bo198214 Wrote:Apart from the Schroeder function, does the fractional Matrix power return the proper fractional iteration?Well, the fractional matrix-power will be computed by that same formula; empirically, the eigensystemsolver of Pari/Gp gives exactly the same results (resp.rescaling and reordering) as the analytically constructed Ut-Eigenmatrices.
bo198214 Wrote:And what means crap? I am not able to guess your computations and where there occured a problem.Really crap...
But you made me recheck the problem and I found the error getting now correct results at least for this base. I was multiplying dV(sigma)*dV(u^h) , but for fractional powers of complex u this is obviously different from dV(sigma*u^h), so , for instance for the fourth entry I had
sigma^3 * (u^0.5^3)
where I should have had
(sigma*u^0.5)^3
which obviously are different from each other if u is complex and the power of u is fractional. I had to use the second version.
[update] To make it more precise:
the difference occurs, since for complex exponentiation
a) (u^3)^0.5 can be different from b) (u^0.5)^3, so commutativity with multiplication in the exponent is not generally given.
What is surprising - considering the principles of diagonalization and computing powers of matrices - that one would expect the correct way to compute the h'th power of a matrix using its eigenvalues would be
a) dV(u)^h ,
but the example indicates that we have to compute
b) dV(u^h)
where b) gives then (u^h)^0, (u^h)^1, (u^h)^2,... which gives the correct result in my example computation.
This is really surprising.
[/update]
Stupid error - all the half year where I dealt with that problem I didn't catch that bug.
Well - I'll recompute my old examples; hope I get consistent results now.
New energy!
Gottfried Helms, Kassel