On Wednesday, July 15, 2015 at 8:23:45 AM UTC+5:30, Waldek Hebisch wrote:
>
> Abhinav Baid wrote:
> >
> > I've added all the functions required to implement factor, but don't use
> > guessHolo yet as I couldn't get how to use it for a list of power
> > series, instead of a list of coefficients, and so use some custom PadĂŠ
> > functions which may be quite inefficient as they make use of Polynomial
> > data type [1]. The factor function does work for some input operators,
> > but for others, it seems to be stuck inside the loop in
> > try_factorization. Could you please see if there's some mistake and
> > suggest what changes I'll have to make?
>
> Some remarks:
>
> 1) In factor_global you unconditionally add infinity as a singularity.
> However, it may happen that infinity is a regular point.
> 2) In factor_global you use 'zerosOf' to find singularities.
> However, given irreducible factor of denominator its zeros
> can not be distingushed using algebraic operations, so
> normally it is enough to work with a single zero (the result
> for other are conjugate via action of Galois group). Also
> 'zeroOf' tries to express zero in terms of radicals. But
> this may lead to troubles here, so it is better to use
> 'rootOf'.
> 3) Computing 'lcm' and then factoring is likely to be more
> expensive than using already known factors. In particular,
> you can use 'gcdBasis' routine to produce list of relatively
> prime factors and factor each separately. Also, it makes
> sense to use 'squareFree' first to simplify computations.
>
I've changed the file according to the above comments.
> 4) In 'compute_bound' you seem to ignore ramified exponential
> parts. This looks wrong.
>
Sorry, I don't get what I'm missing here. First, I check that the degree of
'ram' = 1 and then only the constant coefficient of 'expart' matters in
further computation. So, I think it should work?
> 5) Infinity is used differently in bounds: other singularities
> bound denomiantors of a_i. Infinity bounds differences
> between degrees of mumerators and denominators of a_i (so
> given bound on denominators infinity gives bound for degree
> of numarators).
>
Again, I apologize because I think I don't understand this. What are the
a_i in question?
> Concerning solving Hermite-Pade problem: if you have a solution
> at your disposal (which is easy to obtain from a factor), then
> 'guessHolo' should be quite efficient. More precisely, given
> first order factor delta - r, the solution is exponential of
> the integral of r/x which is an easy power series computations.
> Using substitution S_e you can do this with Taylor series.
> Given Taylor series is solution 'guessHolo' will produce the
> factor (of course then you need to do S_{-e} to undo effect of S_e).
>
So, can guessHolo can be used only with first order factors? If so, then
how do I handle the case of m (as defined in section 3.6) > 1, I think the
same goes for guessAlgDep as well.
> The above is a bit different from what van Hoej wrote.
> If you want to follow van Hoej then you need procedure to solve
> system
>
> a_0v_0 + a_1v_1 + ... + a_dv_d
>
> where v_i are vectors of power series. In case of vectors
> of dimenion 1 (which correspond to factor of order 1) you
> can use 'guessAlgDep' with degree bound of 1 to solve
> such problem. Vectors of arbitrary dimension can be handled
> by underlying routines, but I need to check what is the
> best to call them.
>
> --
> Waldek Hebisch
> [email protected] <javascript:>
>
Thanks,
Abhinav.
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