Re: bug in Root in Version 3?

• To: mathgroup at smc.vnet.net
• Subject: [mg23995] Re: bug in Root in Version 3?
• From: Adam Strzebonski <adams at wolfram.com>
• Date: Mon, 19 Jun 2000 01:45:37 -0400 (EDT)
• Organization: Wolfram Research, Inc.
• References: <8i9q4c\$2m7@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In version 3 the standard root ordering applied only to
inert Root objects, i.e. to roots of _irreducible_
polynomials. Roots of reducible polynomials were ordered
first by the order of irreducible factors, and only then
by the standard root ordering of roots of an irreducible
factor.

In version 4 this has been changed. Now the standard root
ordering applies to all polynomials. This is a more costly
solution: roots of different factors need to be isolated
and compared, however it makes Root funtions containing
parameters more useful. In version 4

Root[f[a, #]&, i]

is guaranteed to give the i-th real root of f[a, #]&
for all values of a for which f[a, #]& has at least
i real roots (counted with multiplicities). Version 4 uses
Root functions containing parameters to describe solution
sets of inequality systems, and for this purpose the above
property is very important. With version 3 root ordering
the property might not be satisfied for values of a for
which f[a, #]& factors.

In your example

In[5]:= FactorList[ 2 + 12#1 - 14#1^3 ]

2
Out[5]= {{-2, 1}, {-1 + #1, 1}, {1 + 7 #1 + 7 #1 , 1}}

so with version 3 root ordering the root of #1-1 comes first,
and then come the roots of the quadratic factor in the
increasing order. Version 4 finds roots of both factors and
orders them in the increasing order.

Best Regards,

Adam Strzebonski
Wolfram Research

Gianluca Gorni wrote:
>
> Hello!
>
> I wrote a program that runs fine in version 4. When I tried it in
> version 3 I saw different results. I traced the problem to
> the function Root. It seems that V.3 Root does not necessarily
> order real roots in increasing order.
>
> Example:
>
> Table[Root[2 + 12#1 - 14#1^3 &, n], {n, 3}] // N
>
> gives
>
> {1., -0.827327, -0.172673}
>
> SetOptions[Root, ExactRootIsolation -> True] does not seem to
> affect the result.
>
> \$Version:  "Solaris 3.0 (August 21, 1997)"
>
>           Gianluca Gorni

```

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