On 1/9/2013 3:36 PM, Stavros Macrakis wrote:
> $realroots already uses psqfr.
ok
> But of course that requires exact coefficients (not floats);
as you may recall, I believe that floats are exact rational numbers
chosen from a set that can be expressed
with a finite fraction and an exponent base 2...
> I suppose it could use factor and do even better.
realroots used to use sturm sequences which should be faster than factoring.
> It could also be a little smarter about top-level multiplications and
> exact coefficients: no good reason that realroots((x-2/3)*(x-1/5))
> should be [x = 6710887/33554432,x = 22369621/33554432].
Note that 33554432 is 2^25.
It's finding approximations by binary subdivision and so willnever come
up with 1//3
>
> A counterargument says: realroots isn't in the business of doing
> psqfr, polynomial factorization, and symbolic analysis: it should be
> purely numerical.
Well no, it is rational and exact root isolation and bounding. Not
numerical in the usual sense of "floating point".
> In that case, though, I'd think we should rename it to make it clear
> that that's what it does, e.g. numerical_realroots_by_sturm.
I think providing a "preface" to allroots that removes multiple roots
could be useful. maybe that's the
cleanest approach.
RJF
>
> -s
>
> On Wed, Jan 9, 2013 at 6:15 PM, Richard Fateman
> <fateman at eecs.berkeley.edu <mailto:fateman at eecs.berkeley.edu>> wrote:
>
> well, not the general issue, but finding roots of polynomials.
>
> These programs tend to have difficulty when there are multiple roots.
> e.g. (x-3)^2.
>
> But we can remove multiple roots and count them by using sqfr.
> It is a feature of computer algebra that we seem to have neglected.
>
>
> Anyone willing to muck around with this? Maybe for bfallroots?
>
> RJF
>
>
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