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