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 Understanding Abel/Schroeder with matrix-expression Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 05/26/2008, 08:45 PM (This post was last modified: 05/26/2008, 10:07 PM by Gottfried.) 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 « Next Oldest | Next Newest »

 Messages In This Thread Understanding Abel/Schroeder with matrix-expression - by Gottfried - 03/30/2008, 05:06 AM more Brainstorming on Abel/Schroeder - by Gottfried - 05/18/2008, 02:36 PM RE: more Brainstorming on Abel/Schroeder - by bo198214 - 05/18/2008, 04:51 PM RE: more Brainstorming on Abel/Schroeder - by Gottfried - 05/18/2008, 07:41 PM RE: more Brainstorming on Abel/Schroeder - by bo198214 - 05/18/2008, 08:00 PM RE: Understanding Schröder with matrix-expression - by Gottfried - 05/19/2008, 12:02 PM RE: Understanding Schröder with matrix-expression - by bo198214 - 05/19/2008, 06:14 PM RE: Understanding Schröder with matrix-expression - by Gottfried - 05/19/2008, 07:45 PM RE: Understanding Schröder with matrix-expression - by Gottfried - 05/19/2008, 07:31 PM RE: Understanding Schroeder with matrix-expression - by Gottfried - 05/20/2008, 10:59 AM RE: Understanding Abel/Schroeder with matrix-expression - by Gottfried - 05/26/2008, 07:02 PM RE: Understanding Abel/Schroeder with matrix-expression - by bo198214 - 05/26/2008, 07:36 PM RE: Understanding Abel/Schroeder with matrix-expression - by Gottfried - 05/26/2008, 08:45 PM

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