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.
4) In 'compute_bound' you seem to ignore ramified exponential
parts. This looks wrong.
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).
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).
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]
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