Understanding Abel/Schroeder with matrixexpression  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: Understanding Abel/Schroeder with matrixexpression (/showthread.php?tid=138) Pages:
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RE: Understanding Abel/Schroeder with matrixexpression  Gottfried  05/26/2008 Here I'm considering fractional heights h for the function Ut°h(x) of complex base t, which implemets x > t^x 1 and hope to get to the source of the persisting problem. Note, that with the previously described method everything is fine with integral heights, so the problem should be somehow systematic. Remember the Schröderfunction, whose powerseriescoefficients I find in the 2'nd column of the eigenvectormatrix of the operator Ut, such that where u=log(t) and WI is the inverse of W. Then the Schröderfunction sigma for parameter t is and the inverse takes the coefficients of WI instead: To compute Ut°1(x) we simply do or, for integer h As I said, this works fine even for complex t, checked with some t where abs(t)<1, if heights are integral. But we'll see, that for fractional h this does not work. With this simple process, for instance t=1/2 + 1/2 I there is also no severe problem with convergence or even divergence  things sum up in a completely easy manner  but anyway, the fractional h in u^h produce obviously complete crap. Currently I suspect, that there must be some conjugacy be involved  but don't have a working idea, where. RE: Understanding Abel/Schroeder with matrixexpression  bo198214  05/26/2008 Gottfried Wrote:With this simple process, for instance t=1/2 + 1/2 I there is also no severe problem with convergence or even divergence  things sum up in a completely easy manner  but anyway, the fractional h in u^h produce obviously complete crap. Apart from the Schroeder function, does the fractional Matrix power return the proper fractional iteration? And what means crap? I am not able to guess your computations and where there occured a problem. RE: Understanding Abel/Schroeder with matrixexpression  Gottfried  05/26/2008 bo198214 Wrote:Apart from the Schroeder function, does the fractional Matrix power return the proper fractional iteration?Well, the fractional matrixpower 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 UtEigenmatrices. 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! 