Quotients of polynomials

[Urs Hackstein]

Hi,

I have the following problem. Given a function f in the complex
variable s. I know that f can be written as the quotient of two
polynomials in s with real coefficients, but it is not possible to
determine the coefficients manually. Thus my question is whether one
can calculate the coefficients using sage.

Thanks a lot in advance!

Urs

[John Cremona]

How is f given -- as a powers series? If so, and you have enough
coefficients relative to the denominator degree, then you can
reconstruct f, by detrermining the linear recurrence satisfied by the
coefficients of the power series.

John Cremona

[Urs Hackstein]

Unfortunately, I can't recognize any nice mathematical structure in
the form f is given. It is given not exactly as an continous fraction,
but it is given as a big (in the sense of many variables and terms)
fraction where both denominator and numerator are "complicated"
functions containing big fractions where both denominator and
numerator are .. and so on.
f is not an object of pure mathematical interest, it arise from some
calculation in electronical engineering. But engineers told me that it
is known by theoretical results that it could be written as the
quotient of polynomials.

On Oct 11, 4:44 pm, John Cremona <john.crem...@gmail.com> wrote:
> How is f given -- as a powers series?  If so, and you have enough
> coefficients relative to the denominator degree, then you can
> reconstruct f, by detrermining the linear recurrence satisfied by the
> coefficients of the power series.
>
> John Cremona
>
> On Tue, Oct 11, 2011 at 2:28 PM, Urs Hackstein
>

[John Cremona]

In that case I cannot help.

I suggest that you ask again on sage-support. Most people who read
sage-algebra are pure algebraists, while you need people with more
expertises in symbolic computer algebra (which is different!)

John Cremona

[luisfe]

Urs Hackstein wrote:
> Unfortunately, I can't recognize any nice mathematical structure in
> the form f is given. It is given not exactly as an continous fraction,
> but it is given as a big (in the sense of many variables and terms)
> fraction where both denominator and numerator are "complicated"
> functions containing big fractions where both denominator and
> numerator are .. and so on.
> f is not an object of pure mathematical interest, it arise from some
> calculation in electronical engineering. But engineers told me that it
> is known by theoretical results that it could be written as the
> quotient of polynomials.

Umm, the solution depends a lot on the exact form that f is given and
what the "complicated" functions are.

Do you have a previous bound of the numerator and denominator of f?
without a theoretical bound on numerator and denominator I guess you
are doomed. You will be able to approximate f though.

Can you evaluate f in many point? Is this evaluation exact or
approximate? That is, if I compute f(2) will it be a rational number,
a float point?

This question is better placed in sage-support.

[Urs Hackstein]

My membership in sage-support ist still pending.

"Complicated" means here that the whole term is more than three pages
long. Starting from a fraction with products of constants, values and
variables in the numerator and denominator you subsitute some of the
values in the numerator and the denominator by fractions with products
of constants, values and variables, then you substitute some of the
variables in all of the numerators and denominators by fractions
with ... again and so on. Thus the only pattern of the form f is given
is that there are only basic arithmetic operations in its definition.

Of course, I could choose a set of numerical values for our constants
and evaluate f in some points (this gives us float points) and get the
coefficients by interpolation. But we are looking for a description of
the coefficients as functions of some other variables, so numerical
values for the coefficients don't help us.

By the way, there is another question more elementary: How can I
compute in \mathbb{C}(t) (i.e. the quotient field of the polynomial
ring) using sage?

[Florent Hivert]

On Wed, Oct 12, 2011 at 05:27:02AM -0700, Urs Hackstein wrote:
> My membership in sage-support ist still pending.
>
> "Complicated" means here that the whole term is more than three pages
> long. Starting from a fraction with products of constants, values and
> variables in the numerator and denominator you subsitute some of the
> values in the numerator and the denominator by fractions with products
> of constants, values and variables, then you substitute some of the
> variables in all of the numerators and denominators by fractions
> with ... again and so on. Thus the only pattern of the form f is given
> is that there are only basic arithmetic operations in its definition.
>
> Of course, I could choose a set of numerical values for our constants
> and evaluate f in some points (this gives us float points) and get the
> coefficients by interpolation. But we are looking for a description of
> the coefficients as functions of some other variables, so numerical
> values for the coefficients don't help us.
>
> By the way, there is another question more elementary: How can I
> compute in \mathbb{C}(t) (i.e. the quotient field of the polynomial
> ring) using sage?

Nothing particular to do: just divide polynomials, the fraction field is
automatically created.

sage: P.<t> = CC[]
sage: P
Univariate Polynomial Ring in t over Complex Field with 53 bits of precision
sage: F = (t/t).parent()
sage: F
Fraction Field of Univariate Polynomial Ring in t over Complex Field with 53 bits of precision

You can alternatively define

sage: P.fraction_field()
Fraction Field of Univariate Polynomial Ring in t over Complex Field with 53
bits of precision

Cheers,

Florent