Re: Finding an n-variable polynomial's root.
- From: "David L. Johnson" <david.johnson@xxxxxxxxxxxxxxxxx>
- Date: Wed, 11 May 2005 08:48:19 -0400
On Wed, 11 May 2005 04:14:40 -0700, himog wrote:
>
> Robert Israel wrote:
>> In article <1115679951.531724.180120@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>> <himog@xxxxxxxxx> wrote:
>> >Say I have a real polynomial in n real variables (on the order of
> from,
>> >say, 100 variables up to whatever, say 50,000) which has only
>> >one(unknown) root in a domain which can be mapped one-to one onto a
>> >closed(with boundary) n-dimensional cube. What methods should I
>> >consider for finding that root if it may be anywhere in the known
>> >domain or on it's boundary?
>>
>> It's very exceptional for a real multivariate polynomial to have an
>> isolated real root. The Jacobian of the polynomial would have to be
>> 0 at the root. I think it may be easiest to find the root by solving
>> the n equations dP/dx_i = 0 (numerically).
>>
>> Robert Israel israel@xxxxxxxxxxx
>> Department of Mathematics http://www.math.ubc.ca/~israel
>> University of British Columbia Vancouver, BC, Canada
>
> Yes, the root is likely not globally isolated. My question pertains to
> when there is exactly one root on the boundary of a given n-box, and
> the polynomial is most easily defined within that closed box.
It's more than "not globally isolated". In general, a polynomial in n
variables has solution sets to the equation f=0 that are
(n-1)-dimensional. Any smaller sets are special cases, and unstable in
that a slight change of the polynomial will turn these sets into
(n-1)-dimensional sets again.
In addition, finding the solutions of such an equation is in general very
difficult. The entire subject of algebraic geometry, a difficult and
subtle (but beautiful) branch of mathematics, has been developed around
this one question. Your problem is in addition complicated by the
restriction to real variables; real algebraic geometry is an even more
subtle subject than when using complex numbers (or any algebraically
closed field).
If you do know, somehow, that there is an isolated solution to your
problem at some point in that box, then try Robert's suggestion. Solving
n equations in n unknowns will, at least in general, yield discrete
solutions. However, there is no simple method to solve such systems. The
hope is, though, that since your problem would have these very unusual
properties, there may well be some method that works.
--
David L. Johnson
__o | A mathematician is a machine for turning coffee into theorems.
_`\(,_ | -- Paul Erdos
(_)/ (_) |
.
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