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Re: How to eliminate noises? A better way perhaps.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg122755] Re: How to eliminate noises? A better way perhaps.
*From*: Noqsi <noqsiaerospace at gmail.com>
*Date*: Thu, 10 Nov 2011 06:49:18 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201111021121.GAA03503@smc.vnet.net> <j8tl5t$f3t$1@smc.vnet.net>
On Nov 9, 4:32 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> On 8 Nov 2011, at 13:15, Noqsi wrote:
>
> > Now, note the the documentation for Root makes the following promise:
> > "The ordering used by Root[f,k] takes real roots to come before
> > complex ones, and takes complex conjugate pairs of roots to be
> > adjacent. " One difficulty with this promise is that it doesn't tell
> > you how to find the break between the real part of the vector and the
> > complex part. The following code assumes that N[Root[f,n]] will
> > reliably have head Real for real roots.
>
> If f is a polynomial, as it is in your case, than this will be true. The
> reason is that the roots of a polynomial can always be completely
> isolated and Mathematica does so the first time a numerical value of a
> root is used. Although Root isolation uses extended precision arithmetic
> (it can also be done exactly, if you use the options ExactRootIsolation
> but that will make the computation slower and should not affect the
> result), applying N with MachinePrecision to a real Root object should
> always produce a number with head Real (as long as the coefficients of
> f are exact; note that Root has the Attribute NHoldAll)
>
> By the way, this is not going to be necessarily true when f is a
> transcendental function. In this case a Root object may represent a
> cluster of roots, some of which could be real and some not, and it may
> require high precision arithmetic to decide. In this situation
> Mathematica will not perform root isolation until sufficient precision
> is specified. But when f is a polynomial this is not an issue.
I should have been more clear. I intended this method to be a way to
take an exact polynomial and put it in a stable approximate form. I
never thought it would be more general than that.
John Doty Noqsi Aerospace, Ltd.
http://www.noqsi.com/
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