Re: Conversion of a function/closure with rational coefficients into one with real coefficients

["Ruud H.G. van Tol" <rvtol@isolution.nl>]

On 2022-11-17 13:42, Gottfried Helms wrote:
> I'm trying to optimize some bivariate function,
> and found, that I can create a vector of polynomials p_r(x).
> Let pp be a vector of closures, so for instance
>
>   ...
>   pp[4](x) =  1/4*x^2-2/3*x+1/6
>   pp[5](x) = -1/8*x^3+31/48*x^2-7/12*x+1/24
>   ...
>
> But going in detail of the function I get vectors pp of
> dimension 128 or 256 or the like, where in the denominators
> are factorials of 128 or 256 and the like with very
> long integer numbers.
>
> By construction I get for all coefficients rational values,
> but the complexity grows much with increasing index r
> for pp[r](x).
> That means, for numerical evaluation it should be much more
> efficient to have that coefficients all in real values,
> and I'm looking for a tool/procedure, to convert the
> existing closures to that same functions, but with
> coefficients expressed in real numbers.
>
> I can't do, for instance, naive attempts like
>   ppr[5]=1.0*pp[5], or ppr[5](x)=1.0*pp[5](x).
>
> What is a good idea to convert the full vector "pp" into
> some -say- "ppr" ?
> (Well, I could recalculate all polynomials again from
> the generating script, but I rather want to learn
> the Pari/GP inherent tools for such a problem.)


I now wonder if it would help anything
to keep such numbers in some deconstructed
(and maybe even fractal/generative) way,
for example:

? print(factor(31/48))
[2, -4; 3, -1; 31, 1]

-- Ruud