Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Half-iteration of x^(n^2) + 1
#1
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
Reply
#2
(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.
Xorter Unizo
Reply
#3
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.
Reply
#4
(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.
Xorter Unizo
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Does tetration take the right half plane to itself? JmsNxn 7 810 05/16/2017, 08:46 PM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 5,500 01/09/2017, 02:41 AM
Last Post: Gottfried
  [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 20,295 03/27/2015, 11:28 PM
Last Post: tommy1729
  [entire exp^0.5] The half logaritm. tommy1729 1 1,184 05/11/2014, 06:10 PM
Last Post: tommy1729
  Does the Mellin transform have a half-iterate ? tommy1729 4 1,586 05/07/2014, 11:52 PM
Last Post: tommy1729
  Simple method for half iterate NOT based on a fixpoint. tommy1729 2 1,811 04/30/2013, 09:33 PM
Last Post: tommy1729
  half-iterates of x^2-x+1 Balarka Sen 2 2,515 04/30/2013, 01:14 AM
Last Post: tommy1729
  Is the half iterate of 2sinh(x) analytic near R ? tommy1729 1 1,369 03/13/2013, 12:13 AM
Last Post: tommy1729
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 1,766 09/04/2011, 05:59 AM
Last Post: Gottfried
  Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 10 10,116 06/09/2011, 05:56 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)