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Understanding Abel/Schroeder with matrix-expression
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... Wink 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


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.

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! Smile
Gottfried Helms, Kassel

Messages In This Thread
RE: Understanding Abel/Schroeder with matrix-expression - by Gottfried - 05/26/2008, 08:45 PM

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