Ralf Hemmecke wrote:
>
> >> http://axiom-wiki.newsynthesis.org/TaylorSeries
>
> > I think that currently MultivariateTaylorSeriesCategory is
> > deliberatly incomplete.
>
> Really? But that makes it somewhat useless.
Note that domain or subcategory may add extra operations.
>
> > Namely in practice one needs some term ordering and for mutivariate
> > series there is no single distinguished order.
>
> Right. But look into the representation. In fact,
> SparseMultivariateTaylorSeries is as stream of things that are given as
> the third parameter P (SparseMultivariatePolynomial in my case). So
> there is a structure, that SMTS is in fact a univariate series of
> homogeneous multivariate polynomials. What "homogenous" means could be
> up to the type P itself.
>
> So, maybe in SMTS we could have a function
>
> smts(s: Stream P): % == ...
>
> where the input s is required to be a stream of homogeneous elements of
> P of degree 0, 1, 2, 3, etc. or (equivalently)
>
> smts(f: NonNegativeInteger -> P): % == ...
>
> That would (at least) remove the need for "pretend" and would be easy to
> understand.
>
> It wouldn't probably cover all possible term orders, but certainly all
> the ones that SMTS covers now.
You overestimate generality of SMTS: AFAICS currently SMTS assumes
that variables have degree 1, so you effectively get total degree
+ order from polynomial domain within given total degree.
> > We could add 'coefficients' and 'series' function to SMTS.
>
> You probably mean by 'series' what I denoted by 'smts'. Yes, I would be
> in favour of such 'series' construction.
I used name 'series' because it is what univariate domains use. This
would be an easy addition.
> > Or maybe have 'addiag' defined on series level.
>
> Possible, but also that requires some concept of (total) degree, so that
> the result can nicely be interpreted as a multivariate series.
> I think direct construction via a stream of homogeneous elements of the
> third parameter of SMTS would be preferrable. I somehow have the
> feeling, that was also the intention by the original author(s), because
> otherwise one could have hardcoded the third parameter inside the domain
> and only offer two parameters to the user.
Hard to comment about intention of original authors. They routinely
have extra parameter even if there is natural candidate, so I would
not draw conclusions from presence of third parameter. Clearly,
in univariate case original authors decided to export 'coefficients'
and 'series'. I find 'addiag' more mathematical -- one could work
with series instead of switching back and forth between series and
streams. 'addiag' should be enough to do all interesting
"infinte" operations on series. 'addiag' has one drawback that
it effectively forces specific convergence rate.
--
Waldek Hebisch
[email protected]
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