because it isn't possible to do so. Hence I would
figure it's not possible to determine the exact range of poly[x] over
the given domain.
However, the exact range is a subset of the Interval in Out[2]. As I
understand it this should always be the case.
In the lines below I used Plot and FindMimimum to convince myself that a
smaller Interval containing the exact range and nearly equal to the
exact range is:
Interval[{ 12.221, 32.082 }]
Questions:
Is it possible to improve on the built in algorithm (developed by
Wolfram Research)?
Is there a known algorithm that can be used to obtain the smaller (
preferred ) Interval?
I am not looking for an approach that uses Numerical methods as I did
below. I want is something that provides gaurenteed results, and works
on high order polynomials.
In[3]:=
Plot[poly[x],{x,-1.0,0.5}]
In[4]:=
FindMinimum[poly[x],{x,0.41}]
Out[4]=
{12.2202, {x -> 0.4154}}
In[5]:=
FindMinimum[ -poly[x], {x,-0.8} ] //MapAt[ Minus, # ,1]&
Out[5]=
{32.0813, {x -> -0.84552}}
Ted Ersek
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