On 07/12/2015 02:24 AM, Waldek Hebisch wrote:
Now, I've also added the factor_minmult1 function [1]. However, I'm
having a doubt regarding how to implement try_factorization. I've
mentioned the input and output specifications of the routine in the file
itself, but am unsure about how PadĂŠ approximation should be used to
reconstruct the factor from the inputs. Here's what I've thought about
so far: First, in section 3.6 of the thesis, van Hoeij talks about
writing D^0,D^1,...,D^order_R as vectors in the vector space generated
by D^0,D^1,...,D^order_r. I think I can do that. This will generate a
list of lists (alternatively, vectors) of vectors in k((x)). How do I
proceed from there? Waldek, you mentioned before that guessHolo with
some modifications should work for this purpose. Could you please expand
on that a little?
To use guessHolo you need:
- bound on degree of coefficients
- sufficiently many coefficients of expansion of apropriate
solution into power series
As result you will get operator vanishing on the solution.
In other words guessHolo will produce the factor.
When I wrote "apropriate solution", I mean that van Hoej gives
conditions when this method works: you need solution that is
anihilated by one of the factors, but not all of them. The
part computing exponential parts in many cases will provide
such a solution. More precisely, given a order 1 factor
you first conjugate it with an exponential to get semi-regular
operator. Then you can use standard FriCAS solver to get
arbitrarily many coefficents of expansion of the solution
into power series.
Bounds tell you how many coefficients you need: if you have
bound say 10 on degree of coefficients and expect order 3
factor, then you need at least 4*(10 + 1) + 1 + 3 coefficients (a few
more is good to speed things up). Note: 4*(10+1) is number
of coefficients of the factor that we want to find. The 3
appears because we need to differentialte power series 3 times
and effectively loose 3 terms. The +1 term is because
we solve homogeneous system.
Hi Waldek,
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 Padé
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?
[1]
https://github.com/fandango-/fricas/blob/lodof/src/algebra/lodof_new.spad
Thanks,
Abhinav.
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