... you iterate over all n. n lives where exactly? Unspecified. Right?
No, I do not iterate over all n. n is a non negative integer, we are
dealing with formal power series only, i.e., no Puiseux or Laurent
expansions are considered.
So, why not making it explicit somewhere? In the paper and in the ++
documentation.
But your sequence is fully given, i.e. you could compute the
coefficient up to any order n.
no, although that's one would hope for. But it may happen that the
equation does not determine a sequence completely.
Does this answer your question?
Not really. You know, I want to have an exact specification of the function.
So an input of guessADE is a list of rational numbers. The output is?
Not right.
Next attempt. Given a formal power series f \in K[[x]]. (If I write
that, it means that (in principle) I know every coefficient.) The input
of guessADE is [f_0, ..., f_k], i.e. the first k+1 coefficients in the
series expansion of f.
The output is... ? You have to help me here. If you say it's an
expression, that is rather unspecific. I would rather like to have
something else than Expression Integer. You could (and I think you
should) rather have
guessADE: List(K) -> ADES(K)
where ADES is a type of "system of algebraic differential equations over
K". Actually, what you write in equation (7) on page 5, you could even say
gusesADE: List(K) -> (SUP(K), INI(K))
(I guess, it's easy enough to figure out what the type INI(K) for the
initial conditions can be.)
Why not simply returning a polynomial as described in (7)?
Wouldn't that even make the result more accessible to further
computation? And, in fact, it would be easier to specify. All you would
claim in the end is that this is a polynomial that works for the given
input list. We all know that with a finite amount of input we can never
be sure of the resulting polynomial. But what one can do, is, to clearly
specify what the returned polynomial actually means.
From what you say, that the result does not always specify the solution
uniquely. OK, but as far as I understood, guessADE is not claiming that.
It is just finding a possible polynomial p where, if the given f is
entered, it vanishes up to order k+2 (at least). That is a clear
specification. And if this is not fulfilled, the function returned
garbage. By no means you can claim that your function figured out the
"right" polynomial. That you have to do in an additional (manual proof)
that involves *all* the coefficients of f.
Ralf
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