+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: Half-iteration of x^(n^2) + 1 (/showthread.php?tid=1155)
Half-iteration of x^(n^2) + 1 - tommy1729 - 02/13/2017
I was intrested in the half-iteration of f_n(x) = x^(n^2) + 1 for Large x.
For instance g_n(x) = f_n ^[1/2](x) - x^n.
H_n(x) = f_n ^[1/2](x) / x^n.
For Large x :
Is abs g_n(x) increasing or decreasing with n ?
Is abs H_n(x) decreasing ?
Probably abs g_n is increasing and abs H_n decreasing.
The focus is on integer n and branch structure.
But also if n is real , are these functions analytic in n ?
Perturbation Theory suggests this.
I wonder how these functions look like on the complex plane , especially with resp to n.
Regards
Tommy1729
RE: Half-iteration of x^(n^2) + 1 - Xorter - 03/04/2017
(02/13/2017, 12:12 PM)tommy1729 Wrote: I was intrested in the half-iteration of f_n(x) = x^(n^2) + 1 for Large x.
For instance g_n(x) = f_n ^[1/2](x) - x^n.
H_n(x) = f_n ^[1/2](x) / x^n.
For Large x :
Is abs g_n(x) increasing or decreasing with n ?
Is abs H_n(x) decreasing ?
Probably abs g_n is increasing and abs H_n decreasing.
The focus is on integer n and branch structure.
But also if n is real , are these functions analytic in n ?
Perturbation Theory suggests this.
I wonder how these functions look like on the complex plane , especially with resp to n.
Regards
Tommy1729
Okey, I got some Taylor series of the half -iteration of f_n(x) = x^(n^2)+1 by a PARI/gp programme code:
I know these are not the best results, but this is that I could get from my programme. Here is the code:
Code:
init()={
default(format,"g0.4");
}
D(z,n)={for(i=0,n-1,z=z');return(z)}
Car(f,dim)={return(subst(matrix(dim,dim,k,j,D(f^(j-1),k-1)/(k-1)!),x,0))}
Decar(M,dim)={
f=0;for(i=1,dim,f+=M[i,2]*x^(i-1));
return(f);
}
Msqrt(B,dim,prec)={
A=matid(dim);
for(i=0,prec,A=(B*A^-1+A)/2);
return(A);
}I hope it helps you, and you can develope this code. If you can, please share it with me.
RE: Half-iteration of x^(n^2) + 1 - tommy1729 - 03/09/2017
Thank u for your reply.
However i have questions
1) your coëfficiënt 1/2048 occurs twice !? Are you sure about that.
2) also the coëfficiënts : i noticed all of them ( though truncated ) are positive.
Does this pattern remain ? Are they correct ?
3) not sure how you computed it. I assume no fixpoint but a kind of carleman matrix method ?
If you used a fixpoint , which one ?
4) im intrested in using the fix with largest real part.
Regards
Tommy1729.
RE: Half-iteration of x^(n^2) + 1 - Xorter - 03/09/2017
(03/09/2017, 01:28 PM)tommy1729 Wrote: Thank u for your reply.
However i have questions
1) your coëfficiënt 1/2048 occurs twice !? Are you sure about that.
2) also the coëfficiënts : i noticed all of them ( though truncated ) are positive.
Does this pattern remain ? Are they correct ?
3) not sure how you computed it. I assume no fixpoint but a kind of carleman matrix method ?
If you used a fixpoint , which one ?
4) im intrested in using the fix with largest real part.
Regards
Tommy1729.
I did not use fixpoint, because by the Carleman matrix it can be calculated, too. You can see above, how I computed. Just save it in gp and open it with gp.exe and enter this code:
Decar(Msqrt(Car(x^4+1,20),20,5),20)*1.0
Where Car makes a 20x20 Carleman matrix from x^4+1, Msqrt get its square root and Decar gets the Taylor series of the function from the matrix. It is simple, because:
M[f]M[g]=M[fog]
thus
sqrt M[f] = M[f^o0.5], right? Of course!
N root of M[f] = M[f^o1÷N]
If you check the code above, you can see it has a lot of (infinity) part with negative sign.
Naturally, it is not perfect, the bigger Carleman matrices you use, the better the results are.