Uniqueness of half-iterate of exp(x) ?

[Gottfried]

Hello!
I found an interesting site about the Carleman matrix: https://en.m.wikipedia.org/wiki/Carleman_matrix
(...)
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?

Well, using the truncated series of the exp(x)-function up to 16 terms (Carleman-matrix-size) I get, using my own routine for matrix-square-root in Pari/GP with arbitrary numerical precision (here 200 decimal digits for internal computation) , the following truncated series-approximation:
\( \exp^{[0.5]}(x) \approx 0.498568472273 + 0.876337510066*x + 0.247418943917*x^2 + 0.0248068936680*x^3 - 0.00112303037149*x^4 \\
+ 0.000361451686885*x^5 + 0.0000337024986252*x^6 - 0.0000517784266699*x^7+ 0.0000259224188256*x^8 - 0.00000189354770473*x^9 \\
- 0.00000360748613972*x^{10} + 0.00000411482178000*x^{11} - 0.00000216221756598*x^{12} + 0.000000558540558241*x^{13} \\
- 0.0000000635173773180*x^{14} + 0.00000000192054352361*x^{15}+ O(x^{16})
\) 
This gives, for \( 0 \le x \lt 1 \) eight correct digits when applying this two times (and should approximate Sheldon's Kneser-implementation).                                   
The reason, why your function is badly misshaped might be: matrix is too small (did you only take size 4x4?) and/or the matrix-squareroot-computation is not optimal.


To crosscheck: one simple approach to the matrix-square-root is the "Newton-iteration".                       

Let M be the original Carleman-matrix and N denote its approximated square-root                 

initialize ...                 
\( N=Id \qquad \qquad \text{ /* Identity matrix of some finite dimension dim */} \)

iterate  ...                                
\( N = (M * N^{-1} + N)/2 \).
until convergence   .

Unfortunately, the matrix N shall not be "Carleman" unless M were of infinite size; nitpicking this means, the function \( \exp^{0.5}(x) \) with coefficients taken from the second row (or in my version:column) is not really well suited for iteration. (But this problem has not yet been discussed systematically here in the forum, to my best knowledge)
Gottfried