10/20/2017, 08:19 PM
(This post was last modified: 10/20/2017, 08:38 PM by sheldonison.)

(10/20/2017, 07:55 PM)Gottf ried Wrote:acoeff is just like an unknown; like "x" in the equations "x^3+2*x^2+3*x+1". In the routine, Pari-gp will treat acoeff as an unknown unassigned variable just like "x", that happens to be the multiplier for the "i'th" coefficient we are trying to determine.(10/20/2017, 06:00 PM)sheldonison Wrote: For example, here is the pari-gp program for the formal inverse schroeder function. I don't know how to turn this into a matrix function, but not many programming languages support the powerful polyonomial functions that pari-gp has.

Code:`formalischroder(fx,n) = {`

local(lambda,i,j,z,f1t,f2t,ns,f1s);

lambda = polcoeff(fx,1);

f1t=x;

i=2;

while (i<=n,

f1s=f1t;

f1t=f1t+acoeff*x^i+O(x^(i+1));

f2t=subst(f1t,x,lambda*x)-subst(fx+O(x^(i+1)),x,f1t);

z = polcoeff(f2t, i);

z = subst(z,acoeff,x);

ns=-polcoeff(z,0)/polcoeff(z,1);

f1t=f1s+ns*x^i;

i++;

);

return(Pol(f1t));

}

fz1=x^2+(1-sqrt(3))*x;

[size=small][font=Monaco, Consolas, Courier, monospace]lambda1=polcoeff(fz1,1);[/font][/size]

[size=small][font=Monaco, Consolas, Courier, monospace]fs1=formalischroder(fz1,20);[/font][/size]

superfunction1(z)=subst(fs2,x,lambda2^z);

Sheldon - I find some unexplained terms: what is "acoeff" / how is this defined before it is queried for "f1t"?

the same with "fz2" in "lambda1=pol..." and "fs2" in "superfuncion...subst(fs2..."

The others are typos: I was originally going to post both superfunctions, from both fixed points.... The other fixed point equation would be fz2=x^2+(1+sqrt(3))*x. Depending on whether |lambda|>1 or <1, determines where the superfunction equation converges, either negative or positive values of .

Actually, the other fixed point also converges better, so that's what I meant to post. Here superfunction1 converges pretty good for

Code:

`fz1=x^2+(1+sqrt(3))*x;`

lambda1=polcoeff(fz1,1);

fs1=formalischroder(fz1,20);

superfunction1(z)=subst(fs1,x,lambda1^z);

- Sheldon