01/07/2017, 11:00 PM
(This post was last modified: 01/07/2017, 11:02 PM by Xorter.
Edit Reason: wrong tex
)
Hello!
I found an interesting site about the Carleman matrix: https://en.m.wikipedia.org/wiki/Carleman_matrix
\( M[f]_i_,_j = 1/k! [D^k f(0)^j] \)
\( f(x) = \sum_{k=0}^{\infty} M[f]_1,k x^k \)
And the most important of this case:
M[fog] = M[f] M[g]
So these matrices convert composition to matrix multiplication.
Thus
\( M[f^o ^N] = M[f]^N \)
Therefor
\( f^o ^0.5 (x) = \sum_{k=0}^{\infty} sqrt(M[f])_1_,_k x^k \)
So we get the M[exp(x)], it was the easy part of the thing. We need to get the squered root of this matrix, and I could find a program for it: http://calculator.vhex.net/calculator/li...quare-root
And I got another matrix, which satisfies that: \( sqrt(M[exp(x)])_1_,_k = [0.606, 0.606, 0.303 ...]^T \)
So the function is:
\( exp^o ^0.5 (x) ~= 0.606 + 0.606x + 0.303x^2 + 0.101x^3 \)
But it is not the half-iterate of exp(x), Could you help me why not, please? What was my mistake?
I found an interesting site about the Carleman matrix: https://en.m.wikipedia.org/wiki/Carleman_matrix
\( M[f]_i_,_j = 1/k! [D^k f(0)^j] \)
\( f(x) = \sum_{k=0}^{\infty} M[f]_1,k x^k \)
And the most important of this case:
M[fog] = M[f] M[g]
So these matrices convert composition to matrix multiplication.
Thus
\( M[f^o ^N] = M[f]^N \)
Therefor
\( f^o ^0.5 (x) = \sum_{k=0}^{\infty} sqrt(M[f])_1_,_k x^k \)
So we get the M[exp(x)], it was the easy part of the thing. We need to get the squered root of this matrix, and I could find a program for it: http://calculator.vhex.net/calculator/li...quare-root
And I got another matrix, which satisfies that: \( sqrt(M[exp(x)])_1_,_k = [0.606, 0.606, 0.303 ...]^T \)
So the function is:
\( exp^o ^0.5 (x) ~= 0.606 + 0.606x + 0.303x^2 + 0.101x^3 \)
But it is not the half-iterate of exp(x), Could you help me why not, please? What was my mistake?
Xorter Unizo