05/27/2020, 11:22 PM

Hey!

Almost everybody knows Schröder's equation:

Psi o y(x) = s × Psi(x)

where s = (d/dx y) o y^o plus/minus infinity

where y^o plus/minus infinity is the fixpoint of the function y.

Almost everybody knows some form of the Abel equation:

Y o (1+x) o Y^o-1 = y(x)

where Y is the superfunction of y.

The connection of the two equation:

Y^o-1 = log Psi(x) / log s

Y = Psi^o-1 o s^x

y(x) = Psi^o-1 o (s × Psi)

y and s are usually given, we look for its iterated function or superfunction Y.

Some of you knows the Carleman matrix representations of analytic functions, I will notate it with [ f(x) ] or just [ f ]. It can be usefull, when you look for a concrate functional power/root of a function, beacuse:

[ y^o n ] = [ y ]^n for every n,

it is derived from:

[f o g] = [f]×[g]

The bigger is the matrix, the more precise is the represantation.

My pari/gp implementation of this conversions:

How to iterate? I also show you my tetration implementation:

It is much less precise as Sheldon's and others' fatou.gp programme; it works without using Schröder and Abel equations but using Carleman matrices and diagonalization. It is not enough for me, lets search a better method. What is about determining the Psi function? We said that:

Y = Psi^o-1 o s^x

So to determine inverse of Psi(x) seems promising. Let me notate P = [ Psi(x) ]. The formula says:

P^-1 × (s × P) = [ y(x) ], here are the first product which is matrix-matrix and the second product which is scalar-matrix.

P^-1 × s^x' × P = [ Y(x') ]

It looks like a diagonalized form, so what if I identify P^-1 as mateigen([ y(x) ])? I have already tested it for y = 2x-1 for which it succeded on a small interval but no success with determining the tetration for base 2.

How would you approximate function Psi?

And sorry not for using latex. I forget how to. :(

Almost everybody knows Schröder's equation:

Psi o y(x) = s × Psi(x)

where s = (d/dx y) o y^o plus/minus infinity

where y^o plus/minus infinity is the fixpoint of the function y.

Almost everybody knows some form of the Abel equation:

Y o (1+x) o Y^o-1 = y(x)

where Y is the superfunction of y.

The connection of the two equation:

Y^o-1 = log Psi(x) / log s

Y = Psi^o-1 o s^x

y(x) = Psi^o-1 o (s × Psi)

y and s are usually given, we look for its iterated function or superfunction Y.

Some of you knows the Carleman matrix representations of analytic functions, I will notate it with [ f(x) ] or just [ f ]. It can be usefull, when you look for a concrate functional power/root of a function, beacuse:

[ y^o n ] = [ y ]^n for every n,

it is derived from:

[f o g] = [f]×[g]

The bigger is the matrix, the more precise is the represantation.

My pari/gp implementation of this conversions:

Code:

`D(f,n)=for(k=1,n,f=deriv(f));f;`

/*n-th derivative of f for integer n*/

M(f,n)=matrix(n,n,j,k,1/(k-1)!*subst(D(f^(j-1),k-1),x,0));

/*n×n Carleman matrix of f*/

T(A,n)=sum(k=1,n,A[2,k]*x^(k-1));

/*Taylor series of an n×n Carleman matrix*/

How to iterate? I also show you my tetration implementation:

Code:

`h=10;`

tetinit(a)=A=M(a^x,h);Xt=mateigen(A);Bt=mateigen(A,1)[1];a;

tet(b)=if(real(b)>0,if(real(b)>=1,a^tet(b-1),return(subst(T(Xt*matdiagonal(Bt^b)/Xt,h),x,1)),a^tet(b+1)),if(b==0,1.,log(tet(b+1))/log(a)));

Dtet(n,{lh=10^-6})=sum(k=0,n,(-1)^k*binomial(n,k)*tet(lh*(n-k)*1.))/lh^n;

Ttet(n)=sum(k=0,n,x^k*Dtet(k)/k!);

itet(z)=if(real(z)>1,itet(log(z)/log(a))+1,if(0>real(z),itet(a^z)-1,-tet(-z)));

It is much less precise as Sheldon's and others' fatou.gp programme; it works without using Schröder and Abel equations but using Carleman matrices and diagonalization. It is not enough for me, lets search a better method. What is about determining the Psi function? We said that:

Y = Psi^o-1 o s^x

So to determine inverse of Psi(x) seems promising. Let me notate P = [ Psi(x) ]. The formula says:

P^-1 × (s × P) = [ y(x) ], here are the first product which is matrix-matrix and the second product which is scalar-matrix.

P^-1 × s^x' × P = [ Y(x') ]

It looks like a diagonalized form, so what if I identify P^-1 as mateigen([ y(x) ])? I have already tested it for y = 2x-1 for which it succeded on a small interval but no success with determining the tetration for base 2.

How would you approximate function Psi?

And sorry not for using latex. I forget how to. :(

Xorter Unizo