Re: Elliptic Integrals?

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Jun 12, 2018 at 09:00:52PM +0200, Eduardo Morras wrote:
> 
> Hello, I'm trying to implement an algorithm that uses Elliptic Integrals. 

Hello Eduardo,

Which kind of elliptic integrals do you need ?
You might be able to use ellperiods.

I have this GP script which computes E and K.
(numE/numK are the slow version, agmE/agmK are the fast ones).
(Unfortunately I forgot where I found the formulae, which avoid
the derivation).

Cheers,
Bill

magm(a,b)=
{
  my(c=0,eps=10^-(default(realprecision)-5));
  while(abs(a-b)>eps,
    my(u = sqrt((a-c)*(b-c))); 
    [a,b,c] = [(a+b)/2,c+u,c-u]);
  (a+b)/2
}

numE(g)=my(g2=g^2);intnum(x=0,[1,-1/2],my(x2=x^2);sqrt((1-g2*x2)/(1-x2)))

numK(g)=my(g2=g^2);intnum(x=0,[1,-1/2],my(x2=x^2);1/sqrt((1-g2*x2)*(1-x2)))

agmE(g)=my(b2=1-g^2);Pi/2/agm(1,sqrt(b2))*magm(1,b2)

agmK(g)=my(b2=1-g^2);Pi/2/agm(1,sqrt(b2))

agmPi(g)=my(b=sqrt(1-g^2));2*agm(1,b)*agm(1,g)/(magm(1,b^2)+magm(1,g^2)-1)