Re: Efficient way to define and evaluate polynomial function

[Jacques Gélinas <jacquesg00@hotmail.com>]

V=[1 .. 2^8];

apply( n -> (2*n^6 + 6*n^5 + 5*n^4 - n^2)\12, V) ==

apply( n -> sum(j=1,n,j^5), V)

Jacques Gélinas

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De : Jérôme Raulin <jerome@raulinfoissac.fr>
Envoyé : 13 mai 2020 12:50
À : pari-users@pari.math.u-bordeaux.fr <pari-users@pari.math.u-bordeaux.fr>
Objet : Efficient way to define and evaluate polynomial function

 

Hi,

 

Let’s suppose that I need to evaluate many times the sum of the 5^th power of the first ‘n’ integers. 1^st implementation is to define :

 

sum_n_5(n) = (2*n^6 + 6*n^5 + 5*n^4 - n^2) \ 12;

 

and call it directly. Still it not efficient and a better way is to use Horner form such as (the difference is small but for larger degree polynomial the difference may be huge) :

 

sumn_5(n) = n^2 * (-1 + n^2 * (5 + n * (6 + 2 * n))) \ 12;

 

Still the best way would be to directly use the polynomial form built from the coefficients so that Pari can use its internal polynomial evaluation function:

 

Pol([2, 6, 5, 0, -1, 0, 0])/12

 

But I fail to define a callable function this way. Surely subst(Pol([2, 6, 5, 0, -1, 0, 0],x),x,n)\12 is not at all what to be done.

Is there an efficient way to define and evaluate a polynomial function ?

 

Regards

 

Jerome