Re: ExactRootIsolation

*To*: mathgroup at smc.vnet.net*Subject*: [mg71033] [mg70997] Re: [mg70927] ExactRootIsolation*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 6 Nov 2006 02:52:32 -0500 (EST)*References*: <200611021146.GAA15558@smc.vnet.net>

On 2 Nov 2006, at 20:46, dimitris wrote: > Options[Root] > (Information[Evaluate[#1[[1]]]] & ) /@ %; > > {ExactRootIsolation -> False} > "ExactRootIsolation is an option for Root, which specifies whether > exact isolating intervals rather then numeric approximations should be > used to identify algebraic numbers." > Attributes[ExactRootIsolation] = {Protected} > > Can someone provide me more information about this undocumented option > of Root with some examples of the setting ExactRootIsolation ->True? > > Thanks > Root isolation is an algorithm that is used to separate ("Isolate") the roots of a polynomial (real and complex) so that they can be ordered and manipulated as individual objects. One method is an exact algebraic one and the other a validated "numerical" one, which means that although it uses numerical methods the answers are as reliable or "exact" as when you use "exact" root isolation. Basically when the numerical method is used, circles in the complex plane containing individual roots are constructed using using arbitrary precision approximations to the roots. With the exact method open rational rectangles (intervals for real roots) are used. The numerical method is generally faster so it is the default. Other than that the methods differ in the way the non-real roots are ordered: in both cases the real roots come before the non-real ones and are ordered naturally, but the orderings of the non-real roots may be different. If you use a function like Solve with the default options setting ExactRootIsolation->True and then with the setting ExactRootIsolation->False, you will see that Root has an invisible third argument which tells you which method of root isolation was used. I think that's all there is to it. I don't think any of it should concern the average user, unless you are really working on something involving algebraic numbers, when knowing which method of root isolation was used may be important. You also need to know this if you want to convert Mathematica root objects to analogous entities in other computer algebra programs. Andrzej Kozlowski Tokyo, Japan

**References**:**ExactRootIsolation***From:*"dimitris" <dimmechan@yahoo.com>