*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

**Follow-Ups**:**Re: Q about Interval arithmetic***From:*Daniel Lichtblau <danl@wolfram.com>