Ralf Hemmecke wrote:
>
> On 11/11/2014 10:54 PM, Waldek Hebisch wrote:
> >> 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.
>
> I wanted to quickly convince myself that this is not true, but failed to
> create an appropriate polynomial domain Q[x,y] where x would have degree
> 1 and y degree 2.
>
> However, maybe I'm wrong, but I don't really see where SMTS relies on
> the variables having degree 1.
>
> Maybe here:
>
> https://github.com/fricas/fricas/blob/master/src/algebra//mts.spad#L127
>
> monomial(r : %, v : Var, n : NNI) ==
> r * monom(monomial(1, v, n)$SMP, n)$STT
>
> n is used for creating the exponent of the monomial and also to give the
> place in the stream.
Yes.
>
> > 'addiag' has one drawback that it effectively forces specific
> > convergence rate.
>
> What exactly do you mean by that? Aren't we dealing with *formal* power
> series?
We have order function on pewer series, that is degree of first
nonvanishing term. For multivariate series this depends on term
order (note that SMTS does not implement such function). Given
a sequence {s_i} of power series we say that series \sum s_i is
convergent if coefficient of each monomial is a sum with only
finite number of nonzero terms. Given order function this
condition is satisfied if order(s_i) -> \infty. For series in
finite number of variables condition on order is necessary for
convergence (in infinte number of variables one may have convergence
without order going to infinity).
Of couse, if oder(s_i) grows faster this means faster convegence, so
it is natural to identify convergence rate of {s_i} with growth of
oder(s_i). When 'addiag' function is directly used to sum series
of power series it forces condition order(s_i) >= i. More precisely,
to use addiag on series we need to drop first i terms from s_i and
convert resulting series to stream (and then convert result of
addiag back to a series). So _directly_ addiag it can not be used
to sum series when order(s_i) grows slower. Of course, for any fixed
growth of order(s_i) one can split sum into packets so that sum
of packet number j has order >= j. But this can only be done
if growth of order(s_i) is known.
--
Waldek Hebisch
[email protected]
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